Abstract group 576.1072: $S_3\times C_{96}$

• Label: 576.1072
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{5} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{5} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $38$
• Trans deg.: $192$
• Rank: $2$

Group information

Description:	$S_3\times C_{96}$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Automorphism group:	$C_{24}:C_2^4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
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
Elements	1	7	8	8	20	16	28	32	56	64	112	224	576
Conjugacy classes	1	3	5	4	9	8	14	16	28	32	56	112	288
Divisions	1	3	3	2	5	2	4	2	4	2	4	4	36
Autjugacy classes	1	2	3	2	4	2	4	2	4	2	4	4	34

Dimension	1	2	4	8	16	32	64
Irr. complex chars.	192	96	0	0	0	0	0	288
Irr. rational chars.	4	8	7	6	6	4	1	36

Minimal presentations

Permutation degree:	$38$
Transitive degree:	$192$
Rank:	$2$
Inequivalent generating pairs:	$192$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	64
Arbitrary	2	4	20

Constructions

Presentation:	$\langle a, b \mid a^{6}=b^{96}=1, b^{a}=b^{65} \rangle$
Permutation group:	Degree $38$ $\langle(33,34)(35,37)(36,38), (1,2,3,7,4,8,11,17,5,9,12,18,14,20,23,27,6,10,13,19,15,21,24,28,16,22,25,29,26,30,31,32) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 36 & 0 \\ 0 & 62 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{97})$
Direct product:	$C_{32}$ $\, \times\, $ $C_3$ $\, \times\, $ $S_3$
Semidirect product:	$C_{96}$ $\,\rtimes\,$ $C_6$	$(C_3\times C_{96})$ $\,\rtimes\,$ $C_2$	$(C_3:C_{96})$ $\,\rtimes\,$ $C_2$	$(C_3:C_{32})$ $\,\rtimes\,$ $C_6$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{48}$ . $D_6$	$D_6$ . $C_{48}$	$(S_3\times C_{48})$ . $C_2$	$(S_3\times C_{24})$ . $C_4$	all 30
Aut. group:	$\Aut(C_2\times C_{194})$	$\Aut(S_3\times C_{97})$

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

Homology

Abelianization:	$C_{2} \times C_{96} \simeq C_{2} \times C_{32} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 124 subgroups in 74 conjugacy classes, 46 normal (42 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{96}$	$G/Z \simeq$ $S_3$
Commutator:	$G' \simeq$ $C_3$	$G/G' \simeq$ $C_2\times C_{96}$
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $C_6\times S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_{96}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $S_3\times C_{96}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times C_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_{32}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 576: $S_3\times C_{96}$

Order 288: $C_3\times C_{96}$, $C_3:C_{96}$, $S_3\times C_{48}$

Order 192: $S_3\times C_{32}$, $C_2\times C_{96}$

Order 144: $S_3\times C_{24}$, $C_3\times C_{48}$, $C_3:C_{48}$

Order 96: $C_{96}$ x 4, $C_2\times C_{48}$, $S_3\times C_{16}$, $C_3:C_{32}$

Order 72: $S_3\times C_{12}$, $C_3\times C_{24}$, $C_3:C_{24}$

Order 64: $C_2\times C_{32}$

Order 48: $C_{48}$ x 4, $S_3\times C_8$, $C_2\times C_{24}$, $C_3:C_{16}$

Order 36: $C_3\times C_{12}$, $C_3:C_{12}$, $C_6\times S_3$

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

Order 24: $C_{24}$ x 4, $C_2\times C_{12}$, $C_4\times S_3$, $C_3:C_8$

Order 18: $C_3\times S_3$ x 2, $C_3\times C_6$

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

Order 12: $C_{12}$ x 4, $C_2\times C_6$, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

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

Order 6: $C_6$ x 5, $S_3$ x 2

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $S_3\times C_{96}$

Order 288: $C_3\times C_{96}$, $C_3:C_{96}$, $S_3\times C_{48}$

Order 192: $S_3\times C_{32}$, $C_2\times C_{96}$

Order 144: $S_3\times C_{24}$, $C_3\times C_{48}$, $C_3:C_{48}$

Order 96: $C_{96}$ x 4, $C_2\times C_{48}$, $S_3\times C_{16}$, $C_3:C_{32}$

Order 72: $S_3\times C_{12}$, $C_3\times C_{24}$, $C_3:C_{24}$

Order 64: $C_2\times C_{32}$

Order 48: $C_{48}$ x 4, $S_3\times C_8$, $C_2\times C_{24}$, $C_3:C_{16}$

Order 36: $C_3\times C_{12}$, $C_3:C_{12}$, $C_6\times S_3$

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

Order 24: $C_{24}$ x 4, $C_2\times C_{12}$, $C_4\times S_3$, $C_3:C_8$

Order 18: $C_3\times C_6$, $C_3\times S_3$

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

Order 12: $C_{12}$ x 4, $C_2\times C_6$, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

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

Order 6: $C_6$ x 4, $S_3$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $S_3\times C_{96}$ ($C_1$)

Order 288: $C_3\times C_{96}$ ($C_2$), $C_3:C_{96}$ ($C_2$), $S_3\times C_{48}$ ($C_2$)

Order 192: $S_3\times C_{32}$ ($C_3$)

Order 144: $S_3\times C_{24}$ ($C_4$), $C_3\times C_{48}$ ($C_2^2$), $C_3:C_{48}$ ($C_4$)

Order 96: $S_3\times C_{16}$ ($C_6$), $C_{96}$ ($C_6$), $C_{96}$ ($S_3$), $C_3:C_{32}$ ($C_6$)

Order 72: $S_3\times C_{12}$ ($C_8$), $C_3\times C_{24}$ ($C_2\times C_4$), $C_3:C_{24}$ ($C_8$)

Order 48: $S_3\times C_8$ ($C_{12}$), $C_{48}$ ($C_2\times C_6$), $C_{48}$ ($D_6$), $C_3:C_{16}$ ($C_{12}$)

Order 36: $C_3\times C_{12}$ ($C_2\times C_8$), $C_3:C_{12}$ ($C_{16}$), $C_6\times S_3$ ($C_{16}$)

Order 32: $C_{32}$ ($C_3\times S_3$)

Order 24: $C_4\times S_3$ ($C_{24}$), $C_{24}$ ($C_2\times C_{12}$), $C_{24}$ ($C_4\times S_3$), $C_3:C_8$ ($C_{24}$)

Order 18: $C_3\times S_3$ ($C_{32}$) x 2, $C_3\times C_6$ ($C_2\times C_{16}$)

Order 16: $C_{16}$ ($C_6\times S_3$)

Order 12: $D_6$ ($C_{48}$), $C_{12}$ ($S_3\times C_8$), $C_{12}$ ($C_2\times C_{24}$), $C_3:C_4$ ($C_{48}$)

Order 9: $C_3^2$ ($C_2\times C_{32}$)

Order 8: $C_8$ ($S_3\times C_{12}$)

Order 6: $S_3$ ($C_{96}$) x 2, $C_6$ ($C_2\times C_{48}$), $C_6$ ($S_3\times C_{16}$)

Order 4: $C_4$ ($S_3\times C_{24}$)

Order 3: $C_3$ ($S_3\times C_{32}$), $C_3$ ($C_2\times C_{96}$)

Order 2: $C_2$ ($S_3\times C_{48}$)

Order 1: $C_1$ ($S_3\times C_{96}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $S_3\times C_{96}$ ($C_1$)

Order 288: $C_3\times C_{96}$ ($C_2$), $C_3:C_{96}$ ($C_2$), $S_3\times C_{48}$ ($C_2$)

Order 192: $S_3\times C_{32}$ ($C_3$)

Order 144: $S_3\times C_{24}$ ($C_4$), $C_3\times C_{48}$ ($C_2^2$), $C_3:C_{48}$ ($C_4$)

Order 96: $S_3\times C_{16}$ ($C_6$), $C_{96}$ ($C_6$), $C_{96}$ ($S_3$), $C_3:C_{32}$ ($C_6$)

Order 72: $S_3\times C_{12}$ ($C_8$), $C_3\times C_{24}$ ($C_2\times C_4$), $C_3:C_{24}$ ($C_8$)

Order 48: $S_3\times C_8$ ($C_{12}$), $C_{48}$ ($C_2\times C_6$), $C_{48}$ ($D_6$), $C_3:C_{16}$ ($C_{12}$)

Order 36: $C_3\times C_{12}$ ($C_2\times C_8$), $C_3:C_{12}$ ($C_{16}$), $C_6\times S_3$ ($C_{16}$)

Order 32: $C_{32}$ ($C_3\times S_3$)

Order 24: $C_4\times S_3$ ($C_{24}$), $C_{24}$ ($C_2\times C_{12}$), $C_{24}$ ($C_4\times S_3$), $C_3:C_8$ ($C_{24}$)

Order 18: $C_3\times C_6$ ($C_2\times C_{16}$), $C_3\times S_3$ ($C_{32}$)

Order 16: $C_{16}$ ($C_6\times S_3$)

Order 12: $D_6$ ($C_{48}$), $C_{12}$ ($S_3\times C_8$), $C_{12}$ ($C_2\times C_{24}$), $C_3:C_4$ ($C_{48}$)

Order 9: $C_3^2$ ($C_2\times C_{32}$)

Order 8: $C_8$ ($S_3\times C_{12}$)

Order 6: $C_6$ ($C_2\times C_{48}$), $C_6$ ($S_3\times C_{16}$), $S_3$ ($C_{96}$)

Order 4: $C_4$ ($S_3\times C_{24}$)

Order 3: $C_3$ ($S_3\times C_{32}$), $C_3$ ($C_2\times C_{96}$)

Order 2: $C_2$ ($S_3\times C_{48}$)

Order 1: $C_1$ ($S_3\times C_{96}$)

Series

Derived series

$S_3\times C_{96}$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$S_3\times C_{96}$

$\rhd$

$C_3\times C_{96}$

$\rhd$

$C_{96}$

$\rhd$

$C_{48}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$S_3\times C_{96}$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_{96}$

Supergroups

This group is a maximal subgroup of 14 larger groups in the database.

This group is a maximal quotient of 10 larger groups in the database.

Character theory

Complex character table

See the $288 \times 288$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $36 \times 36$ rational character table.