Abstract group 81.8: $C_3.\He_3$

• Label: 81.8
• Order: \( 3^{4} \)
• Exponent: \( 3^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 3^{2} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $27$
• Trans deg.: $27$
• Rank: $2$

Group information

Description:	$C_3.\He_3$
Order:	\(81\)\(\medspace = 3^{4} \)
Exponent:	\(9\)\(\medspace = 3^{2} \)
Automorphism group:	$C_3^3:D_6$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \)
Composition factors:	$C_3$ x 4
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	26	54	81
Conjugacy classes	1	6	10	17
Divisions	1	3	3	7
Autjugacy classes	1	3	2	6

Dimension	1	2	3	6	18
Irr. complex chars.	9	0	8	0	0	17
Irr. rational chars.	1	4	0	1	1	7

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$27$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	6	18
Arbitrary	3	6	18

Constructions

Presentation:	$\langle a, b, c \mid a^{3}=b^{3}=c^{9}=[b,c]=1, b^{a}=bc^{3}, c^{a}=bc^{7} \rangle$
Permutation group:	Degree $27$ $\langle(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 11 \\ 3 & 10 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 10 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/27\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 49 & 41 \\ 3 & 40 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/54\Z)$
Transitive group:	27T20	27T26			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3\times C_9)$ $\,\rtimes\,$ $C_3$	$(C_9:C_3)$ $\,\rtimes\,$ $C_3$ (2)			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$\He_3$ . $C_3$	$C_3$ . $\He_3$	$C_3^2$ . $C_3^2$		more information

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

Homology

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

Subgroups

There are 32 subgroups in 14 conjugacy classes, 8 normal (6 characteristic).

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

Special subgroups

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

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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Classes of subgroups up to conjugation

Order 81: $C_3.\He_3$

Order 27: $C_9:C_3$ x 2, $\He_3$, $C_3\times C_9$

Order 9: $C_9$ x 3, $C_3^2$ x 2

Order 3: $C_3$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 81: $C_3.\He_3$

Order 27: $C_9:C_3$, $\He_3$, $C_3\times C_9$

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

Order 3: $C_3$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 81: $C_3.\He_3$ ($C_1$)

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

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

Order 3: $C_3$ ($\He_3$)

Order 1: $C_1$ ($C_3.\He_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 81: $C_3.\He_3$ ($C_1$)

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

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

Order 3: $C_3$ ($\He_3$)

Order 1: $C_1$ ($C_3.\He_3$)

Series

Derived series	$C_3.\He_3$	$\rhd$	$C_3^2$	$\rhd$	$C_1$
Chief series	$C_3.\He_3$	$\rhd$	$C_3\times C_9$	$\rhd$	$C_3^2$	$\rhd$	$C_3$	$\rhd$	$C_1$
Lower central series	$C_3.\He_3$	$\rhd$	$C_3^2$	$\rhd$	$C_3$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_3$	$\lhd$	$C_3^2$	$\lhd$	$C_3.\He_3$

Supergroups

This group is a maximal subgroup of 28 larger groups in the database.

This group is a maximal quotient of 30 larger groups in the database.

Character theory

Complex character table

See the $17 \times 17$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	3A	3B	3C	9A	9B	9C
	Size	1	2	6	18	18	18	18
	3 P	1A	3A	3B	3C	9A	9B	9C
81.8.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$
81.8.1b		$2$	$2$	$2$	$-1$	$-1$	$-1$	$2$
81.8.1c		$2$	$2$	$2$	$-1$	$-1$	$2$	$-1$
81.8.1d		$2$	$2$	$2$	$-1$	$2$	$-1$	$-1$
81.8.1e		$2$	$2$	$2$	$2$	$-1$	$-1$	$-1$
81.8.3a		$6$	$6$	$-3$	$0$	$0$	$0$	$0$
81.8.3b		$18$	$-9$	$0$	$0$	$0$	$0$	$0$