Abstract group 444672.e: $C_{18528}.C_{24}$

• Label: 444672.e
• Order: \( 2^{8} \cdot 3^{2} \cdot 193 \)
• Exponent: \( 2^{6} \cdot 3 \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{8} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{5} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{2} \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \cdot 3 \)
• Perm deg.: not computed
• Trans deg.: $37056$
• Rank: $2$

Group information

Description:	$C_{18528}.C_{24}$
Order:	\(444672\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 193 \)
Exponent:	\(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Automorphism group:	$C_{2316}.C_{24}.C_4^2.C_2^5$, of order \(28459008\)\(\medspace = 2^{14} \cdot 3^{2} \cdot 193 \)
Composition factors:	$C_2$ x 8, $C_3$ x 2, $C_{193}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.

Group statistics

Order	1	2	3	4	6	8	12	16	24	32	48	64	96	192	193	386	579	772	1158	1544	2316	3088	4632	6176	9264	18528
Elements	1	387	1160	1932	4248	2320	17760	4640	23168	9280	46336	24704	92672	197632	192	192	384	384	384	768	768	1536	1536	3072	3072	6144	444672
Conjugacy classes	1	3	8	12	24	16	96	32	128	64	256	128	512	1024	8	8	16	16	16	32	32	64	64	128	128	256	3072
Divisions	1	3	4	6	12	4	24	4	16	4	16	4	16	16	1	1	1	1	1	1	1	1	1	1	1	1	142
Autjugacy classes	1	2	3	4	6	4	12	4	12	4	12	4	12	12	1	1	1	1	1	1	1	1	1	1	1	1	104

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$37056$
Rank:	$2$
Inequivalent generating pairs:	$1536$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid b^{18528}=1, a^{24}=b^{6948}, b^{a}=b^{8161} \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 186 & 0 \\ 0 & 55 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 105 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_3$ $\, \times\, $ $(C_{6176}.C_{24})$
Semidirect product:	$C_{579}$ $\,\rtimes\,$ $(C_4\times C_{192})$	$(C_{6176}.C_8)$ $\,\rtimes\,$ $C_3^2$	$(C_{193}:C_{12})$ $\,\rtimes\,$ $C_{192}$	$(C_{193}:C_{12})$ $\,\rtimes\,$ $C_{192}$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{18528}$ . $C_{24}$	$D_{386}$ . $(C_6\times C_{96})$	$D_{193}$ . $(C_6\times C_{192})$	$C_{9264}$ . $(C_2\times C_{24})$	all 76

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{12} \times C_{192} \simeq C_{4} \times C_{64} \times C_{3}^{2}$
Schur multiplier:	$C_{12}$
Commutator length:	$1$

Subgroups

There are 47748 subgroups in 516 conjugacy classes, 270 normal (81 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{96}$	$G/Z \simeq$ $C_{193}:C_{24}$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_{12}\times C_{192}$
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $C_{193}:(C_6\times C_{24})$
Fitting:	$\operatorname{Fit} \simeq$ $C_{18528}$	$G/\operatorname{Fit} \simeq$ $C_{24}$
Radical:	$R \simeq$ $C_{18528}.C_{24}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{1158}$	$G/\operatorname{soc} \simeq$ $C_4\times C_{96}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times C_{64}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
193-Sylow subgroup:	$P_{ 193 } \simeq$ $C_{193}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 444672: $C_{18528}.C_{24}$

Order 222336: $C_3\times C_{772}.C_{48}.C_2$ x 2, $C_3\times C_{772}.(C_4\times C_{24})$

Order 148224: $C_{772}.(C_4\times C_{48})$ x 3, $C_3\times C_{193}:(C_4\times C_{64})$

Order 111168: $C_3\times C_{772}.C_{48}$ x 4, $C_3\times C_{772}.(C_4\times C_{12})$, $C_3\times C_{772}.C_{24}.C_2$, $C_3\times C_{3088}.C_6.C_2$

Order 74112: $C_{772}.C_{48}.C_2$ x 6, $C_{772}.(C_4\times C_{24})$ x 3, $C_3\times C_{193}:(C_2\times C_{64})$ x 2, $C_3\times C_{193}:(C_4\times C_{32})$

Order 55584: $C_3\times C_{193}:C_{96}$ x 2, $C_3\times C_{193}:(C_2\times C_{48})$, $C_3\times C_{193}:(C_4\times C_{24})$, $C_3\times C_{193}:(C_2\times C_{48})$, $C_{96}\times C_{193}:C_3$, $C_3\times C_{193}:C_{96}$

Order 49408: $C_{6176}.C_8$

Order 37056: $C_{772}.C_{48}$ x 12, $C_3\times C_{193}:C_{64}$ x 4, $C_{772}.(C_4\times C_{12})$ x 3, $C_{3088}.C_6.C_2$ x 3, $C_{772}.C_{24}.C_2$ x 3, $C_3\times C_{193}:(C_2\times C_{32})$, $C_3\times C_{193}:(C_4\times C_{16})$, $C_3\times C_{193}:(C_2\times C_{32})$

Order 27792: $C_3\times C_{193}:C_{48}$ x 2, $C_{12}\times C_{193}:C_{12}$, $C_3\times C_{193}:(C_2\times C_{24})$, $C_3\times C_{193}:C_{48}$, $C_{48}\times C_{193}:C_3$, $C_3\times C_{193}:(C_2\times C_{24})$

Order 24704: $D_{193}:C_{64}$ x 2, $C_{6176}:C_4$

Order 18528: $C_{193}:C_{96}$ x 6, $C_{193}:C_{96}$ x 3, $C_{193}:(C_2\times C_{48})$ x 3, $C_{32}\times C_{193}:C_3$ x 3, $C_{193}:(C_2\times C_{48})$ x 3, $C_{193}:(C_4\times C_{24})$ x 3, $C_3\times C_{193}:C_{32}$ x 2, $C_3\times C_{193}:(C_4\times C_8)$, $C_3\times C_{193}:C_{32}$, $C_3\times C_{193}:(C_2\times C_{16})$, $C_3\times C_{193}:(C_2\times C_{16})$, $C_{18528}$

Order 13896: $C_3\times C_{193}:C_{24}$ x 2, $C_6\times C_{193}:C_{12}$ x 2, $C_3\times C_{193}:(C_2\times C_{12})$, $C_{24}\times C_{193}:C_3$, $C_3\times C_{193}:C_{24}$

Order 12352: $C_{193}:C_{64}$ x 4, $C_{32}\times D_{193}$, $D_{193}:C_{32}$, $C_{3088}:C_4$

Order 9264: $C_{193}:C_{48}$ x 6, $C_{193}:(C_2\times C_{24})$ x 3, $C_{193}:(C_2\times C_{24})$ x 3, $C_{193}:C_{48}$ x 3, $C_{16}\times C_{193}:C_3$ x 3, $C_4\times C_{193}:C_{12}$ x 3, $C_3\times C_{193}:C_{16}$ x 2, $C_3\times C_{193}:(C_2\times C_8)$, $C_3\times C_{193}:(C_2\times C_8)$, $C_{12}\times C_{193}:C_4$, $C_3\times C_{193}:C_{16}$, $C_{9264}$

Order 6948: $C_3\times C_{193}:C_{12}$ x 4, $C_{12}\times C_{193}:C_3$, $C_6\times C_{193}:C_6$, $C_3\times C_{193}:C_{12}$

Order 6176: $C_{193}:C_{32}$ x 2, $C_{16}\times D_{193}$, $C_{193}:C_{32}$, $D_{193}:C_{16}$, $C_{1544}:C_4$, $C_{6176}$

Order 4632: $C_{193}:C_{24}$ x 6, $C_2\times C_{193}:C_{12}$ x 6, $C_8\times C_{193}:C_3$ x 3, $C_{193}:C_{24}$ x 3, $C_{193}:(C_2\times C_{12})$ x 3, $C_6\times C_{193}:C_4$ x 2, $C_3\times C_{193}:C_8$ x 2, $C_3\times C_{193}:C_8$, $C_3\times C_{193}:(C_2\times C_4)$, $C_{4632}$

Order 3474: $C_{579}:C_6$ x 2, $C_{1158}:C_3$

Order 3088: $C_{193}:C_{16}$ x 2, $D_{193}:C_8$, $C_{772}:C_4$, $C_8\times D_{193}$, $C_{193}:C_{16}$, $C_{3088}$

Order 2316: $C_{193}:C_{12}$ x 12, $C_{193}:C_{12}$ x 4, $C_{193}:C_{12}$ x 3, $C_{386}:C_6$ x 3, $C_{193}:C_{12}$ x 3, $C_3\times D_{386}$, $C_{193}:C_{12}$, $C_{2316}$

Order 2304: $C_{12}\times C_{192}$

Order 1737: $C_{579}:C_3$

Order 1544: $C_{386}:C_4$ x 2, $C_{193}:C_8$ x 2, $C_4\times D_{193}$, $C_{193}:C_8$, $C_{1544}$

Order 1158: $C_{193}:C_6$ x 6, $C_{193}:C_6$ x 3, $C_3\times D_{193}$ x 2, $C_{1158}$

Order 1152: $C_6\times C_{192}$ x 2, $C_{12}\times C_{96}$

Order 772: $C_{193}:C_4$ x 4, $D_{386}$, $C_{772}$, $C_{193}:C_4$

Order 768: $C_4\times C_{192}$ x 4

Order 579: $C_{193}:C_3$ x 3, $C_{579}$

Order 576: $C_3\times C_{192}$ x 4, $C_6\times C_{96}$ x 2, $C_{12}\times C_{48}$

Order 386: $D_{193}$ x 2, $C_{386}$

Order 384: $C_2\times C_{192}$ x 8, $C_4\times C_{96}$ x 4

Order 288: $C_3\times C_{96}$ x 4, $C_6\times C_{48}$ x 2, $C_{12}\times C_{24}$

Order 256: $C_4\times C_{64}$

Order 193: $C_{193}$

Order 192: $C_{192}$ x 16, $C_2\times C_{96}$ x 8, $C_4\times C_{48}$ x 4

Order 144: $C_3\times C_{48}$ x 4, $C_6\times C_{24}$ x 2, $C_{12}^2$

Order 128: $C_2\times C_{64}$ x 2, $C_4\times C_{32}$

Order 96: $C_{96}$ x 16, $C_2\times C_{48}$ x 8, $C_4\times C_{24}$ x 4

Order 72: $C_3\times C_{24}$ x 4, $C_6\times C_{12}$ x 3

Order 64: $C_{64}$ x 4, $C_2\times C_{32}$ x 2, $C_4\times C_{16}$

Order 48: $C_{48}$ x 16, $C_2\times C_{24}$ x 8, $C_4\times C_{12}$ x 4

Order 36: $C_3\times C_{12}$ x 6, $C_6^2$

Order 32: $C_{32}$ x 4, $C_2\times C_{16}$ x 2, $C_4\times C_8$

Order 24: $C_{24}$ x 16, $C_2\times C_{12}$ x 12

Order 18: $C_3\times C_6$ x 3

Order 16: $C_{16}$ x 4, $C_2\times C_8$ x 2, $C_4^2$

Order 12: $C_{12}$ x 24, $C_2\times C_6$ x 4

Order 9: $C_3^2$

Order 8: $C_8$ x 4, $C_2\times C_4$ x 3

Order 6: $C_6$ x 12

Order 4: $C_4$ x 6, $C_2^2$

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 444672: $C_{18528}.C_{24}$

Order 222336: $C_3\times C_{772}.C_{48}.C_2$, $C_3\times C_{772}.(C_4\times C_{24})$

Order 148224: $C_{772}.(C_4\times C_{48})$, $C_3\times C_{193}:(C_4\times C_{64})$

Order 111168: $C_3\times C_{772}.(C_4\times C_{12})$, $C_3\times C_{772}.C_{24}.C_2$, $C_3\times C_{772}.C_{48}$, $C_3\times C_{3088}.C_6.C_2$

Order 74112: $C_3\times C_{193}:(C_4\times C_{32})$, $C_{772}.(C_4\times C_{24})$, $C_3\times C_{193}:(C_2\times C_{64})$, $C_{772}.C_{48}.C_2$

Order 55584: $C_3\times C_{193}:(C_2\times C_{48})$, $C_3\times C_{193}:(C_4\times C_{24})$, $C_3\times C_{193}:(C_2\times C_{48})$, $C_{96}\times C_{193}:C_3$, $C_3\times C_{193}:C_{96}$, $C_3\times C_{193}:C_{96}$

Order 49408: $C_{6176}.C_8$

Order 37056: $C_{772}.C_{48}$, $C_{772}.(C_4\times C_{12})$, $C_3\times C_{193}:(C_2\times C_{32})$, $C_3\times C_{193}:(C_4\times C_{16})$, $C_3\times C_{193}:C_{64}$, $C_3\times C_{193}:(C_2\times C_{32})$, $C_{3088}.C_6.C_2$, $C_{772}.C_{24}.C_2$

Order 27792: $C_{12}\times C_{193}:C_{12}$, $C_3\times C_{193}:(C_2\times C_{24})$, $C_3\times C_{193}:C_{48}$, $C_{48}\times C_{193}:C_3$, $C_3\times C_{193}:(C_2\times C_{24})$, $C_3\times C_{193}:C_{48}$

Order 24704: $D_{193}:C_{64}$, $C_{6176}:C_4$

Order 18528: $C_{193}:C_{96}$, $C_{193}:C_{96}$, $C_3\times C_{193}:(C_4\times C_8)$, $C_3\times C_{193}:C_{32}$, $C_{193}:(C_2\times C_{48})$, $C_3\times C_{193}:C_{32}$, $C_{32}\times C_{193}:C_3$, $C_3\times C_{193}:(C_2\times C_{16})$, $C_{193}:(C_2\times C_{48})$, $C_{193}:(C_4\times C_{24})$, $C_3\times C_{193}:(C_2\times C_{16})$, $C_{18528}$

Order 13896: $C_3\times C_{193}:C_{24}$, $C_3\times C_{193}:(C_2\times C_{12})$, $C_{24}\times C_{193}:C_3$, $C_3\times C_{193}:C_{24}$, $C_6\times C_{193}:C_{12}$

Order 12352: $C_{32}\times D_{193}$, $C_{193}:C_{64}$, $D_{193}:C_{32}$, $C_{3088}:C_4$

Order 9264: $C_{193}:(C_2\times C_{24})$, $C_3\times C_{193}:(C_2\times C_8)$, $C_{193}:C_{48}$, $C_{193}:(C_2\times C_{24})$, $C_{193}:C_{48}$, $C_{16}\times C_{193}:C_3$, $C_3\times C_{193}:C_{16}$, $C_3\times C_{193}:(C_2\times C_8)$, $C_{12}\times C_{193}:C_4$, $C_4\times C_{193}:C_{12}$, $C_3\times C_{193}:C_{16}$, $C_{9264}$

Order 6948: $C_{12}\times C_{193}:C_3$, $C_6\times C_{193}:C_6$, $C_3\times C_{193}:C_{12}$, $C_3\times C_{193}:C_{12}$

Order 6176: $C_{16}\times D_{193}$, $C_{193}:C_{32}$, $D_{193}:C_{16}$, $C_{1544}:C_4$, $C_{193}:C_{32}$, $C_{6176}$

Order 4632: $C_6\times C_{193}:C_4$, $C_8\times C_{193}:C_3$, $C_{193}:C_{24}$, $C_3\times C_{193}:C_8$, $C_{193}:C_{24}$, $C_{193}:(C_2\times C_{12})$, $C_3\times C_{193}:C_8$, $C_2\times C_{193}:C_{12}$, $C_3\times C_{193}:(C_2\times C_4)$, $C_{4632}$

Order 3474: $C_{1158}:C_3$, $C_{579}:C_6$

Order 3088: $D_{193}:C_8$, $C_{772}:C_4$, $C_{193}:C_{16}$, $C_8\times D_{193}$, $C_{193}:C_{16}$, $C_{3088}$

Order 2316: $C_3\times D_{386}$, $C_{193}:C_{12}$, $C_{193}:C_{12}$, $C_{386}:C_6$, $C_{193}:C_{12}$, $C_{193}:C_{12}$, $C_{193}:C_{12}$, $C_{2316}$

Order 2304: $C_{12}\times C_{192}$

Order 1737: $C_{579}:C_3$

Order 1544: $C_4\times D_{193}$, $C_{193}:C_8$, $C_{386}:C_4$, $C_{193}:C_8$, $C_{1544}$

Order 1158: $C_{1158}$, $C_3\times D_{193}$, $C_{193}:C_6$, $C_{193}:C_6$

Order 1152: $C_6\times C_{192}$, $C_{12}\times C_{96}$

Order 772: $D_{386}$, $C_{193}:C_4$, $C_{772}$, $C_{193}:C_4$

Order 768: $C_4\times C_{192}$ x 2

Order 579: $C_{579}$, $C_{193}:C_3$

Order 576: $C_6\times C_{96}$ x 2, $C_3\times C_{192}$, $C_{12}\times C_{48}$

Order 386: $C_{386}$, $D_{193}$

Order 384: $C_2\times C_{192}$ x 2, $C_4\times C_{96}$ x 2

Order 288: $C_3\times C_{96}$ x 3, $C_6\times C_{48}$ x 2, $C_{12}\times C_{24}$

Order 256: $C_4\times C_{64}$

Order 193: $C_{193}$

Order 192: $C_2\times C_{96}$ x 4, $C_{192}$ x 2, $C_4\times C_{48}$ x 2

Order 144: $C_3\times C_{48}$ x 3, $C_6\times C_{24}$ x 2, $C_{12}^2$

Order 128: $C_2\times C_{64}$, $C_4\times C_{32}$

Order 96: $C_{96}$ x 6, $C_2\times C_{48}$ x 4, $C_4\times C_{24}$ x 2

Order 72: $C_3\times C_{24}$ x 3, $C_6\times C_{12}$ x 2

Order 64: $C_2\times C_{32}$ x 2, $C_4\times C_{16}$, $C_{64}$

Order 48: $C_{48}$ x 6, $C_2\times C_{24}$ x 4, $C_4\times C_{12}$ x 2

Order 36: $C_3\times C_{12}$ x 3, $C_6^2$

Order 32: $C_{32}$ x 3, $C_2\times C_{16}$ x 2, $C_4\times C_8$

Order 24: $C_{24}$ x 6, $C_2\times C_{12}$ x 4

Order 18: $C_3\times C_6$ x 2

Order 16: $C_{16}$ x 3, $C_2\times C_8$ x 2, $C_4^2$

Order 12: $C_{12}$ x 6, $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 8: $C_8$ x 3, $C_2\times C_4$ x 2

Order 6: $C_6$ x 4

Order 4: $C_4$ x 3, $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 444672: $C_{18528}.C_{24}$ ($C_1$)

Order 222336: $C_3\times C_{772}.C_{48}.C_2$ ($C_2$) x 2, $C_3\times C_{772}.(C_4\times C_{24})$ ($C_2$)

Order 148224: $C_{772}.(C_4\times C_{48})$ ($C_3$) x 3, $C_3\times C_{193}:(C_4\times C_{64})$ ($C_3$)

Order 111168: $C_3\times C_{772}.C_{48}$ ($C_4$) x 4, $C_3\times C_{772}.(C_4\times C_{12})$ ($C_4$), $C_3\times C_{772}.C_{24}.C_2$ ($C_2^2$), $C_3\times C_{3088}.C_6.C_2$ ($C_4$)

Order 74112: $C_{772}.C_{48}.C_2$ ($C_6$) x 6, $C_{772}.(C_4\times C_{24})$ ($C_6$) x 3, $C_3\times C_{193}:(C_2\times C_{64})$ ($C_6$) x 2, $C_3\times C_{193}:(C_4\times C_{32})$ ($C_6$)

Order 55584: $C_3\times C_{193}:C_{96}$ ($C_2\times C_4$) x 2, $C_3\times C_{193}:(C_2\times C_{48})$ ($C_2\times C_4$), $C_3\times C_{193}:(C_4\times C_{24})$ ($C_8$), $C_3\times C_{193}:(C_2\times C_{48})$ ($C_8$), $C_{96}\times C_{193}:C_3$ ($C_8$), $C_3\times C_{193}:C_{96}$ ($C_8$)

Order 49408: $C_{6176}.C_8$ ($C_3^2$)

Order 37056: $C_{772}.C_{48}$ ($C_{12}$) x 12, $C_3\times C_{193}:C_{64}$ ($C_{12}$) x 4, $C_{772}.(C_4\times C_{12})$ ($C_{12}$) x 3, $C_{3088}.C_6.C_2$ ($C_{12}$) x 3, $C_{772}.C_{24}.C_2$ ($C_2\times C_6$) x 3, $C_3\times C_{193}:(C_2\times C_{32})$ ($C_2\times C_6$), $C_3\times C_{193}:(C_4\times C_{16})$ ($C_{12}$), $C_3\times C_{193}:(C_2\times C_{32})$ ($C_{12}$)

Order 27792: $C_3\times C_{193}:C_{48}$ ($C_{16}$) x 2, $C_{12}\times C_{193}:C_{12}$ ($C_{16}$), $C_3\times C_{193}:(C_2\times C_{24})$ ($C_2\times C_8$), $C_3\times C_{193}:C_{48}$ ($C_4^2$), $C_{48}\times C_{193}:C_3$ ($C_2\times C_8$), $C_3\times C_{193}:(C_2\times C_{24})$ ($C_{16}$)

Order 24704: $D_{193}:C_{64}$ ($C_3\times C_6$) x 2, $C_{6176}:C_4$ ($C_3\times C_6$)

Order 18528: $C_{193}:C_{96}$ ($C_2\times C_{12}$) x 6, $C_{193}:C_{96}$ ($C_{24}$) x 3, $C_{193}:(C_2\times C_{48})$ ($C_{24}$) x 3, $C_{32}\times C_{193}:C_3$ ($C_{24}$) x 3, $C_{193}:(C_2\times C_{48})$ ($C_2\times C_{12}$) x 3, $C_{193}:(C_4\times C_{24})$ ($C_{24}$) x 3, $C_3\times C_{193}:C_{32}$ ($C_2\times C_{12}$) x 2, $C_3\times C_{193}:(C_4\times C_8)$ ($C_{24}$), $C_3\times C_{193}:C_{32}$ ($C_{24}$), $C_3\times C_{193}:(C_2\times C_{16})$ ($C_{24}$), $C_3\times C_{193}:(C_2\times C_{16})$ ($C_2\times C_{12}$), $C_{18528}$ ($C_{24}$)

Order 13896: $C_3\times C_{193}:C_{24}$ ($C_{32}$) x 2, $C_6\times C_{193}:C_{12}$ ($C_{32}$) x 2, $C_3\times C_{193}:(C_2\times C_{12})$ ($C_2\times C_{16}$), $C_{24}\times C_{193}:C_3$ ($C_4\times C_8$), $C_3\times C_{193}:C_{24}$ ($C_2\times C_{16}$)

Order 12352: $C_{193}:C_{64}$ ($C_3\times C_{12}$) x 4, $C_{32}\times D_{193}$ ($C_3\times C_{12}$), $D_{193}:C_{32}$ ($C_6^2$), $C_{3088}:C_4$ ($C_3\times C_{12}$)

Order 9264: $C_{193}:C_{48}$ ($C_{48}$) x 6, $C_{193}:(C_2\times C_{24})$ ($C_{48}$) x 3, $C_{193}:(C_2\times C_{24})$ ($C_2\times C_{24}$) x 3, $C_{193}:C_{48}$ ($C_4\times C_{12}$) x 3, $C_{16}\times C_{193}:C_3$ ($C_2\times C_{24}$) x 3, $C_4\times C_{193}:C_{12}$ ($C_{48}$) x 3, $C_3\times C_{193}:C_{16}$ ($C_{48}$) x 2, $C_3\times C_{193}:(C_2\times C_8)$ ($C_2\times C_{24}$), $C_3\times C_{193}:(C_2\times C_8)$ ($C_{48}$), $C_{12}\times C_{193}:C_4$ ($C_{48}$), $C_3\times C_{193}:C_{16}$ ($C_4\times C_{12}$), $C_{9264}$ ($C_2\times C_{24}$)

Order 6948: $C_3\times C_{193}:C_{12}$ ($C_{64}$) x 4, $C_{12}\times C_{193}:C_3$ ($C_4\times C_{16}$), $C_6\times C_{193}:C_6$ ($C_2\times C_{32}$), $C_3\times C_{193}:C_{12}$ ($C_2\times C_{32}$)

Order 6176: $C_{193}:C_{32}$ ($C_6\times C_{12}$) x 2, $C_{16}\times D_{193}$ ($C_6\times C_{12}$), $C_{193}:C_{32}$ ($C_3\times C_{24}$), $D_{193}:C_{16}$ ($C_3\times C_{24}$), $C_{1544}:C_4$ ($C_3\times C_{24}$), $C_{6176}$ ($C_3\times C_{24}$)

Order 4632: $C_{193}:C_{24}$ ($C_{96}$) x 6, $C_2\times C_{193}:C_{12}$ ($C_{96}$) x 6, $C_8\times C_{193}:C_3$ ($C_4\times C_{24}$) x 3, $C_{193}:C_{24}$ ($C_2\times C_{48}$) x 3, $C_{193}:(C_2\times C_{12})$ ($C_2\times C_{48}$) x 3, $C_6\times C_{193}:C_4$ ($C_{96}$) x 2, $C_3\times C_{193}:C_8$ ($C_{96}$) x 2, $C_3\times C_{193}:C_8$ ($C_2\times C_{48}$), $C_3\times C_{193}:(C_2\times C_4)$ ($C_2\times C_{48}$), $C_{4632}$ ($C_4\times C_{24}$)

Order 3474: $C_{579}:C_6$ ($C_2\times C_{64}$) x 2, $C_{1158}:C_3$ ($C_4\times C_{32}$)

Order 3088: $C_{193}:C_{16}$ ($C_3\times C_{48}$) x 2, $D_{193}:C_8$ ($C_3\times C_{48}$), $C_{772}:C_4$ ($C_3\times C_{48}$), $C_8\times D_{193}$ ($C_6\times C_{24}$), $C_{193}:C_{16}$ ($C_{12}^2$), $C_{3088}$ ($C_6\times C_{24}$)

Order 2316: $C_{193}:C_{12}$ ($C_{192}$) x 12, $C_{193}:C_{12}$ ($C_{192}$) x 4, $C_{193}:C_{12}$ ($C_2\times C_{96}$) x 3, $C_{386}:C_6$ ($C_2\times C_{96}$) x 3, $C_{193}:C_{12}$ ($C_4\times C_{48}$) x 3, $C_3\times D_{386}$ ($C_2\times C_{96}$), $C_{193}:C_{12}$ ($C_2\times C_{96}$), $C_{2316}$ ($C_4\times C_{48}$)

Order 1737: $C_{579}:C_3$ ($C_4\times C_{64}$)

Order 1544: $C_{386}:C_4$ ($C_3\times C_{96}$) x 2, $C_{193}:C_8$ ($C_3\times C_{96}$) x 2, $C_4\times D_{193}$ ($C_6\times C_{48}$), $C_{193}:C_8$ ($C_6\times C_{48}$), $C_{1544}$ ($C_{12}\times C_{24}$)

Order 1158: $C_{193}:C_6$ ($C_2\times C_{192}$) x 6, $C_{193}:C_6$ ($C_4\times C_{96}$) x 3, $C_3\times D_{193}$ ($C_2\times C_{192}$) x 2, $C_{1158}$ ($C_4\times C_{96}$)

Order 772: $C_{193}:C_4$ ($C_3\times C_{192}$) x 4, $D_{386}$ ($C_6\times C_{96}$), $C_{772}$ ($C_{12}\times C_{48}$), $C_{193}:C_4$ ($C_6\times C_{96}$)

Order 579: $C_{193}:C_3$ ($C_4\times C_{192}$) x 3, $C_{579}$ ($C_4\times C_{192}$)

Order 386: $D_{193}$ ($C_6\times C_{192}$) x 2, $C_{386}$ ($C_{12}\times C_{96}$)

Order 193: $C_{193}$ ($C_{12}\times C_{192}$)

Order 96: $C_{96}$ ($C_{193}:C_{24}$)

Order 48: $C_{48}$ ($C_{193}:(C_2\times C_{24})$)

Order 32: $C_{32}$ ($C_{193}:(C_3\times C_{24})$)

Order 24: $C_{24}$ ($C_{193}:(C_4\times C_{24})$)

Order 16: $C_{16}$ ($C_{193}:(C_6\times C_{24})$)

Order 12: $C_{12}$ ($C_{1544}.C_{24}$)

Order 8: $C_8$ ($C_{2316}.C_{24}$)

Order 6: $C_6$ ($C_{3088}.C_{24}$)

Order 4: $C_4$ ($C_{4632}.C_{24}$)

Order 3: $C_3$ ($C_{6176}.C_{24}$)

Order 2: $C_2$ ($C_{9264}.C_{24}$)

Order 1: $C_1$ ($C_{18528}.C_{24}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 444672: $C_{18528}.C_{24}$ ($C_1$)

Order 222336: $C_3\times C_{772}.C_{48}.C_2$ ($C_2$), $C_3\times C_{772}.(C_4\times C_{24})$ ($C_2$)

Order 148224: $C_{772}.(C_4\times C_{48})$ ($C_3$), $C_3\times C_{193}:(C_4\times C_{64})$ ($C_3$)

Order 111168: $C_3\times C_{772}.(C_4\times C_{12})$ ($C_4$), $C_3\times C_{772}.C_{24}.C_2$ ($C_2^2$), $C_3\times C_{772}.C_{48}$ ($C_4$), $C_3\times C_{3088}.C_6.C_2$ ($C_4$)

Order 74112: $C_3\times C_{193}:(C_4\times C_{32})$ ($C_6$), $C_{772}.(C_4\times C_{24})$ ($C_6$), $C_3\times C_{193}:(C_2\times C_{64})$ ($C_6$), $C_{772}.C_{48}.C_2$ ($C_6$)

Order 55584: $C_3\times C_{193}:(C_2\times C_{48})$ ($C_2\times C_4$), $C_3\times C_{193}:(C_4\times C_{24})$ ($C_8$), $C_3\times C_{193}:(C_2\times C_{48})$ ($C_8$), $C_{96}\times C_{193}:C_3$ ($C_8$), $C_3\times C_{193}:C_{96}$ ($C_8$), $C_3\times C_{193}:C_{96}$ ($C_2\times C_4$)

Order 49408: $C_{6176}.C_8$ ($C_3^2$)

Order 37056: $C_{772}.C_{48}$ ($C_{12}$), $C_{772}.(C_4\times C_{12})$ ($C_{12}$), $C_3\times C_{193}:(C_2\times C_{32})$ ($C_2\times C_6$), $C_3\times C_{193}:(C_4\times C_{16})$ ($C_{12}$), $C_3\times C_{193}:C_{64}$ ($C_{12}$), $C_3\times C_{193}:(C_2\times C_{32})$ ($C_{12}$), $C_{3088}.C_6.C_2$ ($C_{12}$), $C_{772}.C_{24}.C_2$ ($C_2\times C_6$)

Order 27792: $C_{12}\times C_{193}:C_{12}$ ($C_{16}$), $C_3\times C_{193}:(C_2\times C_{24})$ ($C_2\times C_8$), $C_3\times C_{193}:C_{48}$ ($C_4^2$), $C_{48}\times C_{193}:C_3$ ($C_2\times C_8$), $C_3\times C_{193}:(C_2\times C_{24})$ ($C_{16}$), $C_3\times C_{193}:C_{48}$ ($C_{16}$)

Order 24704: $D_{193}:C_{64}$ ($C_3\times C_6$), $C_{6176}:C_4$ ($C_3\times C_6$)

Order 18528: $C_{193}:C_{96}$ ($C_2\times C_{12}$), $C_{193}:C_{96}$ ($C_{24}$), $C_3\times C_{193}:(C_4\times C_8)$ ($C_{24}$), $C_3\times C_{193}:C_{32}$ ($C_{24}$), $C_{193}:(C_2\times C_{48})$ ($C_{24}$), $C_3\times C_{193}:C_{32}$ ($C_2\times C_{12}$), $C_{32}\times C_{193}:C_3$ ($C_{24}$), $C_3\times C_{193}:(C_2\times C_{16})$ ($C_{24}$), $C_{193}:(C_2\times C_{48})$ ($C_2\times C_{12}$), $C_{193}:(C_4\times C_{24})$ ($C_{24}$), $C_3\times C_{193}:(C_2\times C_{16})$ ($C_2\times C_{12}$), $C_{18528}$ ($C_{24}$)

Order 13896: $C_3\times C_{193}:C_{24}$ ($C_{32}$), $C_3\times C_{193}:(C_2\times C_{12})$ ($C_2\times C_{16}$), $C_{24}\times C_{193}:C_3$ ($C_4\times C_8$), $C_3\times C_{193}:C_{24}$ ($C_2\times C_{16}$), $C_6\times C_{193}:C_{12}$ ($C_{32}$)

Order 12352: $C_{32}\times D_{193}$ ($C_3\times C_{12}$), $C_{193}:C_{64}$ ($C_3\times C_{12}$), $D_{193}:C_{32}$ ($C_6^2$), $C_{3088}:C_4$ ($C_3\times C_{12}$)

Order 9264: $C_{193}:(C_2\times C_{24})$ ($C_{48}$), $C_3\times C_{193}:(C_2\times C_8)$ ($C_2\times C_{24}$), $C_{193}:C_{48}$ ($C_{48}$), $C_{193}:(C_2\times C_{24})$ ($C_2\times C_{24}$), $C_{193}:C_{48}$ ($C_4\times C_{12}$), $C_{16}\times C_{193}:C_3$ ($C_2\times C_{24}$), $C_3\times C_{193}:C_{16}$ ($C_{48}$), $C_3\times C_{193}:(C_2\times C_8)$ ($C_{48}$), $C_{12}\times C_{193}:C_4$ ($C_{48}$), $C_4\times C_{193}:C_{12}$ ($C_{48}$), $C_3\times C_{193}:C_{16}$ ($C_4\times C_{12}$), $C_{9264}$ ($C_2\times C_{24}$)

Order 6948: $C_{12}\times C_{193}:C_3$ ($C_4\times C_{16}$), $C_6\times C_{193}:C_6$ ($C_2\times C_{32}$), $C_3\times C_{193}:C_{12}$ ($C_2\times C_{32}$), $C_3\times C_{193}:C_{12}$ ($C_{64}$)

Order 6176: $C_{16}\times D_{193}$ ($C_6\times C_{12}$), $C_{193}:C_{32}$ ($C_3\times C_{24}$), $D_{193}:C_{16}$ ($C_3\times C_{24}$), $C_{1544}:C_4$ ($C_3\times C_{24}$), $C_{193}:C_{32}$ ($C_6\times C_{12}$), $C_{6176}$ ($C_3\times C_{24}$)

Order 4632: $C_6\times C_{193}:C_4$ ($C_{96}$), $C_8\times C_{193}:C_3$ ($C_4\times C_{24}$), $C_{193}:C_{24}$ ($C_2\times C_{48}$), $C_3\times C_{193}:C_8$ ($C_{96}$), $C_{193}:C_{24}$ ($C_{96}$), $C_{193}:(C_2\times C_{12})$ ($C_2\times C_{48}$), $C_3\times C_{193}:C_8$ ($C_2\times C_{48}$), $C_2\times C_{193}:C_{12}$ ($C_{96}$), $C_3\times C_{193}:(C_2\times C_4)$ ($C_2\times C_{48}$), $C_{4632}$ ($C_4\times C_{24}$)

Order 3474: $C_{1158}:C_3$ ($C_4\times C_{32}$), $C_{579}:C_6$ ($C_2\times C_{64}$)

Order 3088: $D_{193}:C_8$ ($C_3\times C_{48}$), $C_{772}:C_4$ ($C_3\times C_{48}$), $C_{193}:C_{16}$ ($C_3\times C_{48}$), $C_8\times D_{193}$ ($C_6\times C_{24}$), $C_{193}:C_{16}$ ($C_{12}^2$), $C_{3088}$ ($C_6\times C_{24}$)

Order 2316: $C_3\times D_{386}$ ($C_2\times C_{96}$), $C_{193}:C_{12}$ ($C_2\times C_{96}$), $C_{193}:C_{12}$ ($C_2\times C_{96}$), $C_{386}:C_6$ ($C_2\times C_{96}$), $C_{193}:C_{12}$ ($C_4\times C_{48}$), $C_{193}:C_{12}$ ($C_{192}$), $C_{193}:C_{12}$ ($C_{192}$), $C_{2316}$ ($C_4\times C_{48}$)

Order 1737: $C_{579}:C_3$ ($C_4\times C_{64}$)

Order 1544: $C_4\times D_{193}$ ($C_6\times C_{48}$), $C_{193}:C_8$ ($C_6\times C_{48}$), $C_{386}:C_4$ ($C_3\times C_{96}$), $C_{193}:C_8$ ($C_3\times C_{96}$), $C_{1544}$ ($C_{12}\times C_{24}$)

Order 1158: $C_{1158}$ ($C_4\times C_{96}$), $C_3\times D_{193}$ ($C_2\times C_{192}$), $C_{193}:C_6$ ($C_4\times C_{96}$), $C_{193}:C_6$ ($C_2\times C_{192}$)

Order 772: $D_{386}$ ($C_6\times C_{96}$), $C_{193}:C_4$ ($C_3\times C_{192}$), $C_{772}$ ($C_{12}\times C_{48}$), $C_{193}:C_4$ ($C_6\times C_{96}$)

Order 579: $C_{579}$ ($C_4\times C_{192}$), $C_{193}:C_3$ ($C_4\times C_{192}$)

Order 386: $C_{386}$ ($C_{12}\times C_{96}$), $D_{193}$ ($C_6\times C_{192}$)

Order 193: $C_{193}$ ($C_{12}\times C_{192}$)

Order 96: $C_{96}$ ($C_{193}:C_{24}$)

Order 48: $C_{48}$ ($C_{193}:(C_2\times C_{24})$)

Order 32: $C_{32}$ ($C_{193}:(C_3\times C_{24})$)

Order 24: $C_{24}$ ($C_{193}:(C_4\times C_{24})$)

Order 16: $C_{16}$ ($C_{193}:(C_6\times C_{24})$)

Order 12: $C_{12}$ ($C_{1544}.C_{24}$)

Order 8: $C_8$ ($C_{2316}.C_{24}$)

Order 6: $C_6$ ($C_{3088}.C_{24}$)

Order 4: $C_4$ ($C_{4632}.C_{24}$)

Order 3: $C_3$ ($C_{6176}.C_{24}$)

Order 2: $C_2$ ($C_{9264}.C_{24}$)

Order 1: $C_1$ ($C_{18528}.C_{24}$)

Series

Derived series

$C_{18528}.C_{24}$

$\rhd$

$C_{18528}.C_{24}$

$\rhd$

$C_{193}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{18528}.C_{24}$

$\rhd$

$C_{18528}.C_{24}$

$\rhd$

$C_3\times C_{772}.(C_4\times C_{24})$

$\rhd$

$C_3\times C_{772}.(C_4\times C_{24})$

$\rhd$

$C_3\times C_{3088}.C_6.C_2$

$\rhd$

$C_3\times C_{3088}.C_6.C_2$

$\rhd$

$C_{96}\times C_{193}:C_3$

$\rhd$

$C_{96}\times C_{193}:C_3$

$\rhd$

$C_{18528}$

$\rhd$

$C_{18528}$

$\rhd$

$C_{9264}$

$\rhd$

$C_{9264}$

$\rhd$

$C_{4632}$

$\rhd$

$C_{4632}$

$\rhd$

$C_{2316}$

$\rhd$

$C_{2316}$

$\rhd$

$C_{1158}$

$\rhd$

$C_{1158}$

$\rhd$

$C_{579}$

$\rhd$

$C_{579}$

$\rhd$

$C_{193}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{18528}.C_{24}$

$\rhd$

$C_{18528}.C_{24}$

$\rhd$

$C_{193}$

$\rhd$

$C_{193}$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_{96}$

$\lhd$

$C_{96}$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $3072 \times 3072$ character table is not available for this group.

Rational character table

The $142 \times 142$ rational character table is not available for this group.