Abstract group 432.251: $(C_3\times A_4):C_{12}$

• Label: 432.251
• Order: \( 2^{4} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $17$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$(C_3\times A_4):C_{12}$
Order:	\(432\)\(\medspace = 2^{4} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2\times C_6^2:D_6$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Composition factors:	$C_2$ x 4, $C_3$ x 3
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	7	80	72	128	144	432
Conjugacy classes	1	3	6	4	16	8	38
Divisions	1	3	4	2	10	2	22
Autjugacy classes	1	3	4	2	10	2	22

Dimension	1	2	3	4	6	12
Irr. complex chars.	12	6	12	0	8	0	38
Irr. rational chars.	2	5	2	3	7	3	22

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	12
Arbitrary	6	8	8

Constructions

Presentation:	$\langle a, b, c, d \mid a^{4}=b^{3}=c^{6}=d^{6}=[a,d]=[c,d]=1, b^{a}=b^{2}, c^{a}=c^{5}d^{3}, c^{b}=c^{4}d^{3}, d^{b}=c^{5}d \rangle$
Permutation group:	Degree $17$ $\langle(2,4)(3,6)(5,7)(8,9)(11,12)(14,15,16,17), (2,5,9)(4,7,8), (14,16)(15,17) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 29 \\ 3 & 4 \end{array}\right), \left(\begin{array}{rr} 17 & 11 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 18 & 19 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 18 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Transitive group:	36T572	36T574	36T656		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6.S_4)$ $\,\rtimes\,$ $C_3$	$(C_3\times A_4)$ $\,\rtimes\,$ $C_{12}$	$(C_3^2:A_4)$ $\,\rtimes\,$ $C_4$	$C_3^2$ $\,\rtimes\,$ $(A_4:C_4)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3\times C_6)$ . $S_4$	$C_6$ . $(C_3\times S_4)$	$(C_6\times A_4)$ . $C_6$	$C_3$ . $(A_4:C_{12})$	all 10

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

Homology

Abelianization:	$C_{12} \simeq C_{4} \times C_{3}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 449 subgroups in 82 conjugacy classes, 18 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^2:S_4$
Commutator:	$G' \simeq$ $C_3\times A_4$	$G/G' \simeq$ $C_{12}$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_3\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $(C_3\times A_4):C_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_3\times S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3$

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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Subgroup information

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

Classes of subgroups up to conjugation

Order 432: $(C_3\times A_4):C_{12}$

Order 216: $C_6^2:C_6$

Order 144: $C_6^2:C_4$, $C_6.S_4$

Order 108: $C_3^2:C_{12}$, $C_3^2:A_4$

Order 72: $C_6\times A_4$ x 2, $C_6:C_{12}$ x 2, $C_2\times C_6^2$

Order 54: $C_2\times \He_3$

Order 48: $A_4:C_4$, $C_2^2:C_{12}$, $C_6.D_4$

Order 36: $C_6^2$ x 3, $C_3:C_{12}$ x 2, $C_3\times A_4$ x 2, $C_3^2:C_4$

Order 27: $\He_3$

Order 24: $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2, $C_2^2\times C_6$ x 2, $C_2\times A_4$ x 2

Order 18: $C_3\times C_6$ x 5

Order 16: $C_2^2:C_4$

Order 12: $C_2\times C_6$ x 8, $C_3:C_4$ x 3, $A_4$ x 2, $C_{12}$ x 2

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$ x 2, $C_2^3$

Order 6: $C_6$ x 10

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 432: $(C_3\times A_4):C_{12}$

Order 216: $C_6^2:C_6$

Order 144: $C_6^2:C_4$, $C_6.S_4$

Order 108: $C_3^2:C_{12}$, $C_3^2:A_4$

Order 72: $C_6\times A_4$ x 2, $C_6:C_{12}$ x 2, $C_2\times C_6^2$

Order 54: $C_2\times \He_3$

Order 48: $A_4:C_4$, $C_2^2:C_{12}$, $C_6.D_4$

Order 36: $C_6^2$ x 3, $C_3:C_{12}$ x 2, $C_3\times A_4$ x 2, $C_3^2:C_4$

Order 27: $\He_3$

Order 24: $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2, $C_2^2\times C_6$ x 2, $C_2\times A_4$ x 2

Order 18: $C_3\times C_6$ x 5

Order 16: $C_2^2:C_4$

Order 12: $C_2\times C_6$ x 8, $C_3:C_4$ x 3, $A_4$ x 2, $C_{12}$ x 2

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$ x 2, $C_2^3$

Order 6: $C_6$ x 10

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 432: $(C_3\times A_4):C_{12}$ ($C_1$)

Order 216: $C_6^2:C_6$ ($C_2$)

Order 144: $C_6.S_4$ ($C_3$)

Order 108: $C_3^2:A_4$ ($C_4$)

Order 72: $C_2\times C_6^2$ ($S_3$), $C_6\times A_4$ ($C_6$)

Order 36: $C_6^2$ ($C_3:C_4$), $C_3\times A_4$ ($C_{12}$)

Order 24: $C_2^2\times C_6$ ($C_3\times S_3$)

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

Order 12: $C_2\times C_6$ ($C_3:C_{12}$)

Order 9: $C_3^2$ ($A_4:C_4$)

Order 8: $C_2^3$ ($C_3^2:C_6$)

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

Order 4: $C_2^2$ ($C_3^2:C_{12}$)

Order 3: $C_3$ ($A_4:C_{12}$)

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

Order 1: $C_1$ ($(C_3\times A_4):C_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 432: $(C_3\times A_4):C_{12}$ ($C_1$)

Order 216: $C_6^2:C_6$ ($C_2$)

Order 144: $C_6.S_4$ ($C_3$)

Order 108: $C_3^2:A_4$ ($C_4$)

Order 72: $C_2\times C_6^2$ ($S_3$), $C_6\times A_4$ ($C_6$)

Order 36: $C_6^2$ ($C_3:C_4$), $C_3\times A_4$ ($C_{12}$)

Order 24: $C_2^2\times C_6$ ($C_3\times S_3$)

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

Order 12: $C_2\times C_6$ ($C_3:C_{12}$)

Order 9: $C_3^2$ ($A_4:C_4$)

Order 8: $C_2^3$ ($C_3^2:C_6$)

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

Order 4: $C_2^2$ ($C_3^2:C_{12}$)

Order 3: $C_3$ ($A_4:C_{12}$)

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

Order 1: $C_1$ ($(C_3\times A_4):C_{12}$)

Series

Derived series

$(C_3\times A_4):C_{12}$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$(C_3\times A_4):C_{12}$

$\rhd$

$C_6^2:C_6$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$(C_3\times A_4):C_{12}$

$\rhd$

$C_3\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 34 larger groups in the database.

This group is a maximal quotient of 24 larger groups in the database.

Character theory

Complex character table

See the $38 \times 38$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $22 \times 22$ rational character table.