Abstract group 148224.j: $C_{6176}:C_{24}$

• Label: 148224.j
• Order: \( 2^{8} \cdot 3 \cdot 193 \)
• Exponent: \( 2^{5} \cdot 3 \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{8} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{5} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3 \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $225$
• Trans deg.: $6176$
• Rank: $2$

Group information

Description:	$C_{6176}:C_{24}$
Order:	\(148224\)\(\medspace = 2^{8} \cdot 3 \cdot 193 \)
Exponent:	\(18528\)\(\medspace = 2^{5} \cdot 3 \cdot 193 \)
Automorphism group:	$C_{772}.C_{24}.C_8^2.C_2^2$, of order \(4743168\)\(\medspace = 2^{13} \cdot 3 \cdot 193 \)
Composition factors:	$C_2$ x 8, $C_3$, $C_{193}$
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	96	193	386	772	1544	3088	6176
Elements	1	387	386	1932	1158	8496	4632	10816	18528	21632	24704	49408	192	192	384	768	1536	3072	148224
Conjugacy classes	1	3	2	12	6	48	24	64	96	128	128	256	8	8	16	32	64	128	1024
Divisions	1	3	1	6	3	12	6	8	12	8	8	8	1	1	1	1	1	1	82
Autjugacy classes	1	2	2	4	4	8	8	8	16	8	16	16	1	1	1	1	1	1	99

Minimal presentations

Permutation degree:	$225$
Transitive degree:	$6176$
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:	$\langle a, b \mid b^{6176}=1, a^{24}=b^{4632}, b^{a}=b^{4577} \rangle$
Permutation group:	Degree $225$ $\langle(1,2,4,14,54,138,81,42,22,5,19,72,21,78,121,135,50,60,15,57,142,85,177,128,59,63,140,55,139,148,153,67,125,94,183,179,87,178,133,191,184,95,29,7,28,91,86,103,33,8,32,101,164,175,83,23,82,173,169,137,166,114,41,10,39,112,64,16,62,96,131,171,79,170,152,80,172,193,163,157,77,20,76,113,40,34,17,66,149,109,73,158,174,187,127,47,126,70,18,69,154,156,188,108,75,162,124,46,53,25,61,98,30,97,185,189,151,167,105,155,159,104,181,92,130,120,44,118,89,26,6,24,84,176,102,186,150,111,38,110,190,161,74,52,13,51,136,56,132,49,12,48,129,147,144,93,182,115,165,145,134,68,58,146,160,141,122,192,119,116,71,143,88,168,117,43,100,31,99,65,107,36,106,37,9,35,27,90,180,123,45,11,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 64 & 0 \\ 0 & 190 \end{array}\right), \left(\begin{array}{rr} 18 & 0 \\ 0 & 24 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_{32}$ $\, \times\, $ $(C_{193}:C_{24})$
Semidirect product:	$C_{6176}$ $\,\rtimes\,$ $C_{24}$	$(C_{6176}:C_8)$ $\,\rtimes\,$ $C_3$	$(C_{193}:C_8)$ $\,\rtimes\,$ $C_{96}$ (8)	$C_{193}$ $\,\rtimes\,$ $(C_8\times C_{96})$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{386}$ . $(C_4\times C_{48})$	$D_{193}$ . $(C_4\times C_{96})$ (2)	$(C_{3088}:C_8)$ . $C_6$	$(C_{6176}:C_4)$ . $C_6$	all 53
Aut. group:	$\Aut(C_2^5.C_2^8:C_5)$

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

Homology

Abelianization:	$C_{8} \times C_{96} \simeq C_{8} \times C_{32} \times C_{3}$
Schur multiplier:	$C_{8}$
Commutator length:	$1$

Subgroups

There are 24844 subgroups in 268 conjugacy classes, 140 normal (60 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{32}$	$G/Z \simeq$ $C_{193}:C_{24}$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_8\times C_{96}$
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $C_{193}:(C_2\times C_{24})$
Fitting:	$\operatorname{Fit} \simeq$ $C_{6176}$	$G/\operatorname{Fit} \simeq$ $C_{24}$
Radical:	$R \simeq$ $C_{6176}:C_{24}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{386}$	$G/\operatorname{soc} \simeq$ $C_8\times C_{48}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\times C_{32}$
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 148224: $C_{6176}:C_{24}$

Order 74112: $C_{772}.(C_4\times C_{24})$, $C_{16}\times C_{193}:C_{24}$, $C_{772}.(C_4\times C_{24})$

Order 49408: $C_{6176}:C_8$

Order 37056: $C_{772}.C_{24}.C_2$ x 2, $C_{772}.(C_4\times C_{12})$, $C_8\times C_{193}:C_{24}$, $C_{3088}.C_6.C_2$, $C_{772}.C_{24}.C_2$, $C_{386}.(C_4\times C_{24})$

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

Order 18528: $C_{193}:C_{96}$ x 4, $C_{193}:C_{96}$ x 2, $C_4\times C_{193}:C_{24}$ x 2, $C_{193}:(C_2\times C_{48})$ x 2, $C_{193}:C_{96}$, $C_{193}:(C_2\times C_{48})$, $C_{32}\times C_{193}:C_3$, $C_{193}:(C_2\times C_{48})$, $C_8\times C_{193}:C_{12}$

Order 12352: $D_{193}:C_{32}$ 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 4, $C_2\times C_{193}:C_{24}$ x 4, $C_{193}:C_{48}$ x 2, $C_{193}:(C_2\times C_{24})$, $C_{193}:(C_2\times C_{24})$, $C_{193}:C_{48}$, $C_{16}\times C_{193}:C_3$, $C_4\times C_{193}:C_{12}$

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: $C_{193}:C_{24}$ x 8, $C_{193}:C_{24}$ x 2, $C_2\times C_{193}:C_{12}$ x 2, $C_8\times C_{193}:C_3$, $C_{193}:C_{24}$, $C_4\times C_{193}:C_6$

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, $C_{193}:C_{12}$, $C_{386}:C_6$, $C_{193}:C_{12}$

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 1158: $C_{193}:C_6$ x 2, $C_{193}:C_6$

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

Order 768: $C_8\times C_{96}$

Order 579: $C_{193}:C_3$

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

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

Order 256: $C_8\times C_{32}$

Order 193: $C_{193}$

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

Order 128: $C_4\times C_{32}$ 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: $C_2\times C_{32}$ 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 148224: $C_{6176}:C_{24}$

Order 74112: $C_{772}.(C_4\times C_{24})$, $C_{16}\times C_{193}:C_{24}$, $C_{772}.(C_4\times C_{24})$

Order 49408: $C_{6176}:C_8$

Order 37056: $C_{772}.(C_4\times C_{12})$, $C_{772}.C_{24}.C_2$, $C_8\times C_{193}:C_{24}$, $C_{3088}.C_6.C_2$, $C_{772}.C_{24}.C_2$, $C_{386}.(C_4\times C_{24})$

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

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

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

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

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

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: $C_{193}:C_{12}$, $C_{386}:C_6$, $C_{193}:C_{12}$, $C_{193}:C_{12}$

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 1158: $C_{193}:C_6$, $C_{193}:C_6$

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

Order 768: $C_8\times C_{96}$

Order 579: $C_{193}:C_3$

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

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

Order 256: $C_8\times C_{32}$

Order 193: $C_{193}$

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

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

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, $C_8^2$

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 148224: $C_{6176}:C_{24}$ ($C_1$)

Order 74112: $C_{772}.(C_4\times C_{24})$ ($C_2$), $C_{16}\times C_{193}:C_{24}$ ($C_2$), $C_{772}.(C_4\times C_{24})$ ($C_2$)

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

Order 37056: $C_{772}.C_{24}.C_2$ ($C_4$) x 2, $C_{772}.(C_4\times C_{12})$ ($C_2^2$), $C_8\times C_{193}:C_{24}$ ($C_4$), $C_{3088}.C_6.C_2$ ($C_4$), $C_{772}.C_{24}.C_2$ ($C_4$), $C_{386}.(C_4\times C_{24})$ ($C_4$)

Order 24704: $C_{3088}.C_8$ ($C_6$), $C_{3088}:C_8$ ($C_6$), $C_{6176}:C_4$ ($C_6$)

Order 18528: $C_{193}:C_{96}$ ($C_8$) x 4, $C_{193}:C_{96}$ ($C_8$) x 2, $C_4\times C_{193}:C_{24}$ ($C_8$) x 2, $C_{193}:(C_2\times C_{48})$ ($C_8$) x 2, $C_{193}:C_{96}$ ($C_8$), $C_{193}:(C_2\times C_{48})$ ($C_2\times C_4$), $C_{32}\times C_{193}:C_3$ ($C_8$), $C_{193}:(C_2\times C_{48})$ ($C_2\times C_4$), $C_8\times C_{193}:C_{12}$ ($C_2\times C_4$)

Order 12352: $D_{193}:C_{32}$ ($C_{12}$) x 2, $C_{32}\times D_{193}$ ($C_{12}$), $C_{1544}.C_8$ ($C_{12}$), $C_{1544}:C_8$ ($C_{12}$), $D_{193}:C_{32}$ ($C_{12}$), $C_{3088}:C_4$ ($C_2\times C_6$)

Order 9264: $C_{193}:C_{48}$ ($C_{16}$) x 4, $C_2\times C_{193}:C_{24}$ ($C_{16}$) x 4, $C_{193}:C_{48}$ ($C_2\times C_8$) x 2, $C_{193}:(C_2\times C_{24})$ ($C_2\times C_8$), $C_{193}:(C_2\times C_{24})$ ($C_4^2$), $C_{193}:C_{48}$ ($C_2\times C_8$), $C_{16}\times C_{193}:C_3$ ($C_2\times C_8$), $C_4\times C_{193}:C_{12}$ ($C_2\times C_8$)

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

Order 4632: $C_{193}:C_{24}$ ($C_{32}$) x 8, $C_{193}:C_{24}$ ($C_2\times C_{16}$) x 2, $C_2\times C_{193}:C_{12}$ ($C_2\times C_{16}$) x 2, $C_8\times C_{193}:C_3$ ($C_4\times C_8$), $C_{193}:C_{24}$ ($C_4\times C_8$), $C_4\times C_{193}:C_6$ ($C_4\times C_8$)

Order 3088: $C_{386}:C_8$ ($C_{48}$) x 4, $C_{193}:C_{16}$ ($C_{48}$) x 4, $C_{193}:C_{16}$ ($C_2\times C_{24}$) x 2, $D_{193}:C_8$ ($C_2\times C_{24}$), $C_{772}:C_4$ ($C_2\times C_{24}$), $C_8\times D_{193}$ ($C_4\times C_{12}$), $C_{193}:C_{16}$ ($C_2\times C_{24}$), $C_{3088}$ ($C_2\times C_{24}$)

Order 2316: $C_{193}:C_{12}$ ($C_2\times C_{32}$) x 4, $C_{193}:C_{12}$ ($C_4\times C_{16}$), $C_{386}:C_6$ ($C_4\times C_{16}$), $C_{193}:C_{12}$ ($C_8^2$)

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

Order 1158: $C_{193}:C_6$ ($C_4\times C_{32}$) x 2, $C_{193}:C_6$ ($C_8\times C_{16}$)

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

Order 579: $C_{193}:C_3$ ($C_8\times C_{32}$)

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

Order 193: $C_{193}$ ($C_8\times C_{96}$)

Order 32: $C_{32}$ ($C_{193}:C_{24}$)

Order 16: $C_{16}$ ($C_{193}:(C_2\times C_{24})$)

Order 8: $C_8$ ($C_{193}:(C_4\times C_{24})$)

Order 4: $C_4$ ($C_{1544}.C_{24}$)

Order 2: $C_2$ ($C_{3088}.C_{24}$)

Order 1: $C_1$ ($C_{6176}:C_{24}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 148224: $C_{6176}:C_{24}$ ($C_1$)

Order 74112: $C_{772}.(C_4\times C_{24})$ ($C_2$), $C_{16}\times C_{193}:C_{24}$ ($C_2$), $C_{772}.(C_4\times C_{24})$ ($C_2$)

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

Order 37056: $C_{772}.(C_4\times C_{12})$ ($C_2^2$), $C_{772}.C_{24}.C_2$ ($C_4$), $C_8\times C_{193}:C_{24}$ ($C_4$), $C_{3088}.C_6.C_2$ ($C_4$), $C_{772}.C_{24}.C_2$ ($C_4$), $C_{386}.(C_4\times C_{24})$ ($C_4$)

Order 24704: $C_{3088}.C_8$ ($C_6$), $C_{3088}:C_8$ ($C_6$), $C_{6176}:C_4$ ($C_6$)

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

Order 12352: $C_{32}\times D_{193}$ ($C_{12}$), $D_{193}:C_{32}$ ($C_{12}$), $C_{1544}.C_8$ ($C_{12}$), $C_{1544}:C_8$ ($C_{12}$), $D_{193}:C_{32}$ ($C_{12}$), $C_{3088}:C_4$ ($C_2\times C_6$)

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

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

Order 4632: $C_8\times C_{193}:C_3$ ($C_4\times C_8$), $C_{193}:C_{24}$ ($C_4\times C_8$), $C_{193}:C_{24}$ ($C_2\times C_{16}$), $C_{193}:C_{24}$ ($C_{32}$), $C_4\times C_{193}:C_6$ ($C_4\times C_8$), $C_2\times C_{193}:C_{12}$ ($C_2\times C_{16}$)

Order 3088: $C_{386}:C_8$ ($C_{48}$), $C_{193}:C_{16}$ ($C_{48}$), $D_{193}:C_8$ ($C_2\times C_{24}$), $C_{772}:C_4$ ($C_2\times C_{24}$), $C_{193}:C_{16}$ ($C_2\times C_{24}$), $C_8\times D_{193}$ ($C_4\times C_{12}$), $C_{193}:C_{16}$ ($C_2\times C_{24}$), $C_{3088}$ ($C_2\times C_{24}$)

Order 2316: $C_{193}:C_{12}$ ($C_4\times C_{16}$), $C_{386}:C_6$ ($C_4\times C_{16}$), $C_{193}:C_{12}$ ($C_8^2$), $C_{193}:C_{12}$ ($C_2\times C_{32}$)

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

Order 1158: $C_{193}:C_6$ ($C_8\times C_{16}$), $C_{193}:C_6$ ($C_4\times C_{32}$)

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

Order 579: $C_{193}:C_3$ ($C_8\times C_{32}$)

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

Order 193: $C_{193}$ ($C_8\times C_{96}$)

Order 32: $C_{32}$ ($C_{193}:C_{24}$)

Order 16: $C_{16}$ ($C_{193}:(C_2\times C_{24})$)

Order 8: $C_8$ ($C_{193}:(C_4\times C_{24})$)

Order 4: $C_4$ ($C_{1544}.C_{24}$)

Order 2: $C_2$ ($C_{3088}.C_{24}$)

Order 1: $C_1$ ($C_{6176}:C_{24}$)

Series

Derived series

$C_{6176}:C_{24}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Chief series

$C_{6176}:C_{24}$

$\rhd$

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

$\rhd$

$C_{3088}.C_6.C_2$

$\rhd$

$C_{32}\times C_{193}:C_3$

$\rhd$

$C_{6176}$

$\rhd$

$C_{3088}$

$\rhd$

$C_{1544}$

$\rhd$

$C_{772}$

$\rhd$

$C_{386}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Lower central series

$C_{6176}:C_{24}$

$\rhd$

$C_{193}$

Upper central series

$C_1$

$\lhd$

$C_{32}$

Character theory

Complex character table

The $1024 \times 1024$ character table is not available for this group.

Rational character table

The $82 \times 82$ rational character table is not available for this group.