Abstract group 729.355: $C_3^3.C_3^3$

• Label: 729.355
• Order: \( 3^{6} \)
• Exponent: \( 3^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 3^{3} \)
• $\card{Z(G)}$: \( 3^{2} \)
• $\card{\Aut(G)}$: \( 2 \cdot 3^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3^{4} \)
• Perm deg.: $36$
• Trans deg.: $243$
• Rank: $3$

Group information

Description:	$C_3^3.C_3^3$
Order:	\(729\)\(\medspace = 3^{6} \)
Exponent:	\(9\)\(\medspace = 3^{2} \)
Automorphism group:	$C_3^4\times C_3^3:S_3$, of order \(13122\)\(\medspace = 2 \cdot 3^{8} \)
Composition factors:	$C_3$ x 6
Nilpotency class:	$3$
Derived length:	$2$

This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.

Group statistics

Order	1	3	9
Elements	1	188	540	729
Conjugacy classes	1	24	48	73
Divisions	1	12	14	27
Autjugacy classes	1	14	16	31

Dimension	1	2	3	6	9	18
Irr. complex chars.	27	0	42	0	4	0	73
Irr. rational chars.	1	13	0	6	0	7	27

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$243$
Rank:	$3$
Inequivalent generating triples:	$16848$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	6	12	24

Constructions

Presentation:	$\langle a, b, c, d \mid a^{3}=b^{3}=c^{9}=d^{9}=[a,c]=[b,c]=[b,d]=1, b^{a}=bc^{3}d^{6}, d^{a}=bd, d^{c}=d^{4} \rangle$
Permutation group:	Degree $36$ $\langle(1,2,8)(3,12,19)(4,6,20)(5,10,24)(7,11,25)(9,17,16)(13,27,21)(14,22,26) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_9\times \He_3)$ $\,\rtimes\,$ $C_3$	$(\He_3:C_9)$ $\,\rtimes\,$ $C_3$ (6)	$(C_9:\He_3)$ $\,\rtimes\,$ $C_3$ (2)	$\He_3$ $\,\rtimes\,$ $(C_9:C_3)$ (3)	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^3$ . $C_3^3$	$(C_3\times C_9)$ . $\He_3$ (3)	$C_3$ . $(C_9.\He_3)$	$C_3$ . $(C_9.\He_3)$	all 13

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

Homology

Abelianization:	$C_{3}^{3} $
Schur multiplier:	$C_{3}^{3}$
Commutator length:	$1$

Subgroups

There are 562 subgroups in 134 conjugacy classes, 43 normal (20 characteristic).

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

Special subgroups

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

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

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Classes of subgroups up to conjugation

Order 729: $C_3^3.C_3^3$

Order 243: $\He_3:C_9$ x 6, $C_9:\He_3$ x 2, $C_3^3.C_3^2$ x 2, $\He_3:C_3^2$, $C_9\times \He_3$, $C_3^2.C_3^3$

Order 81: $C_3^2:C_9$ x 12, $C_3^2\times C_9$ x 5, $\He_3:C_3$ x 3, $C_9:C_9$ x 3, $C_3\times \He_3$ x 3, $C_9:C_3^2$ x 2

Order 27: $C_3\times C_9$ x 23, $\He_3$ x 11, $C_9:C_3$ x 6, $C_3^3$ x 4

Order 9: $C_3^2$ x 21, $C_9$ x 14

Order 3: $C_3$ x 12

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 729: $C_3^3.C_3^3$

Order 243: $\He_3:C_9$ x 4, $\He_3:C_3^2$, $C_9:\He_3$, $C_9\times \He_3$, $C_3^2.C_3^3$, $C_3^3.C_3^2$

Order 81: $C_3^2:C_9$ x 6, $C_3^2\times C_9$ x 5, $C_9:C_9$ x 2, $C_3\times \He_3$ x 2, $\He_3:C_3$, $C_9:C_3^2$

Order 27: $C_3\times C_9$ x 14, $\He_3$ x 5, $C_3^3$ x 3, $C_9:C_3$

Order 9: $C_3^2$ x 14, $C_9$ x 8

Order 3: $C_3$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 729: $C_3^3.C_3^3$ ($C_1$)

Order 243: $\He_3:C_9$ ($C_3$) x 6, $C_9:\He_3$ ($C_3$) x 2, $C_3^3.C_3^2$ ($C_3$) x 2, $\He_3:C_3^2$ ($C_3$), $C_9\times \He_3$ ($C_3$), $C_3^2.C_3^3$ ($C_3$)

Order 81: $C_3^2:C_9$ ($C_3^2$) x 6, $C_3^2\times C_9$ ($C_3^2$) x 4, $C_3\times \He_3$ ($C_3^2$) x 3

Order 27: $\He_3$ ($C_9:C_3$) x 3, $C_3\times C_9$ ($\He_3$) x 3, $C_3^3$ ($C_3^3$)

Order 9: $C_3^2$ ($C_9.C_3^2$) x 2, $C_3^2$ ($C_9:C_3^2$), $C_3^2$ ($C_3\times \He_3$)

Order 3: $C_3$ ($C_9.\He_3$), $C_3$ ($\He_3:C_3^2$), $C_3$ ($C_9.\He_3$), $C_3$ ($C_9:\He_3$)

Order 1: $C_1$ ($C_3^3.C_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 729: $C_3^3.C_3^3$ ($C_1$)

Order 243: $\He_3:C_9$ ($C_3$) x 4, $\He_3:C_3^2$ ($C_3$), $C_9:\He_3$ ($C_3$), $C_9\times \He_3$ ($C_3$), $C_3^2.C_3^3$ ($C_3$), $C_3^3.C_3^2$ ($C_3$)

Order 81: $C_3^2\times C_9$ ($C_3^2$) x 4, $C_3^2:C_9$ ($C_3^2$) x 3, $C_3\times \He_3$ ($C_3^2$) x 2

Order 27: $C_3\times C_9$ ($\He_3$) x 2, $C_3^3$ ($C_3^3$), $\He_3$ ($C_9:C_3$)

Order 9: $C_3^2$ ($C_9.C_3^2$), $C_3^2$ ($C_9:C_3^2$), $C_3^2$ ($C_3\times \He_3$)

Order 3: $C_3$ ($C_9.\He_3$), $C_3$ ($\He_3:C_3^2$), $C_3$ ($C_9.\He_3$), $C_3$ ($C_9:\He_3$)

Order 1: $C_1$ ($C_3^3.C_3^3$)

Series

Derived series

$C_3^3.C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$C_3^3.C_3^3$

$\rhd$

$C_3^2.C_3^3$

$\rhd$

$C_3^2\times C_9$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^3.C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_3^2$

$\lhd$

$C_3^2\times C_9$

$\lhd$

$C_3^3.C_3^3$

Supergroups

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

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $73 \times 73$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $27 \times 27$ rational character table.