Abstract group 972.564: $C_6^2:\He_3$

• Label: 972.564
• Order: \( 2^{2} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 3^{3} \)
• $\card{Z(G)}$: \( 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{6} \)
• Perm deg.: $22$
• Trans deg.: $108$
• Rank: $3$

Group information

Description:	$C_6^2:\He_3$
Order:	\(972\)\(\medspace = 2^{2} \cdot 3^{5} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$(C_3^4\times C_6^2).S_3^3$, of order \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \)
Composition factors:	$C_2$ x 2, $C_3$ x 5
Derived length:	$2$

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

Group statistics

Order	1	2	3	6
Elements	1	3	728	240	972
Conjugacy classes	1	1	50	32	84
Divisions	1	1	25	16	43
Autjugacy classes	1	1	6	4	12

Dimension	1	2	3	6	9	18
Irr. complex chars.	27	0	51	0	6	0	84
Irr. rational chars.	1	13	1	25	0	3	43

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$108$
Rank:	$3$
Inequivalent generating triples:	$780$

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:	${\langle a, b, c, d, e \mid a^{3}=b^{3}=c^{3}=d^{6}=e^{6}=[a,b]=[a,c]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $22$ $\langle(1,2,5)(3,6,8)(4,7,9)(11,13,12)(15,17,20)(18,22,21), (1,3,4)(2,6,7)(5,8,9) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_6^2$ $\,\rtimes\,$ $\He_3$	$(A_4\times C_3^3)$ $\,\rtimes\,$ $C_3$	$(C_3^3:A_4)$ $\,\rtimes\,$ $C_3$ (2)	$C_3^3$ $\,\rtimes\,$ $(C_3\times A_4)$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $C_3^3$	$C_3^3$ . $(C_3\times A_4)$	$C_3$ . $(C_3^3:A_4)$	$C_3$ . $(A_4\times \He_3)$	all 8

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

Homology

Abelianization:	$C_{3}^{3} $
Schur multiplier:	$C_{3}^{5} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 2655 subgroups in 434 conjugacy classes, 59 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3^2$	$G/Z \simeq$ $C_3^2\times A_4$
Commutator:	$G' \simeq$ $C_6^2$	$G/G' \simeq$ $C_3^3$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_3^2\times A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2:C_6^2$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_6^2:\He_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_3^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4:C_3$

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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Subgroup information

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Classes of subgroups up to conjugation

Order 972: $C_6^2:\He_3$

Order 324: $C_3^3:A_4$ x 11, $A_4\times C_3^3$, $C_3^2:C_6^2$

Order 243: $C_3^4:C_3$

Order 162: $C_6\times \He_3$

Order 108: $C_3^2:A_4$ x 33, $C_3^2\times A_4$ x 21, $C_3\times C_6^2$ x 4, $C_2^2\times \He_3$ x 3

Order 81: $C_3\times \He_3$ x 12, $C_3^4$

Order 54: $C_2\times \He_3$ x 9, $C_3^2\times C_6$ x 4

Order 36: $C_3\times A_4$ x 69, $C_6^2$ x 19

Order 27: $\He_3$ x 36, $C_3^3$ x 25

Order 18: $C_3\times C_6$ x 25

Order 12: $A_4$ x 15, $C_2\times C_6$ x 10

Order 9: $C_3^2$ x 88

Order 6: $C_6$ x 16

Order 4: $C_2^2$

Order 3: $C_3$ x 25

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 972: $C_6^2:\He_3$

Order 324: $C_3^3:A_4$ x 2, $A_4\times C_3^3$, $C_3^2:C_6^2$

Order 243: $C_3^4:C_3$

Order 162: $C_6\times \He_3$

Order 108: $C_3^2\times A_4$ x 4, $C_3\times C_6^2$ x 2, $C_3^2:A_4$ x 2, $C_2^2\times \He_3$

Order 81: $C_3\times \He_3$ x 3, $C_3^4$

Order 54: $C_3^2\times C_6$ x 2, $C_2\times \He_3$

Order 36: $C_3\times A_4$ x 6, $C_6^2$ x 5

Order 27: $C_3^3$ x 6, $\He_3$ x 3

Order 18: $C_3\times C_6$ x 5

Order 12: $C_2\times C_6$ x 4, $A_4$ x 2

Order 9: $C_3^2$ x 11

Order 6: $C_6$ x 4

Order 4: $C_2^2$

Order 3: $C_3$ x 6

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 972: $C_6^2:\He_3$ ($C_1$)

Order 324: $C_3^3:A_4$ ($C_3$) x 11, $A_4\times C_3^3$ ($C_3$), $C_3^2:C_6^2$ ($C_3$)

Order 108: $C_3^2\times A_4$ ($C_3^2$) x 9, $C_3\times C_6^2$ ($C_3^2$) x 4

Order 81: $C_3\times \He_3$ ($A_4$)

Order 36: $C_3\times A_4$ ($\He_3$) x 9, $C_6^2$ ($\He_3$) x 3, $C_6^2$ ($C_3^3$)

Order 27: $C_3^3$ ($C_3\times A_4$) x 4

Order 12: $C_2\times C_6$ ($C_3\times \He_3$) x 4

Order 9: $C_3^2$ ($C_3^2:A_4$) x 3, $C_3^2$ ($C_3^2\times A_4$)

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

Order 3: $C_3$ ($A_4\times \He_3$) x 3, $C_3$ ($C_3^3:A_4$)

Order 1: $C_1$ ($C_6^2:\He_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 972: $C_6^2:\He_3$ ($C_1$)

Order 324: $C_3^3:A_4$ ($C_3$) x 2, $A_4\times C_3^3$ ($C_3$), $C_3^2:C_6^2$ ($C_3$)

Order 108: $C_3\times C_6^2$ ($C_3^2$) x 2, $C_3^2\times A_4$ ($C_3^2$) x 2

Order 81: $C_3\times \He_3$ ($A_4$)

Order 36: $C_6^2$ ($C_3^3$), $C_6^2$ ($\He_3$), $C_3\times A_4$ ($\He_3$)

Order 27: $C_3^3$ ($C_3\times A_4$) x 2

Order 12: $C_2\times C_6$ ($C_3\times \He_3$) x 2

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

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

Order 3: $C_3$ ($C_3^3:A_4$), $C_3$ ($A_4\times \He_3$)

Order 1: $C_1$ ($C_6^2:\He_3$)

Series

Derived series

$C_6^2:\He_3$

$\rhd$

$C_6^2$

$\rhd$

$C_1$

Chief series

$C_6^2:\He_3$

$\rhd$

$A_4\times C_3^3$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_6^2:\He_3$

$\rhd$

$C_6^2$

$\rhd$

$C_2^2$

Upper central series

$C_1$

$\lhd$

$C_3^2$

$\lhd$

$C_3\times \He_3$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $43 \times 43$ rational character table.