Abstract group 55584.b: $C_{4632}:C_{12}$

• Label: 55584.b
• Order: \( 2^{5} \cdot 3^{2} \cdot 193 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3 \)
• Perm deg.: $204$
• Trans deg.: $4632$
• Rank: $2$

Group information

Description:	$C_{4632}:C_{12}$
Order:	\(55584\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 193 \)
Exponent:	\(4632\)\(\medspace = 2^{3} \cdot 3 \cdot 193 \)
Automorphism group:	$C_5^4:(C_2\times C_4)$, of order \(3557376\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 193 \)
Composition factors:	$C_2$ x 5, $C_3$ 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	24	193	386	579	772	1158	1544	2316	4632
Elements	1	387	1160	1932	4248	2320	17760	23168	192	192	384	384	384	768	768	1536	55584
Conjugacy classes	1	3	8	12	24	16	96	128	16	16	32	32	32	64	64	128	672
Divisions	1	3	4	6	12	4	24	16	1	1	1	1	1	1	1	1	78
Autjugacy classes	1	2	3	4	6	4	12	12	1	1	1	1	1	1	1	1	52

Minimal presentations

Permutation degree:	$204$
Transitive degree:	$4632$
Rank:	$2$
Inequivalent generating pairs:	$192$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{12}=b^{4632}=1, b^{a}=b^{49} \rangle$
Permutation group:	Degree $204$ $\langle(2,4,15,8,7,12,3,11,9,10,13,6)(5,18,42,22,21,39,14,38,23,24,40,20)(16,44,56,19,46,36,34,57,47,41,69,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 186 & 0 \\ 0 & 55 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 49 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_8$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_{193}:C_{12})$
Semidirect product:	$C_{4632}$ $\,\rtimes\,$ $C_{12}$	$C_{579}$ $\,\rtimes\,$ $(C_4\times C_{24})$	$(C_{193}:C_{12})$ $\,\rtimes\,$ $C_{24}$	$C_{1544}$ $\,\rtimes\,$ $(C_3\times C_{12})$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{386}$ . $C_{12}^2$	$D_{386}$ . $(C_6\times C_{12})$	$D_{193}$ . $(C_6\times C_{24})$	$C_{772}$ . $(C_6\times C_{12})$	all 33
Aut. group:	$\Aut(D_{193}:C_{64})$

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

Homology

Abelianization:	$C_{12} \times C_{24} \simeq C_{4} \times C_{8} \times C_{3}^{2}$
Schur multiplier:	$C_{12}$
Commutator length:	$1$

Subgroups

There are 24072 subgroups in 264 conjugacy classes, 140 normal (44 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{24}$	$G/Z \simeq$ $C_{193}:C_{12}$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_{12}\times C_{24}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_{193}:(C_6\times C_{12})$
Fitting:	$\operatorname{Fit} \simeq$ $C_{4632}$	$G/\operatorname{Fit} \simeq$ $C_{12}$
Radical:	$R \simeq$ $C_{4632}:C_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{1158}$	$G/\operatorname{soc} \simeq$ $C_4\times C_{12}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times C_8$
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

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

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Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 55584: $C_{4632}:C_{12}$

Order 27792: $C_{12}\times C_{193}:C_{12}$, $C_{24}\times C_{193}:C_6$, $C_3\times C_{193}:(C_2\times C_{24})$

Order 18528: $C_8\times C_{193}:C_{12}$ x 3, $C_{24}\times C_{193}:C_4$

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

Order 9264: $C_{193}:(C_2\times C_{24})$ x 3, $C_8\times C_{193}:C_6$ x 3, $C_4\times C_{193}:C_{12}$ x 3, $C_{24}\times D_{193}$, $C_3\times C_{193}:(C_2\times C_8)$, $C_{12}\times C_{193}:C_4$

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: $C_{1544}:C_4$

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, $C_4\times C_{193}:C_6$ x 3, $C_6\times C_{193}:C_4$ x 2, $C_3\times C_{193}:C_8$ x 2, $C_3\times C_{193}:C_8$, $C_{12}\times D_{193}$, $C_{4632}$

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

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

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

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

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

Order 288: $C_{12}\times C_{24}$

Order 193: $C_{193}$

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

Order 96: $C_4\times C_{24}$ x 4

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

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

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

Order 32: $C_4\times C_8$

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

Order 18: $C_3\times C_6$ x 3

Order 16: $C_2\times C_8$ 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 55584: $C_{4632}:C_{12}$

Order 27792: $C_{12}\times C_{193}:C_{12}$, $C_{24}\times C_{193}:C_6$, $C_3\times C_{193}:(C_2\times C_{24})$

Order 18528: $C_{24}\times C_{193}:C_4$, $C_8\times C_{193}:C_{12}$

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

Order 9264: $C_{193}:(C_2\times C_{24})$, $C_{24}\times D_{193}$, $C_8\times C_{193}:C_6$, $C_3\times C_{193}:(C_2\times C_8)$, $C_{12}\times C_{193}:C_4$, $C_4\times C_{193}:C_{12}$

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: $C_{1544}:C_4$

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

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

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

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

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

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

Order 288: $C_{12}\times C_{24}$

Order 193: $C_{193}$

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

Order 96: $C_4\times C_{24}$ x 2

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

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

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

Order 32: $C_4\times C_8$

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

Order 18: $C_3\times C_6$ x 2

Order 16: $C_2\times C_8$ 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 55584: $C_{4632}:C_{12}$ ($C_1$)

Order 27792: $C_{12}\times C_{193}:C_{12}$ ($C_2$), $C_{24}\times C_{193}:C_6$ ($C_2$), $C_3\times C_{193}:(C_2\times C_{24})$ ($C_2$)

Order 18528: $C_8\times C_{193}:C_{12}$ ($C_3$) x 3, $C_{24}\times C_{193}:C_4$ ($C_3$)

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

Order 9264: $C_{193}:(C_2\times C_{24})$ ($C_6$) x 3, $C_8\times C_{193}:C_6$ ($C_6$) x 3, $C_4\times C_{193}:C_{12}$ ($C_6$) x 3, $C_{24}\times D_{193}$ ($C_6$), $C_3\times C_{193}:(C_2\times C_8)$ ($C_6$), $C_{12}\times C_{193}:C_4$ ($C_6$)

Order 6948: $C_3\times C_{193}:C_{12}$ ($C_8$) x 4, $C_{12}\times C_{193}:C_3$ ($C_2\times C_4$), $C_6\times C_{193}:C_6$ ($C_2\times C_4$), $C_3\times C_{193}:C_{12}$ ($C_2\times C_4$)

Order 6176: $C_{1544}:C_4$ ($C_3^2$)

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

Order 3474: $C_{579}:C_6$ ($C_2\times C_8$) x 2, $C_{1158}:C_3$ ($C_4^2$)

Order 3088: $D_{193}:C_8$ ($C_3\times C_6$), $C_{772}:C_4$ ($C_3\times C_6$), $C_8\times D_{193}$ ($C_3\times C_6$)

Order 2316: $C_{193}:C_{12}$ ($C_{24}$) x 12, $C_{193}:C_{12}$ ($C_{24}$) x 4, $C_{193}:C_{12}$ ($C_2\times C_{12}$) x 3, $C_{386}:C_6$ ($C_2\times C_{12}$) x 3, $C_{193}:C_{12}$ ($C_2\times C_{12}$) x 3, $C_3\times D_{386}$ ($C_2\times C_{12}$), $C_{193}:C_{12}$ ($C_2\times C_{12}$), $C_{2316}$ ($C_2\times C_{12}$)

Order 1737: $C_{579}:C_3$ ($C_4\times C_8$)

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

Order 1158: $C_{193}:C_6$ ($C_2\times C_{24}$) x 6, $C_{193}:C_6$ ($C_4\times C_{12}$) x 3, $C_3\times D_{193}$ ($C_2\times C_{24}$) x 2, $C_{1158}$ ($C_4\times C_{12}$)

Order 772: $C_{193}:C_4$ ($C_3\times C_{24}$) x 4, $D_{386}$ ($C_6\times C_{12}$), $C_{772}$ ($C_6\times C_{12}$), $C_{193}:C_4$ ($C_6\times C_{12}$)

Order 579: $C_{193}:C_3$ ($C_4\times C_{24}$) x 3, $C_{579}$ ($C_4\times C_{24}$)

Order 386: $D_{193}$ ($C_6\times C_{24}$) x 2, $C_{386}$ ($C_{12}^2$)

Order 193: $C_{193}$ ($C_{12}\times C_{24}$)

Order 24: $C_{24}$ ($C_{193}:C_{12}$)

Order 12: $C_{12}$ ($C_{193}:(C_2\times C_{12})$)

Order 8: $C_8$ ($C_{193}:(C_3\times C_{12})$)

Order 6: $C_6$ ($C_{193}:(C_4\times C_{12})$)

Order 4: $C_4$ ($C_{193}:(C_6\times C_{12})$)

Order 3: $C_3$ ($C_{193}:(C_4\times C_{24})$)

Order 2: $C_2$ ($C_{193}:C_{12}^2$)

Order 1: $C_1$ ($C_{4632}:C_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 55584: $C_{4632}:C_{12}$ ($C_1$)

Order 27792: $C_{12}\times C_{193}:C_{12}$ ($C_2$), $C_{24}\times C_{193}:C_6$ ($C_2$), $C_3\times C_{193}:(C_2\times C_{24})$ ($C_2$)

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

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

Order 9264: $C_{193}:(C_2\times C_{24})$ ($C_6$), $C_{24}\times D_{193}$ ($C_6$), $C_8\times C_{193}:C_6$ ($C_6$), $C_3\times C_{193}:(C_2\times C_8)$ ($C_6$), $C_{12}\times C_{193}:C_4$ ($C_6$), $C_4\times C_{193}:C_{12}$ ($C_6$)

Order 6948: $C_{12}\times C_{193}:C_3$ ($C_2\times C_4$), $C_6\times C_{193}:C_6$ ($C_2\times C_4$), $C_3\times C_{193}:C_{12}$ ($C_2\times C_4$), $C_3\times C_{193}:C_{12}$ ($C_8$)

Order 6176: $C_{1544}:C_4$ ($C_3^2$)

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

Order 3474: $C_{1158}:C_3$ ($C_4^2$), $C_{579}:C_6$ ($C_2\times C_8$)

Order 3088: $D_{193}:C_8$ ($C_3\times C_6$), $C_{772}:C_4$ ($C_3\times C_6$), $C_8\times D_{193}$ ($C_3\times C_6$)

Order 2316: $C_3\times D_{386}$ ($C_2\times C_{12}$), $C_{193}:C_{12}$ ($C_2\times C_{12}$), $C_{193}:C_{12}$ ($C_2\times C_{12}$), $C_{386}:C_6$ ($C_2\times C_{12}$), $C_{193}:C_{12}$ ($C_2\times C_{12}$), $C_{193}:C_{12}$ ($C_{24}$), $C_{193}:C_{12}$ ($C_{24}$), $C_{2316}$ ($C_2\times C_{12}$)

Order 1737: $C_{579}:C_3$ ($C_4\times C_8$)

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

Order 1158: $C_{1158}$ ($C_4\times C_{12}$), $C_3\times D_{193}$ ($C_2\times C_{24}$), $C_{193}:C_6$ ($C_4\times C_{12}$), $C_{193}:C_6$ ($C_2\times C_{24}$)

Order 772: $D_{386}$ ($C_6\times C_{12}$), $C_{193}:C_4$ ($C_3\times C_{24}$), $C_{772}$ ($C_6\times C_{12}$), $C_{193}:C_4$ ($C_6\times C_{12}$)

Order 579: $C_{579}$ ($C_4\times C_{24}$), $C_{193}:C_3$ ($C_4\times C_{24}$)

Order 386: $C_{386}$ ($C_{12}^2$), $D_{193}$ ($C_6\times C_{24}$)

Order 193: $C_{193}$ ($C_{12}\times C_{24}$)

Order 24: $C_{24}$ ($C_{193}:C_{12}$)

Order 12: $C_{12}$ ($C_{193}:(C_2\times C_{12})$)

Order 8: $C_8$ ($C_{193}:(C_3\times C_{12})$)

Order 6: $C_6$ ($C_{193}:(C_4\times C_{12})$)

Order 4: $C_4$ ($C_{193}:(C_6\times C_{12})$)

Order 3: $C_3$ ($C_{193}:(C_4\times C_{24})$)

Order 2: $C_2$ ($C_{193}:C_{12}^2$)

Order 1: $C_1$ ($C_{4632}:C_{12}$)

Series

Derived series

$C_{4632}:C_{12}$

$\rhd$

$C_{4632}:C_{12}$

$\rhd$

$C_{193}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{4632}:C_{12}$

$\rhd$

$C_{4632}:C_{12}$

$\rhd$

$C_{24}\times C_{193}:C_6$

$\rhd$

$C_{24}\times C_{193}:C_6$

$\rhd$

$C_{24}\times C_{193}:C_3$

$\rhd$

$C_{24}\times C_{193}:C_3$

$\rhd$

$C_{4632}$

$\rhd$

$C_{4632}$

$\rhd$

$C_{2316}$

$\rhd$

$C_{2316}$

$\rhd$

$C_{1158}$

$\rhd$

$C_{1158}$

$\rhd$

$C_{579}$

$\rhd$

$C_{579}$

$\rhd$

$C_{193}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{4632}:C_{12}$

$\rhd$

$C_{4632}:C_{12}$

$\rhd$

$C_{193}$

$\rhd$

$C_{193}$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_{24}$

$\lhd$

$C_{24}$

Character theory

Complex character table

The $672 \times 672$ character table is not available for this group.

Rational character table

The $78 \times 78$ rational character table is not available for this group.