Abstract group 486.213: $C_3\times C_6.\He_3$

• Label: 486.213
• Order: \( 2 \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{5} \)
• Perm deg.: $32$
• Trans deg.: $162$
• Rank: $3$

Group information

Description:	$C_3\times C_6.\He_3$
Order:	\(486\)\(\medspace = 2 \cdot 3^{5} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^5.(C_3\times S_3^2)$, of order \(26244\)\(\medspace = 2^{2} \cdot 3^{8} \)
Composition factors:	$C_2$, $C_3$ x 5
Nilpotency class:	$3$
Derived length:	$2$

This group is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.

Group statistics

Order	1	2	3	6	9	18
Elements	1	1	26	26	216	216	486
Conjugacy classes	1	1	14	14	36	36	102
Divisions	1	1	7	7	12	12	40
Autjugacy classes	1	1	6	6	2	2	18

Dimension	1	2	3	6	18
Irr. complex chars.	54	0	48	0	0	102
Irr. rational chars.	2	26	0	6	6	40

Minimal presentations

Permutation degree:	$32$
Transitive degree:	$162$
Rank:	$3$
Inequivalent generating triples:	$2184$

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:	$\langle a, b, c, d \mid a^{3}=c^{3}=d^{18}=[a,b]=[a,c]=[a,d]=[c,d]=1, b^{3}=d^{6}, c^{b}=cd^{6}, d^{b}=cd^{7} \rangle$
Permutation group:	Degree $32$ $\langle(1,2), (3,4,7,6,9,16,14,10,20)(5,11,15,13,8,17,24,19,27)(12,22,25,23,21,26,29,18,28) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 26 & 0 \\ 0 & 26 \end{array}\right), \left(\begin{array}{rr} 10 & 0 \\ 0 & 10 \end{array}\right), \left(\begin{array}{rr} 16 & 14 \\ 3 & 19 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 18 & 19 \end{array}\right), \left(\begin{array}{rr} 10 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/27\Z)$
	$\left\langle \left(\begin{array}{rr} 19 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 31 & 50 \\ 30 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 24 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 27 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/54\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_3.\He_3)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^2\times C_9)$ . $C_6$	$(C_9:C_3^2)$ . $C_6$	$(C_9:C_6)$ . $C_3^2$	$C_3^3$ . $(C_3\times C_6)$	all 16

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{27}\Z)$.

Homology

Abelianization:	$C_{3}^{2} \times C_{6} \simeq C_{2} \times C_{3}^{3}$
Schur multiplier:	$C_{3}^{3}$
Commutator length:	$1$

Subgroups

There are 252 subgroups in 132 conjugacy classes, 72 normal (14 characteristic).

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

Special subgroups

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

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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Classes of subgroups up to conjugation

Order 486: $C_3\times C_6.\He_3$

Order 243: $C_3^2.C_3^3$

Order 162: $C_6.\He_3$ x 9, $C_{18}:C_3^2$ x 3, $C_3^2\times C_{18}$

Order 81: $C_3.\He_3$ x 9, $C_9:C_3^2$ x 3, $C_3^2\times C_9$

Order 54: $C_9:C_6$ x 15, $C_3\times C_{18}$ x 9, $C_3^2\times C_6$

Order 27: $C_9:C_3$ x 15, $C_3\times C_9$ x 9, $C_3^3$

Order 18: $C_{18}$ x 12, $C_3\times C_6$ x 7

Order 9: $C_9$ x 12, $C_3^2$ x 7

Order 6: $C_6$ x 7

Order 3: $C_3$ x 7

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 486: $C_3\times C_6.\He_3$

Order 243: $C_3^2.C_3^3$

Order 162: $C_{18}:C_3^2$, $C_3^2\times C_{18}$, $C_6.\He_3$

Order 81: $C_9:C_3^2$, $C_3^2\times C_9$, $C_3.\He_3$

Order 54: $C_3\times C_{18}$ x 4, $C_9:C_6$ x 2, $C_3^2\times C_6$

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

Order 18: $C_3\times C_6$ x 4, $C_{18}$ x 2

Order 9: $C_3^2$ x 4, $C_9$ x 2

Order 6: $C_6$ x 4

Order 3: $C_3$ x 4

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 486: $C_3\times C_6.\He_3$ ($C_1$)

Order 243: $C_3^2.C_3^3$ ($C_2$)

Order 162: $C_6.\He_3$ ($C_3$) x 9, $C_{18}:C_3^2$ ($C_3$) x 3, $C_3^2\times C_{18}$ ($C_3$)

Order 81: $C_3.\He_3$ ($C_6$) x 9, $C_9:C_3^2$ ($C_6$) x 3, $C_3^2\times C_9$ ($C_6$)

Order 54: $C_9:C_6$ ($C_3^2$) x 9, $C_3\times C_{18}$ ($C_3^2$) x 3, $C_3^2\times C_6$ ($C_3^2$)

Order 27: $C_9:C_3$ ($C_3\times C_6$) x 9, $C_3\times C_9$ ($C_3\times C_6$) x 3, $C_3^3$ ($C_3\times C_6$)

Order 18: $C_3\times C_6$ ($\He_3$) x 3, $C_3\times C_6$ ($C_3^3$)

Order 9: $C_3^2$ ($C_2\times \He_3$) x 3, $C_3^2$ ($C_3^2\times C_6$)

Order 6: $C_6$ ($C_3.\He_3$) x 3, $C_6$ ($C_3\times \He_3$)

Order 3: $C_3$ ($C_6.\He_3$) x 3, $C_3$ ($C_6\times \He_3$)

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

Order 1: $C_1$ ($C_3\times C_6.\He_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 486: $C_3\times C_6.\He_3$ ($C_1$)

Order 243: $C_3^2.C_3^3$ ($C_2$)

Order 162: $C_{18}:C_3^2$ ($C_3$), $C_3^2\times C_{18}$ ($C_3$), $C_6.\He_3$ ($C_3$)

Order 81: $C_9:C_3^2$ ($C_6$), $C_3^2\times C_9$ ($C_6$), $C_3.\He_3$ ($C_6$)

Order 54: $C_3\times C_{18}$ ($C_3^2$), $C_3^2\times C_6$ ($C_3^2$), $C_9:C_6$ ($C_3^2$)

Order 27: $C_3^3$ ($C_3\times C_6$), $C_9:C_3$ ($C_3\times C_6$), $C_3\times C_9$ ($C_3\times C_6$)

Order 18: $C_3\times C_6$ ($\He_3$) x 2, $C_3\times C_6$ ($C_3^3$)

Order 9: $C_3^2$ ($C_2\times \He_3$) x 2, $C_3^2$ ($C_3^2\times C_6$)

Order 6: $C_6$ ($C_3\times \He_3$), $C_6$ ($C_3.\He_3$)

Order 3: $C_3$ ($C_6\times \He_3$), $C_3$ ($C_6.\He_3$)

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

Order 1: $C_1$ ($C_3\times C_6.\He_3$)

Series

Derived series

$C_3\times C_6.\He_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_3\times C_6.\He_3$

$\rhd$

$C_3^2.C_3^3$

$\rhd$

$C_9:C_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3\times C_6.\He_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_3\times C_6$

$\lhd$

$C_3^2\times C_6$

$\lhd$

$C_3\times C_6.\He_3$

Supergroups

This group is a maximal subgroup of 62 larger groups in the database.

This group is a maximal quotient of 38 larger groups in the database.

Character theory

Complex character table

See the $102 \times 102$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.