Abstract group 72.49: $C_6:D_6$

• Label: 72.49
• Order: \( 2^{3} \cdot 3^{2} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{2} \)
• Perm deg.: $10$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_6:D_6$
Order:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$S_4\times C_3^2:\GL(2,3)$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \)
Composition factors:	$C_2$ x 3, $C_3$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	3	6
Elements	1	39	8	24	72
Conjugacy classes	1	7	4	12	24
Divisions	1	7	4	12	24
Autjugacy classes	1	2	1	1	5

Dimension	1	2
Irr. complex chars.	8	16	24
Irr. rational chars.	8	16	24

Minimal presentations

Permutation degree:	$10$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$7$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	4	4

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{6}=c^{6}=[b,c]=1, b^{a}=b^{5}, c^{a}=c^{5} \rangle$
Permutation group:	Degree $10$ $\langle(2,3)(4,5)(7,8), (4,5)(9,10), (9,10), (6,7,8), (1,2,3)(6,8,7)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & -1 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\Z)$
	$\left\langle \left(\begin{array}{rrrr} 2 & 1 & 1 & 2 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 1 & 0 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 2 & 0 \\ 0 & 2 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 0 & 2 & 0 & 2 \\ 0 & 1 & 0 & 0 \\ 2 & 2 & 1 & 2 \\ 1 & 1 & 0 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	36T44				more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_3:S_3)$
Semidirect product:	$C_6$ $\,\rtimes\,$ $D_6$ (12)	$C_6^2$ $\,\rtimes\,$ $C_2$	$C_3^2$ $\,\rtimes\,$ $C_2^3$	$(C_2\times C_6)$ $\,\rtimes\,$ $S_3$ (4)	all 6
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} $
Schur multiplier:	$C_{2}^{2} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 272 subgroups in 96 conjugacy classes, 41 normal (5 characteristic).

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

Special subgroups

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

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 72: $C_6:D_6$

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

Order 24: $C_2\times D_6$ x 4

Order 18: $C_3:S_3$ x 4, $C_3\times C_6$ x 3

Order 12: $D_6$ x 24, $C_2\times C_6$ x 4

Order 9: $C_3^2$

Order 8: $C_2^3$

Order 6: $S_3$ x 16, $C_6$ x 12

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 4

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 72: $C_6:D_6$

Order 36: $C_6^2$, $C_6:S_3$

Order 24: $C_2\times D_6$

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

Order 12: $C_2\times C_6$, $D_6$

Order 9: $C_3^2$

Order 8: $C_2^3$

Order 6: $C_6$, $S_3$

Order 4: $C_2^2$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 72: $C_6:D_6$ ($C_1$)

Order 36: $C_6:S_3$ ($C_2$) x 6, $C_6^2$ ($C_2$)

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

Order 12: $C_2\times C_6$ ($S_3$) x 4

Order 9: $C_3^2$ ($C_2^3$)

Order 6: $C_6$ ($D_6$) x 12

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

Order 3: $C_3$ ($C_2\times D_6$) x 4

Order 2: $C_2$ ($C_6:S_3$) x 3

Order 1: $C_1$ ($C_6:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 72: $C_6:D_6$ ($C_1$)

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

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

Order 12: $C_2\times C_6$ ($S_3$)

Order 9: $C_3^2$ ($C_2^3$)

Order 6: $C_6$ ($D_6$)

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

Order 3: $C_3$ ($C_2\times D_6$)

Order 2: $C_2$ ($C_6:S_3$)

Order 1: $C_1$ ($C_6:D_6$)

Series

Derived series

$C_6:D_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_6:D_6$

$\rhd$

$C_6:S_3$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6:D_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 45 larger groups in the database.

This group is a maximal quotient of 45 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $24 \times 24$ rational character table. Alternatively, you may search for characters of this group with desired properties.