Abstract group 432.767: $C_3\times D_6^2$

• Label: 432.767
• Order: \( 2^{4} \cdot 3^{3} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $13$
• Trans deg.: $48$
• Rank: $4$

Group information

Description:	$C_3\times D_6^2$
Order:	\(432\)\(\medspace = 2^{4} \cdot 3^{3} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$D_6^2:(C_2^2\times S_4)$, of order \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \)
Composition factors:	$C_2$ x 4, $C_3$ x 3
Derived length:	$2$

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

Group statistics

Order	1	2	3	6
Elements	1	63	26	342	432
Conjugacy classes	1	15	11	81	108
Divisions	1	15	7	49	72
Autjugacy classes	1	3	5	9	18

Dimension	1	2	4	8
Irr. complex chars.	48	48	12	0	108
Irr. rational chars.	16	32	20	4	72

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$48$
Rank:	$4$
Inequivalent generating quadruples:	$709800$

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:	$\langle a, b, c, d \mid a^{2}=b^{6}=c^{6}=d^{6}=[a,b]=[a,d]=[b,c]=[c,d]=1, c^{a}=c^{5}, d^{b}=d^{5} \rangle$
Permutation group:	Degree $13$ $\langle(2,3)(4,5)(6,7), (8,9)(10,12)(11,13), (4,6)(5,7), (4,7)(5,6), (8,10,11)(9,12,13), (8,11,10)(9,12,13), (1,2,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 7 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 4 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 14 & 18 \\ 9 & 14 \end{array}\right), \left(\begin{array}{rr} 4 & 0 \\ 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/21\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $S_3$ ${}^2$
Semidirect product:	$C_6^2$ $\,\rtimes\,$ $D_6$	$C_3^3$ $\,\rtimes\,$ $C_2^4$	$(C_6\times D_6)$ $\,\rtimes\,$ $C_6$	$(C_6:D_6)$ $\,\rtimes\,$ $C_6$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 2080 subgroups in 642 conjugacy classes, 208 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $S_3^2$
Commutator:	$G' \simeq$ $C_3^2$	$G/G' \simeq$ $C_2^3\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3\times D_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3\times D_6^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4$
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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 432: $C_3\times D_6^2$

Order 216: $C_6\times S_3^2$ x 12, $S_3\times C_6^2$ x 2, $C_6^2:C_6$

Order 144: $C_6^2:C_2^2$ x 2, $D_6^2$

Order 108: $C_3\times S_3^2$ x 16, $C_3^2\times D_6$ x 12, $C_3^2:D_6$ x 6, $C_3\times C_6^2$

Order 72: $C_6\times D_6$ x 33, $S_3\times D_6$ x 12, $C_2\times C_6^2$ x 2, $C_6:D_6$

Order 54: $S_3\times C_3^2$ x 8, $C_3^2:C_6$ x 4, $C_3^2\times C_6$ x 3

Order 48: $C_2^2\times D_6$ x 2, $C_2^3\times C_6$

Order 36: $C_6\times S_3$ x 86, $C_6^2$ x 19, $S_3^2$ x 16, $C_6:S_3$ x 6

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 29, $C_2^2\times C_6$ x 19

Order 18: $C_3\times S_3$ x 36, $C_3\times C_6$ x 29, $C_3:S_3$ x 4

Order 16: $C_2^4$

Order 12: $C_2\times C_6$ x 65, $D_6$ x 62

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 15

Order 6: $C_6$ x 49, $S_3$ x 20

Order 4: $C_2^2$ x 35

Order 3: $C_3$ x 7

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 432: $C_3\times D_6^2$

Order 216: $C_6^2:C_6$, $S_3\times C_6^2$, $C_6\times S_3^2$

Order 144: $C_6^2:C_2^2$, $D_6^2$

Order 108: $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3\times S_3^2$

Order 72: $C_6\times D_6$ x 6, $C_2\times C_6^2$, $C_6:D_6$, $S_3\times D_6$

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

Order 48: $C_2^3\times C_6$, $C_2^2\times D_6$

Order 36: $C_6^2$ x 6, $C_6\times S_3$ x 6, $C_6:S_3$, $S_3^2$

Order 27: $C_3^3$

Order 24: $C_2^2\times C_6$ x 5, $C_2\times D_6$ x 4

Order 18: $C_3\times C_6$ x 6, $C_3\times S_3$ x 5, $C_3:S_3$

Order 16: $C_2^4$

Order 12: $C_2\times C_6$ x 10, $D_6$ x 4

Order 9: $C_3^2$ x 5

Order 8: $C_2^3$ x 3

Order 6: $C_6$ x 9, $S_3$ x 3

Order 4: $C_2^2$ x 4

Order 3: $C_3$ x 5

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 432: $C_3\times D_6^2$ ($C_1$)

Order 216: $C_6\times S_3^2$ ($C_2$) x 12, $S_3\times C_6^2$ ($C_2$) x 2, $C_6^2:C_6$ ($C_2$)

Order 144: $D_6^2$ ($C_3$)

Order 108: $C_3\times S_3^2$ ($C_2^2$) x 16, $C_3^2\times D_6$ ($C_2^2$) x 12, $C_3^2:D_6$ ($C_2^2$) x 6, $C_3\times C_6^2$ ($C_2^2$)

Order 72: $S_3\times D_6$ ($C_6$) x 12, $C_6\times D_6$ ($C_6$) x 2, $C_6\times D_6$ ($S_3$) x 2, $C_6:D_6$ ($C_6$)

Order 54: $S_3\times C_3^2$ ($C_2^3$) x 8, $C_3^2:C_6$ ($C_2^3$) x 4, $C_3^2\times C_6$ ($C_2^3$) x 3

Order 36: $S_3^2$ ($C_2\times C_6$) x 16, $C_6\times S_3$ ($C_2\times C_6$) x 12, $C_6\times S_3$ ($D_6$) x 12, $C_6:S_3$ ($C_2\times C_6$) x 6, $C_6^2$ ($D_6$) x 2, $C_6^2$ ($C_2\times C_6$)

Order 27: $C_3^3$ ($C_2^4$)

Order 24: $C_2\times D_6$ ($C_3\times S_3$) x 2

Order 18: $C_3\times S_3$ ($C_2^2\times C_6$) x 8, $C_3\times S_3$ ($C_2\times D_6$) x 8, $C_3\times C_6$ ($C_2\times D_6$) x 6, $C_3:S_3$ ($C_2^2\times C_6$) x 4, $C_3\times C_6$ ($C_2^2\times C_6$) x 3

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

Order 9: $C_3^2$ ($C_2^2\times D_6$) x 2, $C_3^2$ ($C_2^3\times C_6$)

Order 6: $S_3$ ($C_6\times D_6$) x 8, $C_6$ ($C_6\times D_6$) x 6, $C_6$ ($S_3\times D_6$) x 3

Order 4: $C_2^2$ ($C_3\times S_3^2$)

Order 3: $C_3$ ($C_6^2:C_2^2$) x 2, $C_3$ ($D_6^2$)

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

Order 1: $C_1$ ($C_3\times D_6^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 432: $C_3\times D_6^2$ ($C_1$)

Order 216: $C_6^2:C_6$ ($C_2$), $S_3\times C_6^2$ ($C_2$), $C_6\times S_3^2$ ($C_2$)

Order 144: $D_6^2$ ($C_3$)

Order 108: $C_3\times C_6^2$ ($C_2^2$), $C_3^2:D_6$ ($C_2^2$), $C_3^2\times D_6$ ($C_2^2$), $C_3\times S_3^2$ ($C_2^2$)

Order 72: $C_6:D_6$ ($C_6$), $C_6\times D_6$ ($C_6$), $C_6\times D_6$ ($S_3$), $S_3\times D_6$ ($C_6$)

Order 54: $C_3^2\times C_6$ ($C_2^3$), $C_3^2:C_6$ ($C_2^3$), $S_3\times C_3^2$ ($C_2^3$)

Order 36: $C_6^2$ ($C_2\times C_6$), $C_6^2$ ($D_6$), $C_6:S_3$ ($C_2\times C_6$), $C_6\times S_3$ ($C_2\times C_6$), $C_6\times S_3$ ($D_6$), $S_3^2$ ($C_2\times C_6$)

Order 27: $C_3^3$ ($C_2^4$)

Order 24: $C_2\times D_6$ ($C_3\times S_3$)

Order 18: $C_3\times C_6$ ($C_2^2\times C_6$), $C_3\times C_6$ ($C_2\times D_6$), $C_3:S_3$ ($C_2^2\times C_6$), $C_3\times S_3$ ($C_2^2\times C_6$), $C_3\times S_3$ ($C_2\times D_6$)

Order 12: $C_2\times C_6$ ($C_6\times S_3$), $C_2\times C_6$ ($S_3^2$), $D_6$ ($C_6\times S_3$)

Order 9: $C_3^2$ ($C_2^3\times C_6$), $C_3^2$ ($C_2^2\times D_6$)

Order 6: $C_6$ ($C_6\times D_6$), $C_6$ ($S_3\times D_6$), $S_3$ ($C_6\times D_6$)

Order 4: $C_2^2$ ($C_3\times S_3^2$)

Order 3: $C_3$ ($C_6^2:C_2^2$), $C_3$ ($D_6^2$)

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

Order 1: $C_1$ ($C_3\times D_6^2$)

Series

Derived series

$C_3\times D_6^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_3\times D_6^2$

$\rhd$

$S_3\times C_6^2$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3\times D_6^2$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

Supergroups

This group is a maximal subgroup of 54 larger groups in the database.

This group is a maximal quotient of 52 larger groups in the database.

Character theory

Complex character table

See the $108 \times 108$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $72 \times 72$ rational character table.