Abstract group 729.121: $C_3^3.C_3^3$

• Label: 729.121
• Order: \( 3^{6} \)
• Exponent: \( 3^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 3^{4} \)
• $\card{Z(G)}$: \( 3^{3} \)
• $\card{\Aut(G)}$: \( 2 \cdot 3^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3^{7} \)
• Perm deg.: $27$
• Trans deg.: $729$
• Rank: $3$

Group information

Description:	$C_3^3.C_3^3$
Order:	\(729\)\(\medspace = 3^{6} \)
Exponent:	\(9\)\(\medspace = 3^{2} \)
Automorphism group:	$C_3^6.C_3^4.C_2$, of order \(118098\)\(\medspace = 2 \cdot 3^{10} \)
Composition factors:	$C_3$ x 6
Nilpotency class:	$2$
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	26	702	729
Conjugacy classes	1	26	126	153
Divisions	1	13	33	47
Autjugacy classes	1	11	11	23

Dimension	1	2	3	6	18
Irr. complex chars.	81	0	72	0	0	153
Irr. rational chars.	1	13	0	27	6	47

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$729$
Rank:	$3$
Inequivalent generating triples:	$1872$

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:	$\langle a, b, c \mid a^{9}=b^{9}=c^{9}=[a,c]=1, b^{a}=b^{4}, c^{b}=c^{7} \rangle$
Permutation group:	Degree $27$ $\langle(1,2,5,4,6,9,3,7,8)(10,11,13,12,14,16,15,17,18)(21,24,23)(25,26,27), (2,6,7) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_9:C_9)$ $\,\rtimes\,$ $C_9$	$C_9$ $\,\rtimes\,$ $(C_9:C_9)$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^3$ . $C_3^3$	$(C_3\times C_9^2)$ . $C_3$	$(C_3.C_9^2)$ . $C_3$	$C_3$ . $(C_9^2:C_3)$	all 16

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

Homology

Abelianization:	$C_{3}^{2} \times C_{9} $
Schur multiplier:	$C_{3} \times C_{9}$
Commutator length:	$1$

Subgroups

There are 391 subgroups in 187 conjugacy classes, 103 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3^3$	$G/Z \simeq$ $C_3^3$
Commutator:	$G' \simeq$ $C_3^2$	$G/G' \simeq$ $C_3^2\times C_9$
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:	$\operatorname{soc} \simeq$ $C_3^3$	$G/\operatorname{soc} \simeq$ $C_3^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3.C_3^3$

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

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

Classes of subgroups up to conjugation

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

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

Order 81: $C_9:C_9$ x 24, $C_3^2\times C_9$ x 13, $C_9^2$ x 3

Order 27: $C_3\times C_9$ x 72, $C_3^3$

Order 9: $C_9$ x 33, $C_3^2$ x 13

Order 3: $C_3$ x 13

Order 1: $C_1$

Classes of subgroups up to automorphism

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

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

Order 81: $C_3^2\times C_9$ x 6, $C_9:C_9$ x 4, $C_9^2$

Order 27: $C_3\times C_9$ x 17, $C_3^3$

Order 9: $C_3^2$ x 6, $C_9$ x 6

Order 3: $C_3$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

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

Order 81: $C_3^2\times C_9$ ($C_3^2$) x 13, $C_9:C_9$ ($C_9$) x 9

Order 27: $C_3\times C_9$ ($C_9:C_3$) x 18, $C_3\times C_9$ ($C_3\times C_9$) x 12, $C_3^3$ ($C_3^3$)

Order 9: $C_9$ ($C_9:C_9$) x 9, $C_3^2$ ($C_9.C_3^2$) x 6, $C_3^2$ ($C_9:C_3^2$) x 6, $C_3^2$ ($C_3^2\times C_9$)

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

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

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

Order 27: $C_3\times C_9$ ($C_9:C_3$) x 4, $C_3\times C_9$ ($C_3\times C_9$) x 2, $C_3^3$ ($C_3^3$)

Order 9: $C_3^2$ ($C_9:C_3^2$) x 4, $C_3^2$ ($C_9.C_3^2$), $C_3^2$ ($C_3^2\times C_9$), $C_9$ ($C_9:C_9$)

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

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

Series

Derived series

$C_3^3.C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

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

$C_3^3.C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_3^3$

$\lhd$

$C_3^3.C_3^3$

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 $153 \times 153$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $47 \times 47$ rational character table.