Abstract group 432.624: $C_6^2:C_{12}$

• Label: 432.624
• Order: \( 2^{4} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_6^2:C_{12}$
Order:	\(432\)\(\medspace = 2^{4} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$A_4:D_6^2$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Composition factors:	$C_2$ x 4, $C_3$ x 3
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	7	80	24	128	192	432
Conjugacy classes	1	3	17	4	27	20	72
Divisions	1	3	9	2	15	5	35
Autjugacy classes	1	3	5	2	11	3	25

Dimension	1	2	3	4	6	12
Irr. complex chars.	36	18	12	0	6	0	72
Irr. rational chars.	2	11	2	12	5	3	35

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	12
Arbitrary	5	7	7

Constructions

Presentation:	$\langle a, b, c \mid a^{12}=b^{6}=c^{6}=[b,c]=1, b^{a}=b^{5}c^{3}, c^{a}=b^{5}c^{4} \rangle$
Permutation group:	Degree $14$ $\langle(1,2,3,4)(9,10), (12,14,13), (5,6,7), (1,3)(2,4), (11,12)(13,14), (11,13)(12,14), (8,9,10)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 2 & 21 \\ 21 & 9 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 14 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 8 \\ 6 & 7 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 14 & 15 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/28\Z)$
Transitive group:	36T562				more information
Direct product:	$C_3$ $\, \times\, $ $(C_3:C_4)$ $\, \times\, $ $A_4$
Semidirect product:	$C_6^2$ $\,\rtimes\,$ $C_{12}$	$(C_3\times A_4)$ $\,\rtimes\,$ $C_{12}$ (3)	$C_3$ $\,\rtimes\,$ $(C_{12}\times A_4)$	$(C_3^2\times A_4)$ $\,\rtimes\,$ $C_4$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6\times A_4)$ . $S_3$	$C_6$ . $(S_3\times A_4)$	$(C_6\times A_4)$ . $C_6$ (3)	$C_6$ . $(C_6\times A_4)$	all 12

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

Homology

Abelianization:	$C_{3} \times C_{12} \simeq C_{4} \times C_{3}^{2}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 484 subgroups in 136 conjugacy classes, 40 normal (25 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 432: $C_6^2:C_{12}$

Order 216: $C_6^2:C_6$

Order 144: $A_4\times C_3:C_4$ x 3, $C_6^2:C_4$, $C_{12}\times A_4$

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

Order 72: $C_6\times A_4$ x 8, $C_6:C_{12}$ x 2, $C_2\times C_6^2$

Order 54: $C_3^2\times C_6$

Order 48: $C_4\times A_4$ x 3, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 36: $C_3\times A_4$ x 8, $C_3:C_{12}$ x 5, $C_6^2$ x 3, $C_3\times C_{12}$

Order 27: $C_3^3$

Order 24: $C_2\times A_4$ x 6, $C_2^2\times C_6$ x 3, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2

Order 18: $C_3\times C_6$ x 11

Order 16: $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 9, $A_4$ x 6, $C_{12}$ x 5, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 9

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

Order 6: $C_6$ x 15

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

Order 3: $C_3$ x 9

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 432: $C_6^2:C_{12}$

Order 216: $C_6^2:C_6$

Order 144: $C_6^2:C_4$, $C_{12}\times A_4$, $A_4\times C_3:C_4$

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

Order 72: $C_6\times A_4$ x 4, $C_6:C_{12}$ x 2, $C_2\times C_6^2$

Order 54: $C_3^2\times C_6$

Order 48: $C_2^2\times C_{12}$, $C_6.C_2^3$, $C_4\times A_4$

Order 36: $C_3\times A_4$ x 4, $C_3:C_{12}$ x 3, $C_6^2$ x 3, $C_3\times C_{12}$

Order 27: $C_3^3$

Order 24: $C_2^2\times C_6$ x 3, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2, $C_2\times A_4$ x 2

Order 18: $C_3\times C_6$ x 7

Order 16: $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 9, $C_{12}$ x 3, $A_4$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 5

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

Order 6: $C_6$ x 11

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

Order 3: $C_3$ x 5

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 432: $C_6^2:C_{12}$ ($C_1$)

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

Order 144: $A_4\times C_3:C_4$ ($C_3$) x 3, $C_6^2:C_4$ ($C_3$)

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

Order 72: $C_6\times A_4$ ($C_6$) x 3, $C_2\times C_6^2$ ($C_6$), $C_6\times A_4$ ($S_3$)

Order 48: $C_6.C_2^3$ ($C_3^2$)

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

Order 24: $C_2\times A_4$ ($C_3\times S_3$) x 3, $C_2^2\times C_6$ ($C_3\times C_6$), $C_2^2\times C_6$ ($C_3\times S_3$)

Order 18: $C_3\times C_6$ ($C_2\times A_4$)

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

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

Order 8: $C_2^3$ ($S_3\times C_3^2$)

Order 6: $C_6$ ($C_6\times A_4$), $C_6$ ($S_3\times A_4$)

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

Order 3: $C_3$ ($C_{12}\times A_4$), $C_3$ ($A_4\times C_3:C_4$)

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

Order 1: $C_1$ ($C_6^2:C_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 432: $C_6^2:C_{12}$ ($C_1$)

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

Order 144: $C_6^2:C_4$ ($C_3$), $A_4\times C_3:C_4$ ($C_3$)

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

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

Order 48: $C_6.C_2^3$ ($C_3^2$)

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

Order 24: $C_2^2\times C_6$ ($C_3\times C_6$), $C_2^2\times C_6$ ($C_3\times S_3$), $C_2\times A_4$ ($C_3\times S_3$)

Order 18: $C_3\times C_6$ ($C_2\times A_4$)

Order 12: $C_2\times C_6$ ($C_3\times C_{12}$), $C_2\times C_6$ ($C_3:C_{12}$), $A_4$ ($C_3:C_{12}$), $C_3:C_4$ ($C_3\times A_4$)

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

Order 8: $C_2^3$ ($S_3\times C_3^2$)

Order 6: $C_6$ ($C_6\times A_4$), $C_6$ ($S_3\times A_4$)

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

Order 3: $C_3$ ($C_{12}\times A_4$), $C_3$ ($A_4\times C_3:C_4$)

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

Order 1: $C_1$ ($C_6^2:C_{12}$)

Series

Derived series

$C_6^2:C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$C_6^2:C_{12}$

$\rhd$

$C_6^2:C_6$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^2:C_{12}$

$\rhd$

$C_2\times C_6$

Upper central series

$C_1$

$\lhd$

$C_6$

Supergroups

This group is a maximal subgroup of 48 larger groups in the database.

This group is a maximal quotient of 26 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 $35 \times 35$ rational character table.