Abstract group 296448.l: $C_{18528}.C_{16}$

• Label: 296448.l
• Order: \( 2^{9} \cdot 3 \cdot 193 \)
• Exponent: \( 2^{6} \cdot 3 \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{9} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{5} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3 \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \cdot 3 \)
• Perm deg.: $260$
• Trans deg.: $37056$
• Rank: $2$

Group information

Description:	$C_{18528}.C_{16}$
Order:	\(296448\)\(\medspace = 2^{9} \cdot 3 \cdot 193 \)
Exponent:	\(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Automorphism group:	$C_{1544}.C_{24}.C_4^2.C_2^5$, of order \(18972672\)\(\medspace = 2^{15} \cdot 3 \cdot 193 \)
Composition factors:	$C_2$ x 9, $C_3$, $C_{193}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, 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	2	1932	774	8496	3864	10816	16992	21632	21632	49408	43264	98816	192	192	384	384	384	768	768	1536	1536	3072	3072	6144	296448
Conjugacy classes	1	3	2	12	6	48	24	64	96	128	128	256	256	512	12	12	24	24	24	48	48	96	96	192	192	384	2688
Divisions	1	3	1	6	3	12	6	8	12	8	8	8	8	8	1	1	1	1	1	1	1	1	1	1	1	1	104
Autjugacy classes	1	2	1	4	2	8	4	8	8	8	8	8	8	8	1	1	1	1	1	1	1	1	1	1	1	1	90

Minimal presentations

Permutation degree:	$260$
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	16	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid b^{18528}=1, a^{16}=b^{16984}, b^{a}=b^{18145} \rangle$
Permutation group:	Degree $260$ $\langle(2,4,6,14,21,34,13,11,3,9,10,30,28,25,8,7)(5,16,17,46,49,67,37,23,12,35,36,95,55,52,19,18) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 64 & 0 \\ 0 & 190 \end{array}\right), \left(\begin{array}{rr} 153 & 0 \\ 0 & 120 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_3$ $\, \times\, $ $(C_{6176}.C_{16})$
Semidirect product:	$C_{579}$ $\,\rtimes\,$ $(C_8\times C_{64})$	$(C_{193}:C_{64})$ $\,\rtimes\,$ $C_{24}$ (8)	$(C_{193}:C_8)$ $\,\rtimes\,$ $C_{192}$ (8)	$C_{193}$ $\,\rtimes\,$ $(C_8\times C_{192})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{6176}$ . $C_{48}$	$C_{18528}$ . $C_{16}$	$D_{386}$ . $(C_4\times C_{96})$	$(C_{6176}:C_8)$ . $C_6$	all 73

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

Homology

Abelianization:	$C_{8} \times C_{192} \simeq C_{8} \times C_{64} \times C_{3}$
Schur multiplier:	$C_{8}$
Commutator length:	$1$

Subgroups

There are 29512 subgroups in 328 conjugacy classes, 176 normal (68 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{96}$	$G/Z \simeq$ $C_{193}:C_{16}$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_8\times C_{192}$
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $C_{579}:(C_2\times C_{16})$
Fitting:	$\operatorname{Fit} \simeq$ $C_{18528}$	$G/\operatorname{Fit} \simeq$ $C_{16}$
Radical:	$R \simeq$ $C_{18528}.C_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{1158}$	$G/\operatorname{soc} \simeq$ $C_8\times C_{32}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\times C_{64}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 296448: $C_{18528}.C_{16}$

Order 148224: $C_3\times C_{193}:(C_4\times C_{64})$ x 2, $C_3\times C_{193}:(C_8\times C_{32})$

Order 98816: $C_{193}:(C_8\times C_{64})$

Order 74112: $C_3\times C_{193}:(C_2\times C_{64})$ x 4, $C_3\times C_{193}:(C_4\times C_{32})$, $C_3\times C_{193}:(C_4\times C_{32})$, $C_{48}\times C_{193}:C_8$

Order 49408: $C_{3088}.C_{16}$ x 2, $C_{6176}:C_8$

Order 37056: $C_3\times C_{193}:C_{64}$ x 8, $C_3\times C_{193}:(C_2\times C_{32})$ x 2, $C_{24}\times C_{193}:C_8$, $C_3\times C_{193}:(C_4\times C_{16})$, $C_3\times C_{193}:(C_2\times C_{32})$, $C_3\times C_{193}:(C_4\times C_{16})$, $C_3\times C_{193}:(C_2\times C_{32})$

Order 24704: $D_{193}:C_{64}$ x 4, $C_{3088}.C_8$, $C_{3088}:C_8$, $C_{6176}:C_4$

Order 18528: $C_3\times C_{193}:C_{32}$ x 4, $C_3\times C_{193}:(C_2\times C_{16})$ x 2, $C_3\times C_{193}:C_{32}$ x 2, $C_{12}\times C_{193}:C_8$ x 2, $C_{24}\times C_{193}:C_4$, $C_3\times C_{193}:C_{32}$, $C_3\times C_{193}:(C_2\times C_{16})$, $C_3\times C_{193}:(C_2\times C_{16})$, $C_{18528}$

Order 12352: $C_{193}:C_{64}$ x 8, $D_{193}:C_{32}$ x 2, $C_{32}\times D_{193}$, $C_{1544}.C_8$, $C_{1544}:C_8$, $D_{193}:C_{32}$, $C_{3088}:C_4$

Order 9264: $C_3\times C_{193}:C_{16}$ x 4, $C_6\times C_{193}:C_8$ x 4, $C_3\times C_{193}:C_{16}$ 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 6176: $C_{193}:C_{32}$ x 4, $D_{193}:C_{16}$ x 2, $C_{772}:C_8$ x 2, $C_{193}:C_{32}$ x 2, $C_{16}\times D_{193}$, $C_{193}:C_{32}$, $D_{193}:C_{16}$, $C_{1544}:C_4$, $C_{6176}$

Order 4632: $C_3\times C_{193}:C_8$ x 8, $C_6\times C_{193}:C_4$ x 2, $C_3\times C_{193}:C_8$ x 2, $C_3\times C_{193}:C_8$, $C_{12}\times D_{193}$, $C_{4632}$

Order 3088: $C_{386}:C_8$ x 4, $C_{193}:C_{16}$ x 4, $C_{193}:C_{16}$ 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 4, $C_3\times D_{386}$, $C_{193}:C_{12}$, $C_{2316}$

Order 1544: $C_{193}:C_8$ x 8, $C_{386}:C_4$ x 2, $C_{193}:C_8$ x 2, $C_4\times D_{193}$, $C_{193}:C_8$, $C_{1544}$

Order 1536: $C_8\times C_{192}$

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

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

Order 768: $C_4\times C_{192}$ x 2, $C_8\times C_{96}$

Order 579: $C_{579}$

Order 512: $C_8\times C_{64}$

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

Order 384: $C_2\times C_{192}$ x 4, $C_4\times C_{96}$ x 2, $C_8\times C_{48}$

Order 256: $C_4\times C_{64}$ x 2, $C_8\times C_{32}$

Order 193: $C_{193}$

Order 192: $C_{192}$ x 8, $C_2\times C_{96}$ x 4, $C_4\times C_{48}$ x 2, $C_8\times C_{24}$

Order 128: $C_2\times C_{64}$ x 4, $C_4\times C_{32}$ x 2, $C_8\times C_{16}$

Order 96: $C_{96}$ x 8, $C_2\times C_{48}$ x 4, $C_4\times C_{24}$ x 3

Order 64: $C_{64}$ x 8, $C_2\times C_{32}$ x 4, $C_4\times C_{16}$ x 2, $C_8^2$

Order 48: $C_{48}$ x 8, $C_2\times C_{24}$ x 6, $C_4\times C_{12}$

Order 32: $C_{32}$ x 8, $C_2\times C_{16}$ x 4, $C_4\times C_8$ x 3

Order 24: $C_{24}$ x 12, $C_2\times C_{12}$ x 3

Order 16: $C_{16}$ x 8, $C_2\times C_8$ x 6, $C_4^2$

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

Order 8: $C_8$ x 12, $C_2\times C_4$ x 3

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 296448: $C_{18528}.C_{16}$

Order 148224: $C_3\times C_{193}:(C_8\times C_{32})$, $C_3\times C_{193}:(C_4\times C_{64})$

Order 98816: $C_{193}:(C_8\times C_{64})$

Order 74112: $C_3\times C_{193}:(C_4\times C_{32})$, $C_3\times C_{193}:(C_2\times C_{64})$, $C_3\times C_{193}:(C_4\times C_{32})$, $C_{48}\times C_{193}:C_8$

Order 49408: $C_{3088}.C_{16}$, $C_{6176}:C_8$

Order 37056: $C_{24}\times C_{193}:C_8$, $C_3\times C_{193}:(C_4\times C_{16})$, $C_3\times C_{193}:(C_2\times C_{32})$, $C_3\times C_{193}:(C_2\times C_{32})$, $C_3\times C_{193}:(C_4\times C_{16})$, $C_3\times C_{193}:(C_2\times C_{32})$, $C_3\times C_{193}:C_{64}$

Order 24704: $D_{193}:C_{64}$, $C_{3088}.C_8$, $C_{3088}:C_8$, $C_{6176}:C_4$

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

Order 12352: $C_{32}\times D_{193}$, $C_{193}:C_{64}$, $D_{193}:C_{32}$, $C_{1544}.C_8$, $C_{1544}:C_8$, $D_{193}:C_{32}$, $C_{3088}:C_4$

Order 9264: $C_3\times C_{193}:(C_2\times C_8)$, $C_3\times C_{193}:C_{16}$, $C_3\times C_{193}:C_{16}$, $C_3\times C_{193}:(C_2\times C_8)$, $C_6\times C_{193}:C_8$, $C_{12}\times C_{193}:C_4$, $C_3\times C_{193}:C_{16}$, $C_{9264}$

Order 6176: $C_{16}\times D_{193}$, $C_{193}:C_{32}$, $D_{193}:C_{16}$, $C_{772}:C_8$, $C_{193}:C_{32}$, $D_{193}:C_{16}$, $C_{1544}:C_4$, $C_{193}:C_{32}$, $C_{6176}$

Order 4632: $C_6\times C_{193}:C_4$, $C_3\times C_{193}:C_8$, $C_3\times C_{193}:C_8$, $C_3\times C_{193}:C_8$, $C_{12}\times D_{193}$, $C_{4632}$

Order 3088: $C_{386}:C_8$, $C_{193}:C_{16}$, $D_{193}:C_8$, $C_{772}:C_4$, $C_{193}:C_{16}$, $C_8\times D_{193}$, $C_{193}:C_{16}$, $C_{3088}$

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

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

Order 1536: $C_8\times C_{192}$

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

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

Order 768: $C_4\times C_{192}$, $C_8\times C_{96}$

Order 579: $C_{579}$

Order 512: $C_8\times C_{64}$

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

Order 384: $C_4\times C_{96}$ x 2, $C_2\times C_{192}$, $C_8\times C_{48}$

Order 256: $C_4\times C_{64}$, $C_8\times C_{32}$

Order 193: $C_{193}$

Order 192: $C_2\times C_{96}$ x 3, $C_4\times C_{48}$ x 2, $C_{192}$, $C_8\times C_{24}$

Order 128: $C_4\times C_{32}$ x 2, $C_8\times C_{16}$, $C_2\times C_{64}$

Order 96: $C_{96}$ x 4, $C_2\times C_{48}$ x 3, $C_4\times C_{24}$ x 2

Order 64: $C_2\times C_{32}$ x 3, $C_4\times C_{16}$ x 2, $C_8^2$, $C_{64}$

Order 48: $C_{48}$ x 4, $C_2\times C_{24}$ x 3, $C_4\times C_{12}$

Order 32: $C_{32}$ x 4, $C_2\times C_{16}$ x 3, $C_4\times C_8$ x 2

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

Order 16: $C_{16}$ x 4, $C_2\times C_8$ x 3, $C_4^2$

Order 12: $C_{12}$ x 3, $C_2\times C_6$

Order 8: $C_8$ x 4, $C_2\times C_4$ x 2

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 296448: $C_{18528}.C_{16}$ ($C_1$)

Order 148224: $C_3\times C_{193}:(C_4\times C_{64})$ ($C_2$) x 2, $C_3\times C_{193}:(C_8\times C_{32})$ ($C_2$)

Order 98816: $C_{193}:(C_8\times C_{64})$ ($C_3$)

Order 74112: $C_3\times C_{193}:(C_2\times C_{64})$ ($C_4$) x 4, $C_3\times C_{193}:(C_4\times C_{32})$ ($C_4$), $C_3\times C_{193}:(C_4\times C_{32})$ ($C_2^2$), $C_{48}\times C_{193}:C_8$ ($C_4$)

Order 49408: $C_{3088}.C_{16}$ ($C_6$) x 2, $C_{6176}:C_8$ ($C_6$)

Order 37056: $C_3\times C_{193}:C_{64}$ ($C_8$) x 8, $C_3\times C_{193}:(C_2\times C_{32})$ ($C_2\times C_4$) x 2, $C_{24}\times C_{193}:C_8$ ($C_8$), $C_3\times C_{193}:(C_4\times C_{16})$ ($C_8$), $C_3\times C_{193}:(C_2\times C_{32})$ ($C_8$), $C_3\times C_{193}:(C_4\times C_{16})$ ($C_2\times C_4$), $C_3\times C_{193}:(C_2\times C_{32})$ ($C_8$)

Order 24704: $D_{193}:C_{64}$ ($C_{12}$) x 4, $C_{3088}.C_8$ ($C_2\times C_6$), $C_{3088}:C_8$ ($C_{12}$), $C_{6176}:C_4$ ($C_{12}$)

Order 18528: $C_3\times C_{193}:C_{32}$ ($C_2\times C_8$) x 4, $C_3\times C_{193}:(C_2\times C_{16})$ ($C_{16}$) x 2, $C_3\times C_{193}:C_{32}$ ($C_{16}$) x 2, $C_{12}\times C_{193}:C_8$ ($C_{16}$) x 2, $C_{24}\times C_{193}:C_4$ ($C_2\times C_8$), $C_3\times C_{193}:C_{32}$ ($C_{16}$), $C_3\times C_{193}:(C_2\times C_{16})$ ($C_4^2$), $C_3\times C_{193}:(C_2\times C_{16})$ ($C_2\times C_8$), $C_{18528}$ ($C_{16}$)

Order 12352: $C_{193}:C_{64}$ ($C_{24}$) x 8, $D_{193}:C_{32}$ ($C_2\times C_{12}$) x 2, $C_{32}\times D_{193}$ ($C_{24}$), $C_{1544}.C_8$ ($C_{24}$), $C_{1544}:C_8$ ($C_{24}$), $D_{193}:C_{32}$ ($C_{24}$), $C_{3088}:C_4$ ($C_2\times C_{12}$)

Order 9264: $C_3\times C_{193}:C_{16}$ ($C_{32}$) x 4, $C_6\times C_{193}:C_8$ ($C_{32}$) x 4, $C_3\times C_{193}:C_{16}$ ($C_4\times C_8$) x 2, $C_3\times C_{193}:(C_2\times C_8)$ ($C_4\times C_8$), $C_3\times C_{193}:(C_2\times C_8)$ ($C_2\times C_{16}$), $C_{12}\times C_{193}:C_4$ ($C_2\times C_{16}$), $C_3\times C_{193}:C_{16}$ ($C_2\times C_{16}$), $C_{9264}$ ($C_2\times C_{16}$)

Order 6176: $C_{193}:C_{32}$ ($C_2\times C_{24}$) x 4, $D_{193}:C_{16}$ ($C_{48}$) x 2, $C_{772}:C_8$ ($C_{48}$) x 2, $C_{193}:C_{32}$ ($C_{48}$) x 2, $C_{16}\times D_{193}$ ($C_2\times C_{24}$), $C_{193}:C_{32}$ ($C_{48}$), $D_{193}:C_{16}$ ($C_4\times C_{12}$), $C_{1544}:C_4$ ($C_2\times C_{24}$), $C_{6176}$ ($C_{48}$)

Order 4632: $C_3\times C_{193}:C_8$ ($C_{64}$) x 8, $C_6\times C_{193}:C_4$ ($C_2\times C_{32}$) x 2, $C_3\times C_{193}:C_8$ ($C_2\times C_{32}$) x 2, $C_3\times C_{193}:C_8$ ($C_8^2$), $C_{12}\times D_{193}$ ($C_4\times C_{16}$), $C_{4632}$ ($C_4\times C_{16}$)

Order 3088: $C_{386}:C_8$ ($C_{96}$) x 4, $C_{193}:C_{16}$ ($C_{96}$) x 4, $C_{193}:C_{16}$ ($C_4\times C_{24}$) x 2, $D_{193}:C_8$ ($C_2\times C_{48}$), $C_{772}:C_4$ ($C_2\times C_{48}$), $C_8\times D_{193}$ ($C_4\times C_{24}$), $C_{193}:C_{16}$ ($C_2\times C_{48}$), $C_{3088}$ ($C_2\times C_{48}$)

Order 2316: $C_{193}:C_{12}$ ($C_2\times C_{64}$) x 4, $C_3\times D_{386}$ ($C_4\times C_{32}$), $C_{193}:C_{12}$ ($C_4\times C_{32}$), $C_{2316}$ ($C_8\times C_{16}$)

Order 1544: $C_{193}:C_8$ ($C_{192}$) x 8, $C_{386}:C_4$ ($C_2\times C_{96}$) x 2, $C_{193}:C_8$ ($C_2\times C_{96}$) x 2, $C_4\times D_{193}$ ($C_4\times C_{48}$), $C_{193}:C_8$ ($C_8\times C_{24}$), $C_{1544}$ ($C_4\times C_{48}$)

Order 1158: $C_3\times D_{193}$ ($C_4\times C_{64}$) x 2, $C_{1158}$ ($C_8\times C_{32}$)

Order 772: $C_{193}:C_4$ ($C_2\times C_{192}$) x 4, $D_{386}$ ($C_4\times C_{96}$), $C_{772}$ ($C_8\times C_{48}$), $C_{193}:C_4$ ($C_4\times C_{96}$)

Order 579: $C_{579}$ ($C_8\times C_{64}$)

Order 386: $D_{193}$ ($C_4\times C_{192}$) x 2, $C_{386}$ ($C_8\times C_{96}$)

Order 193: $C_{193}$ ($C_8\times C_{192}$)

Order 96: $C_{96}$ ($C_{193}:C_{16}$)

Order 48: $C_{48}$ ($C_{386}:C_{16}$)

Order 32: $C_{32}$ ($C_{579}:C_{16}$)

Order 24: $C_{24}$ ($C_{772}:C_{16}$)

Order 16: $C_{16}$ ($C_{579}:(C_2\times C_{16})$)

Order 12: $C_{12}$ ($C_{1544}:C_{16}$)

Order 8: $C_8$ ($C_{579}:(C_4\times C_{16})$)

Order 6: $C_6$ ($C_{3088}.C_{16}$)

Order 4: $C_4$ ($C_{579}:(C_8\times C_{16})$)

Order 3: $C_3$ ($C_{193}:(C_8\times C_{64})$)

Order 2: $C_2$ ($C_{579}:(C_8\times C_{32})$)

Order 1: $C_1$ ($C_{18528}.C_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 296448: $C_{18528}.C_{16}$ ($C_1$)

Order 148224: $C_3\times C_{193}:(C_8\times C_{32})$ ($C_2$), $C_3\times C_{193}:(C_4\times C_{64})$ ($C_2$)

Order 98816: $C_{193}:(C_8\times C_{64})$ ($C_3$)

Order 74112: $C_3\times C_{193}:(C_4\times C_{32})$ ($C_4$), $C_3\times C_{193}:(C_2\times C_{64})$ ($C_4$), $C_3\times C_{193}:(C_4\times C_{32})$ ($C_2^2$), $C_{48}\times C_{193}:C_8$ ($C_4$)

Order 49408: $C_{3088}.C_{16}$ ($C_6$), $C_{6176}:C_8$ ($C_6$)

Order 37056: $C_{24}\times C_{193}:C_8$ ($C_8$), $C_3\times C_{193}:(C_4\times C_{16})$ ($C_8$), $C_3\times C_{193}:(C_2\times C_{32})$ ($C_2\times C_4$), $C_3\times C_{193}:(C_2\times C_{32})$ ($C_8$), $C_3\times C_{193}:(C_4\times C_{16})$ ($C_2\times C_4$), $C_3\times C_{193}:(C_2\times C_{32})$ ($C_8$), $C_3\times C_{193}:C_{64}$ ($C_8$)

Order 24704: $D_{193}:C_{64}$ ($C_{12}$), $C_{3088}.C_8$ ($C_2\times C_6$), $C_{3088}:C_8$ ($C_{12}$), $C_{6176}:C_4$ ($C_{12}$)

Order 18528: $C_{24}\times C_{193}:C_4$ ($C_2\times C_8$), $C_3\times C_{193}:(C_2\times C_{16})$ ($C_{16}$), $C_3\times C_{193}:C_{32}$ ($C_{16}$), $C_3\times C_{193}:C_{32}$ ($C_{16}$), $C_3\times C_{193}:(C_2\times C_{16})$ ($C_4^2$), $C_{12}\times C_{193}:C_8$ ($C_{16}$), $C_3\times C_{193}:C_{32}$ ($C_2\times C_8$), $C_3\times C_{193}:(C_2\times C_{16})$ ($C_2\times C_8$), $C_{18528}$ ($C_{16}$)

Order 12352: $C_{32}\times D_{193}$ ($C_{24}$), $C_{193}:C_{64}$ ($C_{24}$), $D_{193}:C_{32}$ ($C_2\times C_{12}$), $C_{1544}.C_8$ ($C_{24}$), $C_{1544}:C_8$ ($C_{24}$), $D_{193}:C_{32}$ ($C_{24}$), $C_{3088}:C_4$ ($C_2\times C_{12}$)

Order 9264: $C_3\times C_{193}:(C_2\times C_8)$ ($C_4\times C_8$), $C_3\times C_{193}:C_{16}$ ($C_{32}$), $C_3\times C_{193}:C_{16}$ ($C_4\times C_8$), $C_3\times C_{193}:(C_2\times C_8)$ ($C_2\times C_{16}$), $C_6\times C_{193}:C_8$ ($C_{32}$), $C_{12}\times C_{193}:C_4$ ($C_2\times C_{16}$), $C_3\times C_{193}:C_{16}$ ($C_2\times C_{16}$), $C_{9264}$ ($C_2\times C_{16}$)

Order 6176: $C_{16}\times D_{193}$ ($C_2\times C_{24}$), $C_{193}:C_{32}$ ($C_{48}$), $D_{193}:C_{16}$ ($C_{48}$), $C_{772}:C_8$ ($C_{48}$), $C_{193}:C_{32}$ ($C_2\times C_{24}$), $D_{193}:C_{16}$ ($C_4\times C_{12}$), $C_{1544}:C_4$ ($C_2\times C_{24}$), $C_{193}:C_{32}$ ($C_{48}$), $C_{6176}$ ($C_{48}$)

Order 4632: $C_6\times C_{193}:C_4$ ($C_2\times C_{32}$), $C_3\times C_{193}:C_8$ ($C_2\times C_{32}$), $C_3\times C_{193}:C_8$ ($C_8^2$), $C_3\times C_{193}:C_8$ ($C_{64}$), $C_{12}\times D_{193}$ ($C_4\times C_{16}$), $C_{4632}$ ($C_4\times C_{16}$)

Order 3088: $C_{386}:C_8$ ($C_{96}$), $C_{193}:C_{16}$ ($C_{96}$), $D_{193}:C_8$ ($C_2\times C_{48}$), $C_{772}:C_4$ ($C_2\times C_{48}$), $C_{193}:C_{16}$ ($C_4\times C_{24}$), $C_8\times D_{193}$ ($C_4\times C_{24}$), $C_{193}:C_{16}$ ($C_2\times C_{48}$), $C_{3088}$ ($C_2\times C_{48}$)

Order 2316: $C_3\times D_{386}$ ($C_4\times C_{32}$), $C_{193}:C_{12}$ ($C_4\times C_{32}$), $C_{193}:C_{12}$ ($C_2\times C_{64}$), $C_{2316}$ ($C_8\times C_{16}$)

Order 1544: $C_4\times D_{193}$ ($C_4\times C_{48}$), $C_{193}:C_8$ ($C_8\times C_{24}$), $C_{193}:C_8$ ($C_{192}$), $C_{386}:C_4$ ($C_2\times C_{96}$), $C_{193}:C_8$ ($C_2\times C_{96}$), $C_{1544}$ ($C_4\times C_{48}$)

Order 1158: $C_{1158}$ ($C_8\times C_{32}$), $C_3\times D_{193}$ ($C_4\times C_{64}$)

Order 772: $D_{386}$ ($C_4\times C_{96}$), $C_{193}:C_4$ ($C_2\times C_{192}$), $C_{772}$ ($C_8\times C_{48}$), $C_{193}:C_4$ ($C_4\times C_{96}$)

Order 579: $C_{579}$ ($C_8\times C_{64}$)

Order 386: $C_{386}$ ($C_8\times C_{96}$), $D_{193}$ ($C_4\times C_{192}$)

Order 193: $C_{193}$ ($C_8\times C_{192}$)

Order 96: $C_{96}$ ($C_{193}:C_{16}$)

Order 48: $C_{48}$ ($C_{386}:C_{16}$)

Order 32: $C_{32}$ ($C_{579}:C_{16}$)

Order 24: $C_{24}$ ($C_{772}:C_{16}$)

Order 16: $C_{16}$ ($C_{579}:(C_2\times C_{16})$)

Order 12: $C_{12}$ ($C_{1544}:C_{16}$)

Order 8: $C_8$ ($C_{579}:(C_4\times C_{16})$)

Order 6: $C_6$ ($C_{3088}.C_{16}$)

Order 4: $C_4$ ($C_{579}:(C_8\times C_{16})$)

Order 3: $C_3$ ($C_{193}:(C_8\times C_{64})$)

Order 2: $C_2$ ($C_{579}:(C_8\times C_{32})$)

Order 1: $C_1$ ($C_{18528}.C_{16}$)

Series

Derived series

$C_{18528}.C_{16}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Chief series

$C_{18528}.C_{16}$

$\rhd$

$C_3\times C_{193}:(C_4\times C_{64})$

$\rhd$

$C_3\times C_{193}:(C_4\times C_{32})$

$\rhd$

$C_3\times C_{193}:(C_2\times C_{32})$

$\rhd$

$C_3\times C_{193}:(C_2\times C_{16})$

$\rhd$

$C_3\times C_{193}:C_{16}$

$\rhd$

$C_3\times C_{193}:C_8$

$\rhd$

$C_{2316}$

$\rhd$

$C_{1158}$

$\rhd$

$C_{579}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Lower central series

$C_{18528}.C_{16}$

$\rhd$

$C_{193}$

Upper central series

$C_1$

$\lhd$

$C_{96}$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

The $2688 \times 2688$ character table is not available for this group.

Rational character table

The $104 \times 104$ rational character table is not available for this group.