Abstract group 288.602: $C_2^3.S_3^2$

• Label: 288.602
• Order: \( 2^{5} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $16$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$C_2^3.S_3^2$
Order:	\(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^2.C_2^6.C_2^4.C_2$, of order \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Composition factors:	$C_2$ x 5, $C_3$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Group statistics

Order	1	2	3	4	6	12
Elements	1	7	8	120	56	96	288
Conjugacy classes	1	7	3	24	21	16	72
Divisions	1	7	3	12	21	8	52
Autjugacy classes	1	3	2	2	7	1	16

Dimension	1	2	4
Irr. complex chars.	32	32	8	72
Irr. rational chars.	8	28	16	52

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$96$
Rank:	$3$
Inequivalent generating triples:	$336$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{4}=b^{12}=c^{6}=[a,c]=1, b^{a}=b^{5}, c^{b}=c^{5} \rangle$
Permutation group:	Degree $16$ $\langle(1,2,3,4)(5,6)(8,9)(13,14)(15,16), (1,3)(2,4)(5,6)(11,12)(13,15,14,16), (1,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 22 & 15 \\ 15 & 2 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 13 & 15 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_3:C_4)$ ${}^2$
Semidirect product:	$(C_6.D_6)$ $\,\rtimes\,$ $C_4$	$(C_6:C_{12})$ $\,\rtimes\,$ $C_4$	$C_6$ $\,\rtimes\,$ $(C_{12}:C_4)$	$(C_3\times C_6)$ $\,\rtimes\,$ $C_4^2$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^3$ . $S_3^2$	$C_6^2$ . $C_2^3$	$C_2$ . $(D_6.D_6)$	$(C_6:C_4)$ . $D_6$	all 23

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

Homology

Abelianization:	$C_{2} \times C_{4}^{2} $
Schur multiplier:	$C_{2}^{2} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 594 subgroups in 243 conjugacy classes, 124 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $S_3^2$
Commutator:	$G' \simeq$ $C_3^2$	$G/G' \simeq$ $C_2\times C_4^2$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $S_3\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_2^3.S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Subgroup information

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

Classes of subgroups up to conjugation

Order 288: $C_2^3.S_3^2$

Order 144: $C_3^2:C_4^2$ x 4, $C_6^2:C_4$ x 2, $C_6^2:C_4$

Order 96: $C_6:C_4^2$ x 2

Order 72: $C_6:C_{12}$ x 12, $C_6.D_6$ x 6, $C_2\times C_6^2$

Order 48: $C_{12}:C_4$ x 8, $C_6.C_2^3$ x 5, $C_2^2\times C_{12}$ x 2

Order 36: $C_3:C_{12}$ x 8, $C_6^2$ x 7, $C_3^2:C_4$ x 4

Order 32: $C_2\times C_4^2$

Order 24: $C_6:C_4$ x 30, $C_2\times C_{12}$ x 12, $C_2^2\times C_6$ x 3

Order 18: $C_3\times C_6$ x 7

Order 16: $C_4^2$ x 4, $C_2^2\times C_4$ x 3

Order 12: $C_2\times C_6$ x 21, $C_3:C_4$ x 20, $C_{12}$ x 8

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 18, $C_2^3$

Order 6: $C_6$ x 21

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 288: $C_2^3.S_3^2$

Order 144: $C_3^2:C_4^2$, $C_6^2:C_4$, $C_6^2:C_4$

Order 96: $C_6:C_4^2$

Order 72: $C_6.D_6$ x 2, $C_6:C_{12}$ x 2, $C_2\times C_6^2$

Order 48: $C_6.C_2^3$ x 3, $C_2^2\times C_{12}$, $C_{12}:C_4$

Order 36: $C_6^2$ x 3, $C_3^2:C_4$, $C_3:C_{12}$

Order 32: $C_2\times C_4^2$

Order 24: $C_6:C_4$ x 6, $C_2\times C_{12}$ x 2, $C_2^2\times C_6$ x 2

Order 18: $C_3\times C_6$ x 3

Order 16: $C_2^2\times C_4$ x 2, $C_4^2$

Order 12: $C_2\times C_6$ x 7, $C_3:C_4$ x 3, $C_{12}$

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 4, $C_2^3$

Order 6: $C_6$ x 7

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 288: $C_2^3.S_3^2$ ($C_1$)

Order 144: $C_3^2:C_4^2$ ($C_2$) x 4, $C_6^2:C_4$ ($C_2$) x 2, $C_6^2:C_4$ ($C_2$)

Order 72: $C_6:C_{12}$ ($C_4$) x 8, $C_6.D_6$ ($C_4$) x 4, $C_6:C_{12}$ ($C_2^2$) x 4, $C_6.D_6$ ($C_2^2$) x 2, $C_2\times C_6^2$ ($C_2^2$)

Order 48: $C_6.C_2^3$ ($S_3$) x 2

Order 36: $C_3:C_{12}$ ($C_2\times C_4$) x 8, $C_6^2$ ($C_2\times C_4$) x 6, $C_3^2:C_4$ ($C_2\times C_4$) x 4, $C_6^2$ ($C_2^3$)

Order 24: $C_6:C_4$ ($C_3:C_4$) x 8, $C_6:C_4$ ($D_6$) x 4, $C_2^2\times C_6$ ($D_6$) x 2

Order 18: $C_3\times C_6$ ($C_4^2$) x 4, $C_3\times C_6$ ($C_2^2\times C_4$) x 3

Order 12: $C_2\times C_6$ ($C_4\times S_3$) x 8, $C_3:C_4$ ($C_6:C_4$) x 8, $C_2\times C_6$ ($C_6:C_4$) x 4, $C_2\times C_6$ ($C_2\times D_6$) x 2

Order 9: $C_3^2$ ($C_2\times C_4^2$)

Order 8: $C_2^3$ ($S_3^2$)

Order 6: $C_6$ ($C_{12}:C_4$) x 8, $C_6$ ($C_4\times D_6$) x 4, $C_6$ ($C_6.C_2^3$) x 2

Order 4: $C_2^2$ ($C_6.D_6$) x 4, $C_2^2$ ($C_6.D_6$) x 2, $C_2^2$ ($S_3\times D_6$)

Order 3: $C_3$ ($C_6:C_4^2$) x 2

Order 2: $C_2$ ($C_3^2:C_4^2$) x 4, $C_2$ ($D_6.D_6$) x 2, $C_2$ ($C_2^2.S_3^2$)

Order 1: $C_1$ ($C_2^3.S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 288: $C_2^3.S_3^2$ ($C_1$)

Order 144: $C_3^2:C_4^2$ ($C_2$), $C_6^2:C_4$ ($C_2$), $C_6^2:C_4$ ($C_2$)

Order 72: $C_2\times C_6^2$ ($C_2^2$), $C_6.D_6$ ($C_2^2$), $C_6.D_6$ ($C_4$), $C_6:C_{12}$ ($C_2^2$), $C_6:C_{12}$ ($C_4$)

Order 48: $C_6.C_2^3$ ($S_3$)

Order 36: $C_6^2$ ($C_2\times C_4$) x 2, $C_3^2:C_4$ ($C_2\times C_4$), $C_3:C_{12}$ ($C_2\times C_4$), $C_6^2$ ($C_2^3$)

Order 24: $C_6:C_4$ ($D_6$), $C_6:C_4$ ($C_3:C_4$), $C_2^2\times C_6$ ($D_6$)

Order 18: $C_3\times C_6$ ($C_2^2\times C_4$) x 2, $C_3\times C_6$ ($C_4^2$)

Order 12: $C_2\times C_6$ ($C_4\times S_3$) x 2, $C_2\times C_6$ ($C_6:C_4$), $C_2\times C_6$ ($C_2\times D_6$), $C_3:C_4$ ($C_6:C_4$)

Order 9: $C_3^2$ ($C_2\times C_4^2$)

Order 8: $C_2^3$ ($S_3^2$)

Order 6: $C_6$ ($C_4\times D_6$) x 2, $C_6$ ($C_6.C_2^3$), $C_6$ ($C_{12}:C_4$)

Order 4: $C_2^2$ ($S_3\times D_6$), $C_2^2$ ($C_6.D_6$), $C_2^2$ ($C_6.D_6$)

Order 3: $C_3$ ($C_6:C_4^2$)

Order 2: $C_2$ ($C_3^2:C_4^2$), $C_2$ ($C_2^2.S_3^2$), $C_2$ ($D_6.D_6$)

Order 1: $C_1$ ($C_2^3.S_3^2$)

Series

Derived series

$C_2^3.S_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_2^3.S_3^2$

$\rhd$

$C_6^2:C_4$

$\rhd$

$C_2\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2^3.S_3^2$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 122 larger groups in the database.

This group is a maximal quotient of 29 larger groups in the database.

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $52 \times 52$ rational character table.