Abstract group 1185792.d: $C_{18528}:C_{64}$

• Label: 1185792.d
• Order: \( 2^{11} \cdot 3 \cdot 193 \)
• Exponent: \( 2^{6} \cdot 3 \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{11} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{5} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3 \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 3 \)
• Perm deg.: not computed
• Trans deg.: $18528$
• Rank: $2$

Group information

Description:	$C_{18528}:C_{64}$
Order:	\(1185792\)\(\medspace = 2^{11} \cdot 3 \cdot 193 \)
Exponent:	\(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Automorphism group:	$C_{3088}.C_{24}.C_4^2.C_2^5$, of order \(37945344\)\(\medspace = 2^{16} \cdot 3 \cdot 193 \)
Composition factors:	$C_2$ x 11, $C_3$, $C_{193}$
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	35520	16992	145152	71040	197632	290304	395264	192	192	384	384	384	768	768	1536	1536	3072	3072	6144	1185792
Conjugacy classes	1	3	2	12	6	48	24	192	96	768	384	1024	1536	2048	3	3	6	6	6	12	12	24	24	48	48	96	6432
Divisions	1	3	1	6	3	12	6	24	12	48	24	32	48	32	1	1	1	1	1	1	1	1	1	1	1	1	264

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$18528$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Presentation:	$\langle a, b \mid a^{192}=b^{6176}=1, b^{a}=b^{4257} \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 185 & 0 \\ 0 & 24 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_{32}$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_{193}:C_{64})$
Semidirect product:	$C_{18528}$ $\,\rtimes\,$ $C_{64}$	$C_{6176}$ $\,\rtimes\,$ $C_{192}$	$(C_{193}:C_{96})$ $\,\rtimes\,$ $C_{64}$	$(C_{193}:C_{96})$ $\,\rtimes\,$ $C_{64}$	all 14
Trans. wreath product:	not computed
Non-split product:	$(C_{96}\times D_{193})$ . $C_{32}$	$(D_{193}:C_{96})$ . $C_{32}$	$(D_{193}:C_{96})$ . $C_{32}$	$(D_{193}:C_{96})$ . $C_{32}$	all 144

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

Homology

Abelianization:	$C_{32} \times C_{192} \simeq C_{32} \times C_{64} \times C_{3}$
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 90816 subgroups in 960 conjugacy classes, 492 normal (96 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{96}$	$G/Z \simeq$ $C_{193}:C_{64}$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_{32}\times C_{192}$
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $C_{1158}:C_{64}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{18528}$	$G/\operatorname{Fit} \simeq$ $C_{64}$
Radical:	$R \simeq$ $C_{18528}:C_{64}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{1158}$	$G/\operatorname{soc} \simeq$ $C_{16}\times C_{64}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{32}\times C_{64}$
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

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1185792: $C_{18528}:C_{64}$

Order 592896: $C_{3088}:C_{192}$ x 2, $C_{18528}:C_{32}$

Order 395264: $C_{6176}:C_{64}$

Order 296448: $C_{1544}:C_{192}$ x 4, $C_{3088}:C_{96}$ x 2, $C_{18528}:C_{16}$

Order 197632: $C_{3088}:C_{64}$ x 2, $C_{6176}:C_{32}$

Order 148224: $C_{2316}:C_{64}$ x 8, $C_{1544}:C_{96}$ x 4, $C_{3088}:C_{48}$, $C_{3088}.C_{48}$, $C_{18528}:C_8$

Order 98816: $C_{1544}:C_{64}$ x 4, $C_{3088}:C_{32}$ x 2, $C_{6176}:C_{16}$

Order 74112: $C_{1158}:C_{64}$ x 16, $C_{2316}:C_{32}$ x 8, $C_{1544}.C_{48}$ x 2, $C_{1544}:C_{48}$ x 2, $C_{3088}:C_{24}$, $C_{18528}.C_4$, $C_{3088}.C_{24}$

Order 49408: $C_{772}:C_{64}$ x 8, $C_{1544}:C_{32}$ x 4, $C_{3088}.C_{16}$, $C_{3088}:C_{16}$, $C_{6176}:C_8$

Order 37056: $C_{193}:C_{192}$ x 32, $C_{1158}:C_{32}$ x 16, $C_{2316}:C_{16}$ x 4, $D_{193}:C_{96}$ x 4, $D_{193}:C_{96}$ x 2, $C_{1544}:C_{24}$, $C_{3088}:C_{12}$, $C_{96}\times D_{193}$, $C_{1544}.C_{24}$, $D_{193}:C_{96}$

Order 24704: $C_{386}:C_{64}$ x 16, $C_{772}:C_{32}$ x 8, $C_{1544}.C_{16}$ x 2, $C_{1544}:C_{16}$ x 2, $C_{3088}.C_8$, $C_{3088}:C_8$, $C_{6176}:C_4$

Order 18528: $C_{193}:C_{96}$ x 32, $C_{1158}:C_{16}$ x 8, $C_{193}:C_{96}$ x 8, $C_{193}:C_{96}$ x 4, $C_{2316}:C_8$ x 2, $D_{193}:C_{48}$ x 2, $C_{193}:C_{96}$ x 2, $C_{1544}:C_{12}$, $C_{48}\times D_{193}$, $D_{193}:C_{48}$, $C_{18528}$, $C_{193}:C_{96}$

Order 12352: $C_{193}:C_{64}$ x 32, $C_{386}:C_{32}$ x 16, $D_{193}:C_{32}$ x 4, $C_{772}:C_{16}$ x 4, $D_{193}:C_{32}$ 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: $C_{193}:C_{48}$ x 16, $C_{1158}:C_8$ x 4, $C_{193}:C_{48}$ x 4, $C_{193}:C_{48}$ x 2, $C_{24}\times D_{193}$, $C_{193}:C_{48}$, $C_{2316}:C_4$, $C_{9264}$, $D_{193}:C_{24}$

Order 6176: $C_{193}:C_{32}$ x 32, $C_{386}:C_{16}$ x 8, $C_{193}:C_{32}$ x 8, $C_{193}:C_{32}$ 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 6144: $C_{32}\times C_{192}$

Order 4632: $C_{193}:C_{24}$ x 8, $C_{1158}:C_4$ x 2, $C_{193}:C_{24}$ x 2, $C_{4632}$, $C_{12}\times D_{193}$, $C_{193}:C_{24}$

Order 3088: $C_{193}:C_{16}$ x 16, $C_{386}:C_8$ 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 3072: $C_{16}\times C_{192}$ x 2, $C_{32}\times C_{96}$

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

Order 2048: $C_{32}\times C_{64}$

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: $C_8\times C_{192}$ x 4, $C_{16}\times C_{96}$ x 3

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

Order 1024: $C_{16}\times C_{64}$ x 2, $C_{32}^2$

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

Order 768: $C_4\times C_{192}$ x 8, $C_8\times C_{96}$ x 6, $C_{16}\times C_{48}$

Order 579: $C_{579}$

Order 512: $C_8\times C_{64}$ x 4, $C_{16}\times C_{32}$ x 3

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

Order 384: $C_2\times C_{192}$ x 16, $C_4\times C_{96}$ x 12, $C_8\times C_{48}$ x 3

Order 256: $C_4\times C_{64}$ x 8, $C_8\times C_{32}$ x 6, $C_{16}^2$

Order 193: $C_{193}$

Order 192: $C_{192}$ x 32, $C_2\times C_{96}$ x 24, $C_4\times C_{48}$ x 6, $C_8\times C_{24}$

Order 128: $C_2\times C_{64}$ x 16, $C_4\times C_{32}$ x 12, $C_8\times C_{16}$ x 3

Order 96: $C_{96}$ x 48, $C_2\times C_{48}$ x 12, $C_4\times C_{24}$ x 3

Order 64: $C_{64}$ x 32, $C_2\times C_{32}$ x 24, $C_4\times C_{16}$ x 6, $C_8^2$

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

Order 32: $C_{32}$ x 48, $C_2\times C_{16}$ x 12, $C_4\times C_8$ x 3

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

Order 16: $C_{16}$ x 24, $C_2\times C_8$ 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 1185792: $C_{18528}:C_{64}$

Order 592896: $C_{18528}:C_{32}$, $C_{3088}:C_{192}$

Order 395264: $C_{6176}:C_{64}$

Order 296448: $C_{3088}:C_{96}$ x 2, $C_{18528}:C_{16}$, $C_{1544}:C_{192}$

Order 197632: $C_{6176}:C_{32}$, $C_{3088}:C_{64}$

Order 148224: $C_{1544}:C_{96}$ x 2, $C_{3088}:C_{48}$, $C_{3088}.C_{48}$, $C_{2316}:C_{64}$, $C_{18528}:C_8$

Order 98816: $C_{3088}:C_{32}$ x 2, $C_{6176}:C_{16}$, $C_{1544}:C_{64}$

Order 74112: $C_{2316}:C_{32}$ x 2, $C_{1544}:C_{48}$ x 2, $C_{3088}:C_{24}$, $C_{18528}.C_4$, $C_{1544}.C_{48}$, $C_{1158}:C_{64}$, $C_{3088}.C_{24}$

Order 49408: $C_{1544}:C_{32}$ x 2, $C_{772}:C_{64}$, $C_{3088}.C_{16}$, $C_{3088}:C_{16}$, $C_{6176}:C_8$

Order 37056: $C_{2316}:C_{16}$ x 2, $C_{1158}:C_{32}$ x 2, $C_{1544}:C_{24}$, $C_{3088}:C_{12}$, $C_{96}\times D_{193}$, $D_{193}:C_{96}$, $C_{193}:C_{192}$, $C_{1544}.C_{24}$, $D_{193}:C_{96}$, $D_{193}:C_{96}$

Order 24704: $C_{772}:C_{32}$ x 2, $C_{1544}:C_{16}$ x 2, $C_{386}:C_{64}$, $C_{1544}.C_{16}$, $C_{3088}.C_8$, $C_{3088}:C_8$, $C_{6176}:C_4$

Order 18528: $C_{2316}:C_8$ x 2, $C_{1158}:C_{16}$ x 2, $C_{193}:C_{96}$ x 2, $C_{1544}:C_{12}$, $C_{48}\times D_{193}$, $D_{193}:C_{48}$, $D_{193}:C_{48}$, $C_{18528}$, $C_{193}:C_{96}$, $C_{193}:C_{96}$, $C_{193}:C_{96}$, $C_{193}:C_{96}$

Order 12352: $C_{386}:C_{32}$ x 2, $C_{772}:C_{16}$ x 2, $C_{32}\times D_{193}$, $C_{193}:C_{64}$, $D_{193}:C_{32}$, $D_{193}:C_{32}$, $C_{1544}.C_8$, $C_{1544}:C_8$, $D_{193}:C_{32}$, $C_{3088}:C_4$

Order 9264: $C_{1158}:C_8$ x 2, $C_{193}:C_{48}$ x 2, $C_{24}\times D_{193}$, $C_{193}:C_{48}$, $C_{2316}:C_4$, $C_{9264}$, $C_{193}:C_{48}$, $C_{193}:C_{48}$, $D_{193}:C_{24}$

Order 6176: $C_{193}:C_{32}$ x 2, $C_{386}:C_{16}$ x 2, $C_{772}:C_8$ x 2, $C_{16}\times D_{193}$, $C_{193}:C_{32}$, $C_{193}:C_{32}$, $D_{193}:C_{16}$, $C_{193}:C_{32}$, $D_{193}:C_{16}$, $C_{1544}:C_4$, $C_{193}:C_{32}$, $C_{6176}$

Order 6144: $C_{32}\times C_{192}$

Order 4632: $C_{1158}:C_4$ x 2, $C_{193}:C_{24}$ x 2, $C_{4632}$, $C_{12}\times D_{193}$, $C_{193}:C_{24}$, $C_{193}:C_{24}$

Order 3088: $C_{193}:C_{16}$ x 2, $C_{386}:C_8$ x 2, $C_{193}:C_{16}$, $D_{193}:C_8$, $C_{772}:C_4$, $C_{193}:C_{16}$, $C_8\times D_{193}$, $C_{193}:C_{16}$, $C_{3088}$

Order 3072: $C_{16}\times C_{192}$, $C_{32}\times C_{96}$

Order 2316: $C_{193}:C_{12}$ x 2, $C_3\times D_{386}$, $C_{193}:C_{12}$, $C_{2316}$

Order 2048: $C_{32}\times C_{64}$

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

Order 1536: $C_{16}\times C_{96}$ x 3, $C_8\times C_{192}$

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

Order 1024: $C_{16}\times C_{64}$, $C_{32}^2$

Order 772: $C_{193}:C_4$ x 2, $D_{386}$, $C_{772}$, $C_{193}:C_4$

Order 768: $C_8\times C_{96}$ x 4, $C_4\times C_{192}$, $C_{16}\times C_{48}$

Order 579: $C_{579}$

Order 512: $C_{16}\times C_{32}$ x 3, $C_8\times C_{64}$

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

Order 384: $C_4\times C_{96}$ x 5, $C_8\times C_{48}$ x 3, $C_2\times C_{192}$

Order 256: $C_8\times C_{32}$ x 4, $C_4\times C_{64}$, $C_{16}^2$

Order 193: $C_{193}$

Order 192: $C_2\times C_{96}$ x 6, $C_4\times C_{48}$ x 4, $C_{192}$, $C_8\times C_{24}$

Order 128: $C_4\times C_{32}$ x 5, $C_8\times C_{16}$ x 3, $C_2\times C_{64}$

Order 96: $C_{96}$ x 7, $C_2\times C_{48}$ x 5, $C_4\times C_{24}$ x 3

Order 64: $C_2\times C_{32}$ x 6, $C_4\times C_{16}$ x 4, $C_8^2$, $C_{64}$

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

Order 32: $C_{32}$ x 7, $C_2\times C_{16}$ x 5, $C_4\times C_8$ x 3

Order 24: $C_{24}$ x 5, $C_2\times C_{12}$ x 3

Order 16: $C_{16}$ x 6, $C_2\times C_8$ x 4, $C_4^2$

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

Order 8: $C_8$ x 5, $C_2\times C_4$ x 3

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1185792: $C_{18528}:C_{64}$ ($C_1$)

Order 592896: $C_{3088}:C_{192}$ ($C_2$) x 2, $C_{18528}:C_{32}$ ($C_2$)

Order 395264: $C_{6176}:C_{64}$ ($C_3$)

Order 296448: $C_{1544}:C_{192}$ ($C_4$) x 4, $C_{18528}:C_{16}$ ($C_4$), $C_{3088}:C_{96}$ ($C_2^2$), $C_{3088}:C_{96}$ ($C_4$)

Order 197632: $C_{3088}:C_{64}$ ($C_6$) x 2, $C_{6176}:C_{32}$ ($C_6$)

Order 148224: $C_{2316}:C_{64}$ ($C_8$) x 8, $C_{1544}:C_{96}$ ($C_2\times C_4$) x 2, $C_{1544}:C_{96}$ ($C_8$) x 2, $C_{3088}:C_{48}$ ($C_2\times C_4$), $C_{3088}.C_{48}$ ($C_8$), $C_{18528}:C_8$ ($C_8$)

Order 98816: $C_{1544}:C_{64}$ ($C_{12}$) x 4, $C_{6176}:C_{16}$ ($C_{12}$), $C_{3088}:C_{32}$ ($C_2\times C_6$), $C_{3088}:C_{32}$ ($C_{12}$)

Order 74112: $C_{1158}:C_{64}$ ($C_{16}$) x 16, $C_{2316}:C_{32}$ ($C_2\times C_8$) x 4, $C_{2316}:C_{32}$ ($C_{16}$) x 4, $C_{1544}.C_{48}$ ($C_{16}$) x 2, $C_{3088}:C_{24}$ ($C_2\times C_8$), $C_{18528}.C_4$ ($C_{16}$), $C_{1544}:C_{48}$ ($C_2\times C_8$), $C_{1544}:C_{48}$ ($C_4^2$), $C_{3088}.C_{24}$ ($C_{16}$)

Order 49408: $C_{772}:C_{64}$ ($C_{24}$) x 8, $C_{1544}:C_{32}$ ($C_2\times C_{12}$) x 2, $C_{1544}:C_{32}$ ($C_{24}$) x 2, $C_{3088}.C_{16}$ ($C_{24}$), $C_{3088}:C_{16}$ ($C_2\times C_{12}$), $C_{6176}:C_8$ ($C_{24}$)

Order 37056: $C_{193}:C_{192}$ ($C_{32}$) x 32, $C_{1158}:C_{32}$ ($C_2\times C_{16}$) x 8, $C_{1158}:C_{32}$ ($C_{32}$) x 8, $D_{193}:C_{96}$ ($C_{32}$) x 4, $C_{2316}:C_{16}$ ($C_4\times C_8$) x 2, $C_{2316}:C_{16}$ ($C_2\times C_{16}$) x 2, $D_{193}:C_{96}$ ($C_{32}$) x 2, $C_{1544}:C_{24}$ ($C_4\times C_8$), $C_{3088}:C_{12}$ ($C_2\times C_{16}$), $C_{96}\times D_{193}$ ($C_{32}$), $C_{1544}.C_{24}$ ($C_2\times C_{16}$), $D_{193}:C_{96}$ ($C_{32}$)

Order 24704: $C_{386}:C_{64}$ ($C_{48}$) x 16, $C_{772}:C_{32}$ ($C_2\times C_{24}$) x 4, $C_{772}:C_{32}$ ($C_{48}$) x 4, $C_{1544}.C_{16}$ ($C_{48}$) x 2, $C_{1544}:C_{16}$ ($C_2\times C_{24}$), $C_{1544}:C_{16}$ ($C_4\times C_{12}$), $C_{3088}.C_8$ ($C_{48}$), $C_{3088}:C_8$ ($C_2\times C_{24}$), $C_{6176}:C_4$ ($C_{48}$)

Order 18528: $C_{193}:C_{96}$ ($C_2\times C_{32}$) x 16, $C_{193}:C_{96}$ ($C_{64}$) x 16, $C_{193}:C_{96}$ ($C_{64}$) x 8, $C_{1158}:C_{16}$ ($C_2\times C_{32}$) x 4, $C_{1158}:C_{16}$ ($C_4\times C_{16}$) x 4, $C_{193}:C_{96}$ ($C_{64}$) x 4, $D_{193}:C_{48}$ ($C_2\times C_{32}$) x 2, $C_{193}:C_{96}$ ($C_{64}$) x 2, $C_{1544}:C_{12}$ ($C_4\times C_{16}$), $C_{48}\times D_{193}$ ($C_2\times C_{32}$), $C_{2316}:C_8$ ($C_4\times C_{16}$), $C_{2316}:C_8$ ($C_8^2$), $D_{193}:C_{48}$ ($C_2\times C_{32}$), $C_{18528}$ ($C_{64}$), $C_{193}:C_{96}$ ($C_{64}$)

Order 12352: $C_{193}:C_{64}$ ($C_{96}$) x 32, $C_{386}:C_{32}$ ($C_2\times C_{48}$) x 8, $C_{386}:C_{32}$ ($C_{96}$) x 8, $D_{193}:C_{32}$ ($C_{96}$) x 4, $C_{772}:C_{16}$ ($C_2\times C_{48}$) x 2, $C_{772}:C_{16}$ ($C_4\times C_{24}$) x 2, $D_{193}:C_{32}$ ($C_{96}$) x 2, $C_{32}\times D_{193}$ ($C_{96}$), $C_{1544}.C_8$ ($C_2\times C_{48}$), $C_{1544}:C_8$ ($C_4\times C_{24}$), $D_{193}:C_{32}$ ($C_{96}$), $C_{3088}:C_4$ ($C_2\times C_{48}$)

Order 9264: $C_{193}:C_{48}$ ($C_2\times C_{64}$) x 8, $C_{193}:C_{48}$ ($C_4\times C_{32}$) x 8, $C_{193}:C_{48}$ ($C_2\times C_{64}$) x 4, $C_{1158}:C_8$ ($C_8\times C_{16}$) x 2, $C_{1158}:C_8$ ($C_4\times C_{32}$) x 2, $C_{193}:C_{48}$ ($C_2\times C_{64}$) x 2, $C_{24}\times D_{193}$ ($C_4\times C_{32}$), $C_{193}:C_{48}$ ($C_2\times C_{64}$), $C_{2316}:C_4$ ($C_8\times C_{16}$), $C_{9264}$ ($C_2\times C_{64}$), $D_{193}:C_{24}$ ($C_4\times C_{32}$)

Order 6176: $C_{193}:C_{32}$ ($C_{192}$) x 16, $C_{193}:C_{32}$ ($C_2\times C_{96}$) x 16, $C_{193}:C_{32}$ ($C_{192}$) x 8, $C_{386}:C_{16}$ ($C_2\times C_{96}$) x 4, $C_{386}:C_{16}$ ($C_4\times C_{48}$) x 4, $C_{193}:C_{32}$ ($C_{192}$) x 4, $D_{193}:C_{16}$ ($C_2\times C_{96}$) x 2, $C_{193}:C_{32}$ ($C_{192}$) x 2, $C_{16}\times D_{193}$ ($C_2\times C_{96}$), $C_{193}:C_{32}$ ($C_{192}$), $C_{772}:C_8$ ($C_4\times C_{48}$), $C_{772}:C_8$ ($C_8\times C_{24}$), $D_{193}:C_{16}$ ($C_2\times C_{96}$), $C_{1544}:C_4$ ($C_4\times C_{48}$), $C_{6176}$ ($C_{192}$)

Order 4632: $C_{193}:C_{24}$ ($C_4\times C_{64}$) x 4, $C_{193}:C_{24}$ ($C_8\times C_{32}$) x 4, $C_{193}:C_{24}$ ($C_4\times C_{64}$) x 2, $C_{4632}$ ($C_4\times C_{64}$), $C_{12}\times D_{193}$ ($C_8\times C_{32}$), $C_{1158}:C_4$ ($C_{16}^2$), $C_{1158}:C_4$ ($C_8\times C_{32}$), $C_{193}:C_{24}$ ($C_4\times C_{64}$)

Order 3088: $C_{193}:C_{16}$ ($C_2\times C_{192}$) x 8, $C_{193}:C_{16}$ ($C_4\times C_{96}$) x 8, $C_{193}:C_{16}$ ($C_2\times C_{192}$) x 4, $C_{386}:C_8$ ($C_4\times C_{96}$) x 2, $C_{386}:C_8$ ($C_8\times C_{48}$) x 2, $C_{193}:C_{16}$ ($C_2\times C_{192}$) x 2, $D_{193}:C_8$ ($C_4\times C_{96}$), $C_{772}:C_4$ ($C_8\times C_{48}$), $C_8\times D_{193}$ ($C_4\times C_{96}$), $C_{193}:C_{16}$ ($C_2\times C_{192}$), $C_{3088}$ ($C_2\times C_{192}$)

Order 2316: $C_{193}:C_{12}$ ($C_{16}\times C_{32}$) x 2, $C_{193}:C_{12}$ ($C_8\times C_{64}$) x 2, $C_3\times D_{386}$ ($C_{16}\times C_{32}$), $C_{193}:C_{12}$ ($C_8\times C_{64}$), $C_{2316}$ ($C_8\times C_{64}$)

Order 1544: $C_{193}:C_8$ ($C_4\times C_{192}$) x 4, $C_{193}:C_8$ ($C_8\times C_{96}$) x 4, $C_{193}:C_8$ ($C_4\times C_{192}$) x 2, $C_4\times D_{193}$ ($C_8\times C_{96}$), $C_{193}:C_8$ ($C_4\times C_{192}$), $C_{386}:C_4$ ($C_{16}\times C_{48}$), $C_{386}:C_4$ ($C_8\times C_{96}$), $C_{1544}$ ($C_4\times C_{192}$)

Order 1158: $C_{1158}$ ($C_{16}\times C_{64}$), $C_3\times D_{193}$ ($C_{16}\times C_{64}$), $C_3\times D_{193}$ ($C_{32}^2$)

Order 772: $C_{193}:C_4$ ($C_{16}\times C_{96}$) x 2, $C_{193}:C_4$ ($C_8\times C_{192}$) x 2, $D_{386}$ ($C_{16}\times C_{96}$), $C_{772}$ ($C_8\times C_{192}$), $C_{193}:C_4$ ($C_8\times C_{192}$)

Order 579: $C_{579}$ ($C_{32}\times C_{64}$)

Order 386: $C_{386}$ ($C_{16}\times C_{192}$), $D_{193}$ ($C_{16}\times C_{192}$), $D_{193}$ ($C_{32}\times C_{96}$)

Order 193: $C_{193}$ ($C_{32}\times C_{192}$)

Order 96: $C_{96}$ ($C_{193}:C_{64}$)

Order 48: $C_{48}$ ($C_{386}:C_{64}$)

Order 32: $C_{32}$ ($C_{193}:C_{192}$)

Order 24: $C_{24}$ ($C_{772}:C_{64}$)

Order 16: $C_{16}$ ($C_{1158}:C_{64}$)

Order 12: $C_{12}$ ($C_{1544}:C_{64}$)

Order 8: $C_8$ ($C_{2316}:C_{64}$)

Order 6: $C_6$ ($C_{3088}:C_{64}$)

Order 4: $C_4$ ($C_{1544}:C_{192}$)

Order 3: $C_3$ ($C_{6176}:C_{64}$)

Order 2: $C_2$ ($C_{3088}:C_{192}$)

Order 1: $C_1$ ($C_{18528}:C_{64}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1185792: $C_{18528}:C_{64}$ ($C_1$)

Order 592896: $C_{18528}:C_{32}$ ($C_2$), $C_{3088}:C_{192}$ ($C_2$)

Order 395264: $C_{6176}:C_{64}$ ($C_3$)

Order 296448: $C_{18528}:C_{16}$ ($C_4$), $C_{3088}:C_{96}$ ($C_2^2$), $C_{3088}:C_{96}$ ($C_4$), $C_{1544}:C_{192}$ ($C_4$)

Order 197632: $C_{6176}:C_{32}$ ($C_6$), $C_{3088}:C_{64}$ ($C_6$)

Order 148224: $C_{3088}:C_{48}$ ($C_2\times C_4$), $C_{3088}.C_{48}$ ($C_8$), $C_{2316}:C_{64}$ ($C_8$), $C_{18528}:C_8$ ($C_8$), $C_{1544}:C_{96}$ ($C_2\times C_4$), $C_{1544}:C_{96}$ ($C_8$)

Order 98816: $C_{6176}:C_{16}$ ($C_{12}$), $C_{3088}:C_{32}$ ($C_2\times C_6$), $C_{3088}:C_{32}$ ($C_{12}$), $C_{1544}:C_{64}$ ($C_{12}$)

Order 74112: $C_{3088}:C_{24}$ ($C_2\times C_8$), $C_{18528}.C_4$ ($C_{16}$), $C_{1544}.C_{48}$ ($C_{16}$), $C_{2316}:C_{32}$ ($C_2\times C_8$), $C_{2316}:C_{32}$ ($C_{16}$), $C_{1158}:C_{64}$ ($C_{16}$), $C_{1544}:C_{48}$ ($C_2\times C_8$), $C_{1544}:C_{48}$ ($C_4^2$), $C_{3088}.C_{24}$ ($C_{16}$)

Order 49408: $C_{772}:C_{64}$ ($C_{24}$), $C_{1544}:C_{32}$ ($C_2\times C_{12}$), $C_{1544}:C_{32}$ ($C_{24}$), $C_{3088}.C_{16}$ ($C_{24}$), $C_{3088}:C_{16}$ ($C_2\times C_{12}$), $C_{6176}:C_8$ ($C_{24}$)

Order 37056: $C_{1544}:C_{24}$ ($C_4\times C_8$), $C_{3088}:C_{12}$ ($C_2\times C_{16}$), $C_{2316}:C_{16}$ ($C_4\times C_8$), $C_{2316}:C_{16}$ ($C_2\times C_{16}$), $C_{96}\times D_{193}$ ($C_{32}$), $C_{1158}:C_{32}$ ($C_2\times C_{16}$), $C_{1158}:C_{32}$ ($C_{32}$), $D_{193}:C_{96}$ ($C_{32}$), $C_{193}:C_{192}$ ($C_{32}$), $C_{1544}.C_{24}$ ($C_2\times C_{16}$), $D_{193}:C_{96}$ ($C_{32}$), $D_{193}:C_{96}$ ($C_{32}$)

Order 24704: $C_{386}:C_{64}$ ($C_{48}$), $C_{772}:C_{32}$ ($C_2\times C_{24}$), $C_{772}:C_{32}$ ($C_{48}$), $C_{1544}.C_{16}$ ($C_{48}$), $C_{1544}:C_{16}$ ($C_2\times C_{24}$), $C_{1544}:C_{16}$ ($C_4\times C_{12}$), $C_{3088}.C_8$ ($C_{48}$), $C_{3088}:C_8$ ($C_2\times C_{24}$), $C_{6176}:C_4$ ($C_{48}$)

Order 18528: $C_{1544}:C_{12}$ ($C_4\times C_{16}$), $C_{48}\times D_{193}$ ($C_2\times C_{32}$), $C_{2316}:C_8$ ($C_4\times C_{16}$), $C_{2316}:C_8$ ($C_8^2$), $D_{193}:C_{48}$ ($C_2\times C_{32}$), $D_{193}:C_{48}$ ($C_2\times C_{32}$), $C_{18528}$ ($C_{64}$), $C_{193}:C_{96}$ ($C_{64}$), $C_{1158}:C_{16}$ ($C_2\times C_{32}$), $C_{1158}:C_{16}$ ($C_4\times C_{16}$), $C_{193}:C_{96}$ ($C_{64}$), $C_{193}:C_{96}$ ($C_{64}$), $C_{193}:C_{96}$ ($C_{64}$), $C_{193}:C_{96}$ ($C_2\times C_{32}$), $C_{193}:C_{96}$ ($C_{64}$)

Order 12352: $C_{32}\times D_{193}$ ($C_{96}$), $C_{193}:C_{64}$ ($C_{96}$), $C_{386}:C_{32}$ ($C_2\times C_{48}$), $C_{386}:C_{32}$ ($C_{96}$), $D_{193}:C_{32}$ ($C_{96}$), $C_{772}:C_{16}$ ($C_2\times C_{48}$), $C_{772}:C_{16}$ ($C_4\times C_{24}$), $D_{193}:C_{32}$ ($C_{96}$), $C_{1544}.C_8$ ($C_2\times C_{48}$), $C_{1544}:C_8$ ($C_4\times C_{24}$), $D_{193}:C_{32}$ ($C_{96}$), $C_{3088}:C_4$ ($C_2\times C_{48}$)

Order 9264: $C_{24}\times D_{193}$ ($C_4\times C_{32}$), $C_{193}:C_{48}$ ($C_2\times C_{64}$), $C_{2316}:C_4$ ($C_8\times C_{16}$), $C_{9264}$ ($C_2\times C_{64}$), $C_{1158}:C_8$ ($C_8\times C_{16}$), $C_{1158}:C_8$ ($C_4\times C_{32}$), $C_{193}:C_{48}$ ($C_2\times C_{64}$), $C_{193}:C_{48}$ ($C_2\times C_{64}$), $D_{193}:C_{24}$ ($C_4\times C_{32}$), $C_{193}:C_{48}$ ($C_2\times C_{64}$), $C_{193}:C_{48}$ ($C_4\times C_{32}$)

Order 6176: $C_{16}\times D_{193}$ ($C_2\times C_{96}$), $C_{193}:C_{32}$ ($C_{192}$), $C_{193}:C_{32}$ ($C_{192}$), $C_{193}:C_{32}$ ($C_2\times C_{96}$), $C_{386}:C_{16}$ ($C_2\times C_{96}$), $C_{386}:C_{16}$ ($C_4\times C_{48}$), $C_{193}:C_{32}$ ($C_{192}$), $D_{193}:C_{16}$ ($C_2\times C_{96}$), $C_{772}:C_8$ ($C_4\times C_{48}$), $C_{772}:C_8$ ($C_8\times C_{24}$), $C_{193}:C_{32}$ ($C_{192}$), $D_{193}:C_{16}$ ($C_2\times C_{96}$), $C_{1544}:C_4$ ($C_4\times C_{48}$), $C_{193}:C_{32}$ ($C_{192}$), $C_{6176}$ ($C_{192}$)

Order 4632: $C_{4632}$ ($C_4\times C_{64}$), $C_{12}\times D_{193}$ ($C_8\times C_{32}$), $C_{1158}:C_4$ ($C_{16}^2$), $C_{1158}:C_4$ ($C_8\times C_{32}$), $C_{193}:C_{24}$ ($C_4\times C_{64}$), $C_{193}:C_{24}$ ($C_4\times C_{64}$), $C_{193}:C_{24}$ ($C_8\times C_{32}$), $C_{193}:C_{24}$ ($C_4\times C_{64}$)

Order 3088: $C_{193}:C_{16}$ ($C_2\times C_{192}$), $C_{193}:C_{16}$ ($C_4\times C_{96}$), $C_{386}:C_8$ ($C_4\times C_{96}$), $C_{386}:C_8$ ($C_8\times C_{48}$), $C_{193}:C_{16}$ ($C_2\times C_{192}$), $D_{193}:C_8$ ($C_4\times C_{96}$), $C_{772}:C_4$ ($C_8\times C_{48}$), $C_{193}:C_{16}$ ($C_2\times C_{192}$), $C_8\times D_{193}$ ($C_4\times C_{96}$), $C_{193}:C_{16}$ ($C_2\times C_{192}$), $C_{3088}$ ($C_2\times C_{192}$)

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

Order 1544: $C_4\times D_{193}$ ($C_8\times C_{96}$), $C_{193}:C_8$ ($C_4\times C_{192}$), $C_{193}:C_8$ ($C_4\times C_{192}$), $C_{193}:C_8$ ($C_8\times C_{96}$), $C_{386}:C_4$ ($C_{16}\times C_{48}$), $C_{386}:C_4$ ($C_8\times C_{96}$), $C_{193}:C_8$ ($C_4\times C_{192}$), $C_{1544}$ ($C_4\times C_{192}$)

Order 1158: $C_{1158}$ ($C_{16}\times C_{64}$), $C_3\times D_{193}$ ($C_{16}\times C_{64}$), $C_3\times D_{193}$ ($C_{32}^2$)

Order 772: $D_{386}$ ($C_{16}\times C_{96}$), $C_{193}:C_4$ ($C_{16}\times C_{96}$), $C_{193}:C_4$ ($C_8\times C_{192}$), $C_{772}$ ($C_8\times C_{192}$), $C_{193}:C_4$ ($C_8\times C_{192}$)

Order 579: $C_{579}$ ($C_{32}\times C_{64}$)

Order 386: $C_{386}$ ($C_{16}\times C_{192}$), $D_{193}$ ($C_{16}\times C_{192}$), $D_{193}$ ($C_{32}\times C_{96}$)

Order 193: $C_{193}$ ($C_{32}\times C_{192}$)

Order 96: $C_{96}$ ($C_{193}:C_{64}$)

Order 48: $C_{48}$ ($C_{386}:C_{64}$)

Order 32: $C_{32}$ ($C_{193}:C_{192}$)

Order 24: $C_{24}$ ($C_{772}:C_{64}$)

Order 16: $C_{16}$ ($C_{1158}:C_{64}$)

Order 12: $C_{12}$ ($C_{1544}:C_{64}$)

Order 8: $C_8$ ($C_{2316}:C_{64}$)

Order 6: $C_6$ ($C_{3088}:C_{64}$)

Order 4: $C_4$ ($C_{1544}:C_{192}$)

Order 3: $C_3$ ($C_{6176}:C_{64}$)

Order 2: $C_2$ ($C_{3088}:C_{192}$)

Order 1: $C_1$ ($C_{18528}:C_{64}$)

Series

Derived series

$C_{18528}:C_{64}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Chief series

$C_{18528}:C_{64}$

$\rhd$

$C_{18528}:C_{32}$

$\rhd$

$C_{18528}:C_{16}$

$\rhd$

$C_{18528}:C_8$

$\rhd$

$C_{18528}.C_4$

$\rhd$

$C_{96}\times D_{193}$

$\rhd$

$C_{18528}$

$\rhd$

$C_{6176}$

$\rhd$

$C_{3088}$

$\rhd$

$C_{1544}$

$\rhd$

$C_{772}$

$\rhd$

$C_{386}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Lower central series

$C_{18528}:C_{64}$

$\rhd$

$C_{193}$

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 1 larger groups in the database.

Character theory

Complex character table

The $6432 \times 6432$ character table is not available for this group.

Rational character table

The $264 \times 264$ rational character table is not available for this group.