Abstract group 972.713: $\He_3:C_6^2$

• Label: 972.713
• Order: \( 2^{2} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{3} \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{6} \)
• Perm deg.: $34$
• Trans deg.: $324$
• Rank: $3$

Group information

Description:	$\He_3:C_6^2$
Order:	\(972\)\(\medspace = 2^{2} \cdot 3^{5} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$S_3\times C_3^4.C_3^4.C_2^3$, of order \(314928\)\(\medspace = 2^{4} \cdot 3^{9} \)
Composition factors:	$C_2$ x 2, $C_3$ x 5
Nilpotency class:	$3$
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	3	188	564	54	162	972
Conjugacy classes	1	3	32	96	18	54	204
Divisions	1	3	16	48	3	9	80
Autjugacy classes	1	1	5	5	1	1	14

Dimension	1	2	3	6	18
Irr. complex chars.	108	0	96	0	0	204
Irr. rational chars.	4	52	0	12	12	80

Minimal presentations

Permutation degree:	$34$
Transitive degree:	$324$
Rank:	$3$
Inequivalent generating triples:	$1092$

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:	$\langle a, b, c, d \mid a^{3}=b^{3}=c^{6}=d^{18}=[a,b]=[a,c]=[a,d]=[c,d]=1, c^{b}=cd^{6}, d^{b}=c^{4}d^{13} \rangle$
Permutation group:	Degree $34$ $\langle(1,2), (3,4), (5,6,9)(7,13,17)(8,11,18)(10,19,15)(12,22,16)(14,24,27)(20,30,31) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 53 & 0 \\ 0 & 53 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 24 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 27 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 31 & 38 \\ 30 & 31 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/54\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $(\He_3:C_3)$
Semidirect product:	$\He_3$ $\,\rtimes\,$ $C_6^2$	$(C_3\times C_9)$ $\,\rtimes\,$ $C_6^2$	$(C_6\times \He_3)$ $\,\rtimes\,$ $C_6$	$(C_3^2\times C_{18})$ $\,\rtimes\,$ $C_6$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $C_3^3$	$C_3^3$ . $C_6^2$	$C_6^2$ . $\He_3$ (2)	$C_6$ . $(C_6\times \He_3)$	all 12

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

Homology

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

Subgroups

There are 1440 subgroups in 420 conjugacy classes, 180 normal (14 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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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: $\He_3:C_6^2$

Order 486: $C_3\times \He_3:C_6$ x 3

Order 324: $C_3^2.C_6^2$ x 9, $C_3^2:C_6^2$ x 3, $C_3\times C_6\times C_{18}$

Order 243: $\He_3:C_3^2$

Order 162: $\He_3:C_6$ x 27, $C_6\times \He_3$ x 9, $C_3^2\times C_{18}$ x 3

Order 108: $C_2^2\times \He_3$ x 15, $C_6\times C_{18}$ x 6, $C_3\times C_6^2$ x 4

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

Order 54: $C_2\times \He_3$ x 45, $C_3\times C_{18}$ x 18, $C_3^2\times C_6$ x 12

Order 36: $C_6^2$ x 25, $C_2\times C_{18}$ x 3

Order 27: $\He_3$ x 15, $C_3\times C_9$ x 6, $C_3^3$ x 4

Order 18: $C_3\times C_6$ x 75, $C_{18}$ x 9

Order 12: $C_2\times C_6$ x 16

Order 9: $C_3^2$ x 25, $C_9$ x 3

Order 6: $C_6$ x 48

Order 4: $C_2^2$

Order 3: $C_3$ x 16

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 972: $\He_3:C_6^2$

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

Order 324: $C_3^2.C_6^2$, $C_3^2:C_6^2$, $C_3\times C_6\times C_{18}$

Order 243: $\He_3:C_3^2$

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

Order 108: $C_6\times C_{18}$ x 3, $C_3\times C_6^2$ x 2, $C_2^2\times \He_3$ x 2

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

Order 54: $C_3\times C_{18}$ x 3, $C_3^2\times C_6$ x 2, $C_2\times \He_3$ x 2

Order 36: $C_6^2$ x 6, $C_2\times C_{18}$

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

Order 18: $C_3\times C_6$ x 6, $C_{18}$

Order 12: $C_2\times C_6$ x 5

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

Order 6: $C_6$ x 5

Order 4: $C_2^2$

Order 3: $C_3$ x 5

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 486: $C_3\times \He_3:C_6$ ($C_2$) x 3

Order 324: $C_3^2.C_6^2$ ($C_3$) x 9, $C_3^2:C_6^2$ ($C_3$) x 3, $C_3\times C_6\times C_{18}$ ($C_3$)

Order 243: $\He_3:C_3^2$ ($C_2^2$)

Order 162: $\He_3:C_6$ ($C_6$) x 27, $C_6\times \He_3$ ($C_6$) x 9, $C_3^2\times C_{18}$ ($C_6$) x 3

Order 108: $C_2^2\times \He_3$ ($C_3^2$) x 9, $C_6\times C_{18}$ ($C_3^2$) x 3, $C_3\times C_6^2$ ($C_3^2$)

Order 81: $\He_3:C_3$ ($C_2\times C_6$) x 9, $C_3\times \He_3$ ($C_2\times C_6$) x 3, $C_3^2\times C_9$ ($C_2\times C_6$)

Order 54: $C_2\times \He_3$ ($C_3\times C_6$) x 27, $C_3\times C_{18}$ ($C_3\times C_6$) x 9, $C_3^2\times C_6$ ($C_3\times C_6$) x 3

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

Order 27: $\He_3$ ($C_6^2$) x 9, $C_3\times C_9$ ($C_6^2$) x 3, $C_3^3$ ($C_6^2$)

Order 18: $C_3\times C_6$ ($C_2\times \He_3$) x 9, $C_3\times C_6$ ($C_3^2\times C_6$) x 3

Order 12: $C_2\times C_6$ ($\He_3:C_3$) x 3, $C_2\times C_6$ ($C_3\times \He_3$)

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

Order 6: $C_6$ ($\He_3:C_6$) x 9, $C_6$ ($C_6\times \He_3$) x 3

Order 4: $C_2^2$ ($\He_3:C_3^2$)

Order 3: $C_3$ ($C_3^2.C_6^2$) x 3, $C_3$ ($C_3^2:C_6^2$)

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

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

Normal subgroups up to automorphism (quotient in parentheses)

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

Order 486: $C_3\times \He_3:C_6$ ($C_2$)

Order 324: $C_3^2.C_6^2$ ($C_3$), $C_3^2:C_6^2$ ($C_3$), $C_3\times C_6\times C_{18}$ ($C_3$)

Order 243: $\He_3:C_3^2$ ($C_2^2$)

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

Order 108: $C_3\times C_6^2$ ($C_3^2$), $C_2^2\times \He_3$ ($C_3^2$), $C_6\times C_{18}$ ($C_3^2$)

Order 81: $\He_3:C_3$ ($C_2\times C_6$), $C_3\times \He_3$ ($C_2\times C_6$), $C_3^2\times C_9$ ($C_2\times C_6$)

Order 54: $C_3\times C_{18}$ ($C_3\times C_6$), $C_3^2\times C_6$ ($C_3\times C_6$), $C_2\times \He_3$ ($C_3\times C_6$)

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

Order 27: $C_3^3$ ($C_6^2$), $\He_3$ ($C_6^2$), $C_3\times C_9$ ($C_6^2$)

Order 18: $C_3\times C_6$ ($C_2\times \He_3$) x 2, $C_3\times C_6$ ($C_3^2\times C_6$)

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

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

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

Order 4: $C_2^2$ ($\He_3:C_3^2$)

Order 3: $C_3$ ($C_3^2.C_6^2$), $C_3$ ($C_3^2:C_6^2$)

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

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

Series

Derived series

$\He_3:C_6^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$\He_3:C_6^2$

$\rhd$

$C_3\times \He_3:C_6$

$\rhd$

$\He_3:C_3^2$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$\He_3:C_6^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_6^2$

$\lhd$

$C_3\times C_6^2$

$\lhd$

$\He_3:C_6^2$

Supergroups

This group is a maximal subgroup of 24 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $204 \times 204$ 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 $80 \times 80$ rational character table.