Abstract group 432.620: $C_3^2:\GL(2,3)$

• Label: 432.620
• Order: \( 2^{4} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3^{3} \)
• Perm deg.: $14$
• Trans deg.: $72$
• Rank: $4$

Group information

Description:	$C_3^2:\GL(2,3)$
Order:	\(432\)\(\medspace = 2^{4} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_3^4.Q_8.A_4.D_6.C_2$, of order \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \)
Composition factors:	$C_2$ x 4, $C_3$ x 3
Derived length:	$4$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	109	80	6	80	108	48	432
Conjugacy classes	1	2	13	1	13	2	4	36
Divisions	1	2	13	1	13	1	4	35
Autjugacy classes	1	2	2	1	2	1	1	10

Dimension	1	2	3	4	6
Irr. complex chars.	2	15	2	13	4	36
Irr. rational chars.	2	13	2	14	4	35

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$72$
Rank:	$4$
Inequivalent generating quadruples:	$98280$

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:	${\langle a, b, c, d \mid a^{2}=b^{3}=d^{12}=1, c^{6}=d^{6}, b^{a}=b^{2}, c^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(2,3)(5,7)(6,8)(10,11)(13,14), (2,5,8)(3,6,7)(12,13,14), (12,14,13), (9,10,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 1 & 0 & 0 \\ -1 & 0 & 1 & 1 & 0 & 0 \\ 1 & -1 & -1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 20 \\ 20 & 11 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 9 & 10 \\ 1 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 20 \\ 20 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 6 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^2$ $\,\rtimes\,$ $\GL(2,3)$	$(Q_8\times C_3^2)$ $\,\rtimes\,$ $S_3$	$Q_8$ $\,\rtimes\,$ $(C_3^2:S_3)$	$(C_3\times \SL(2,3))$ $\,\rtimes\,$ $S_3$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3\times C_6)$ . $S_4$	$C_6$ . $(C_3:S_4)$	$C_2$ . $(C_3^2:S_4)$		more information

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

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_{3}^{3}$
Commutator length:	$1$

Subgroups

There are 2362 subgroups in 208 conjugacy classes, 41 normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^2:S_4$
Commutator:	$G' \simeq$ $C_3^2\times \SL(2,3)$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_3^2:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_3^2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_3^2:\GL(2,3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

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

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Classes of subgroups up to conjugation

Order 432: $C_3^2:\GL(2,3)$

Order 216: $C_3^2\times \SL(2,3)$

Order 144: $C_3:\GL(2,3)$ x 12, $C_{12}.D_6$

Order 108: $C_3^2:D_6$

Order 72: $C_3\times \SL(2,3)$ x 12, $Q_8\times C_3^2$, $C_3:D_{12}$, $C_3^2:C_8$

Order 54: $C_3^2:S_3$ x 2, $C_3^2\times C_6$

Order 48: $\GL(2,3)$ x 9, $Q_8:S_3$ x 4

Order 36: $C_6:S_3$ x 13, $C_3\times C_{12}$

Order 27: $C_3^3$

Order 24: $\SL(2,3)$ x 9, $D_{12}$ x 4, $C_3\times Q_8$ x 4, $C_3:C_8$ x 4

Order 18: $C_3:S_3$ x 25, $C_3\times C_6$ x 13

Order 16: $\SD_{16}$

Order 12: $D_6$ x 13, $C_{12}$ x 4

Order 9: $C_3^2$ x 13

Order 8: $Q_8$, $D_4$, $C_8$

Order 6: $S_3$ x 22, $C_6$ x 13

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

Order 3: $C_3$ x 13

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 432: $C_3^2:\GL(2,3)$

Order 216: $C_3^2\times \SL(2,3)$

Order 144: $C_{12}.D_6$, $C_3:\GL(2,3)$

Order 108: $C_3^2:D_6$

Order 72: $Q_8\times C_3^2$, $C_3:D_{12}$, $C_3\times \SL(2,3)$, $C_3^2:C_8$

Order 54: $C_3^2\times C_6$, $C_3^2:S_3$

Order 48: $\GL(2,3)$, $Q_8:S_3$

Order 36: $C_6:S_3$ x 2, $C_3\times C_{12}$

Order 27: $C_3^3$

Order 24: $D_{12}$, $\SL(2,3)$, $C_3\times Q_8$, $C_3:C_8$

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

Order 16: $\SD_{16}$

Order 12: $D_6$ x 2, $C_{12}$

Order 9: $C_3^2$ x 2

Order 8: $Q_8$, $D_4$, $C_8$

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

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 432: $C_3^2:\GL(2,3)$ ($C_1$)

Order 216: $C_3^2\times \SL(2,3)$ ($C_2$)

Order 72: $C_3\times \SL(2,3)$ ($S_3$) x 12, $Q_8\times C_3^2$ ($S_3$)

Order 24: $\SL(2,3)$ ($C_3:S_3$) x 9, $C_3\times Q_8$ ($C_3:S_3$) x 4

Order 18: $C_3\times C_6$ ($S_4$)

Order 9: $C_3^2$ ($\GL(2,3)$)

Order 8: $Q_8$ ($C_3^2:S_3$)

Order 6: $C_6$ ($C_3:S_4$) x 4

Order 3: $C_3$ ($C_3:\GL(2,3)$) x 4

Order 2: $C_2$ ($C_3^2:S_4$)

Order 1: $C_1$ ($C_3^2:\GL(2,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 432: $C_3^2:\GL(2,3)$ ($C_1$)

Order 216: $C_3^2\times \SL(2,3)$ ($C_2$)

Order 72: $Q_8\times C_3^2$ ($S_3$), $C_3\times \SL(2,3)$ ($S_3$)

Order 24: $\SL(2,3)$ ($C_3:S_3$), $C_3\times Q_8$ ($C_3:S_3$)

Order 18: $C_3\times C_6$ ($S_4$)

Order 9: $C_3^2$ ($\GL(2,3)$)

Order 8: $Q_8$ ($C_3^2:S_3$)

Order 6: $C_6$ ($C_3:S_4$)

Order 3: $C_3$ ($C_3:\GL(2,3)$)

Order 2: $C_2$ ($C_3^2:S_4$)

Order 1: $C_1$ ($C_3^2:\GL(2,3)$)

Series

Derived series

$C_3^2:\GL(2,3)$

$\rhd$

$C_3^2\times \SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_3^2:\GL(2,3)$

$\rhd$

$C_3^2\times \SL(2,3)$

$\rhd$

$C_3\times \SL(2,3)$

$\rhd$

$\SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_3^2:\GL(2,3)$

$\rhd$

$C_3^2\times \SL(2,3)$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 23 larger groups in the database.

This group is a maximal quotient of 12 larger groups in the database.

Character theory

Complex character table

See the $36 \times 36$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $35 \times 35$ rational character table.