Abstract group 972.599: $C_3^3.C_6^2$

• Label: 972.599
• Order: \( 2^{2} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{2} \)
• Perm deg.: $20$
• Trans deg.: $108$
• Rank: $2$

Group information

Description:	$C_3^3.C_6^2$
Order:	\(972\)\(\medspace = 2^{2} \cdot 3^{5} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$(C_3^2\times \He_3).C_2^4$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \)
Composition factors:	$C_2$ x 2, $C_3$ x 5
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	19	80	224	162	486	972
Conjugacy classes	1	3	20	36	42	78	180
Divisions	1	3	11	19	7	13	54
Autjugacy classes	1	2	7	9	5	7	31

Dimension	1	2	4	6	12	36
Irr. complex chars.	108	54	0	18	0	0	180
Irr. rational chars.	4	18	8	14	8	2	54

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$108$
Rank:	$2$
Inequivalent generating pairs:	$36$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	36
Arbitrary	6	8	12

Constructions

Presentation:	$\langle a, b, c, d \mid a^{6}=b^{3}=c^{6}=d^{9}=[a,d]=[b,c]=[b,d]=[c,d]=1, b^{a}=b^{2}c^{2}, c^{a}=c^{5} \rangle$
Permutation group:	Degree $20$ $\langle(2,4)(3,6)(5,7)(8,9), (19,20), (2,5,9)(4,7,8)(10,11,13)(12,14,16)(15,17,18) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 4 & 26 \\ 21 & 19 \end{array}\right), \left(\begin{array}{rr} 26 & 0 \\ 0 & 26 \end{array}\right), \left(\begin{array}{rr} 10 & 0 \\ 0 & 10 \end{array}\right), \left(\begin{array}{rr} 26 & 25 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 4 & 0 \\ 0 & 4 \end{array}\right), \left(\begin{array}{rr} 10 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/27\Z)$
	$\left\langle \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 25 & 8 \\ 12 & 31 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 53 & 13 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 27 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/54\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_9$ $\, \times\, $ $(C_3^2:C_6)$
Semidirect product:	$(C_3^2\times C_9)$ $\,\rtimes\,$ $D_6$	$(C_3^2\times C_{18})$ $\,\rtimes\,$ $S_3$	$C_3^2$ $\,\rtimes\,$ $(S_3\times C_{18})$ (3)	$(C_3^2\times C_{18})$ $\,\rtimes\,$ $C_6$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^3$ . $C_6^2$	$C_3^3$ . $(C_6\times S_3)$	$(C_3^3:C_6)$ . $C_6$ (2)	$C_6$ . $(C_3^3:C_6)$	all 18

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

Homology

Abelianization:	$C_{6} \times C_{18} \simeq C_{2}^{2} \times C_{3} \times C_{9}$
Schur multiplier:	$C_{3} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 849 subgroups in 233 conjugacy classes, 76 normal (31 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{18}$	$G/Z \simeq$ $C_3^2:C_6$
Commutator:	$G' \simeq$ $C_3^2$	$G/G' \simeq$ $C_6\times C_{18}$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_3^2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{18}\times \He_3$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_3^3.C_6^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $S_3\times C_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9\times \He_3$

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

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Classes of subgroups up to conjugation

Order 972: $C_3^3.C_6^2$

Order 486: $\He_3:C_{18}$ x 2, $C_{18}\times \He_3$

Order 324: $C_2\times C_3^2:C_{18}$ x 2, $C_3:S_3\times C_{18}$, $C_3^2:C_6^2$, $C_3^2.C_6^2$

Order 243: $C_9\times \He_3$

Order 162: $C_3^2:C_{18}$ x 5, $C_3^2:C_{18}$ x 4, $C_3^2\times C_{18}$ x 3, $C_3^2:C_{18}$ x 2, $C_3^3:C_6$ x 2, $C_3^2:C_{18}$ x 2, $C_6\times \He_3$

Order 108: $S_3\times C_{18}$ x 4, $C_3^2:D_6$ x 3, $C_3^2:D_6$, $C_3^2\times D_6$, $C_6\times C_{18}$

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

Order 54: $C_3\times C_{18}$ x 15, $S_3\times C_9$ x 8, $C_3^2:C_6$ x 6, $C_2\times \He_3$ x 6, $C_3^2\times C_6$ x 3, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2

Order 36: $C_6\times S_3$ x 5, $C_2\times C_{18}$ x 3, $C_6^2$, $C_6:S_3$

Order 27: $C_3\times C_9$ x 13, $\He_3$ x 6, $C_3^3$ x 3

Order 18: $C_3\times C_6$ x 18, $C_{18}$ x 13, $C_3\times S_3$ x 10, $C_3:S_3$ x 2

Order 12: $C_2\times C_6$ x 4, $D_6$ x 2

Order 9: $C_3^2$ x 16, $C_9$ x 7

Order 6: $C_6$ x 19, $S_3$ x 4

Order 4: $C_2^2$

Order 3: $C_3$ x 11

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 972: $C_3^3.C_6^2$

Order 486: $\He_3:C_{18}$, $C_{18}\times \He_3$

Order 324: $C_2\times C_3^2:C_{18}$, $C_3:S_3\times C_{18}$, $C_3^2:C_6^2$, $C_3^2.C_6^2$

Order 243: $C_9\times \He_3$

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

Order 108: $S_3\times C_{18}$ x 3, $C_3^2:D_6$, $C_3^2\times D_6$, $C_6\times C_{18}$, $C_3^2:D_6$

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

Order 54: $C_3\times C_{18}$ x 11, $S_3\times C_9$ x 3, $C_3^2\times C_6$ x 3, $C_2\times \He_3$ x 2, $C_3^2:C_6$, $C_3^2:C_6$, $S_3\times C_3^2$

Order 36: $C_6\times S_3$ x 3, $C_2\times C_{18}$ x 2, $C_6^2$, $C_6:S_3$

Order 27: $C_3\times C_9$ x 10, $C_3^3$ x 3, $\He_3$ x 2

Order 18: $C_3\times C_6$ x 12, $C_{18}$ x 7, $C_3\times S_3$ x 3, $C_3:S_3$

Order 12: $C_2\times C_6$ x 2, $D_6$ x 2

Order 9: $C_3^2$ x 11, $C_9$ x 5

Order 6: $C_6$ x 9, $S_3$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 7

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 972: $C_3^3.C_6^2$ ($C_1$)

Order 486: $\He_3:C_{18}$ ($C_2$) x 2, $C_{18}\times \He_3$ ($C_2$)

Order 324: $C_2\times C_3^2:C_{18}$ ($C_3$) x 2, $C_3:S_3\times C_{18}$ ($C_3$), $C_3^2:C_6^2$ ($C_3$)

Order 243: $C_9\times \He_3$ ($C_2^2$)

Order 162: $C_3^2:C_{18}$ ($C_6$) x 4, $C_3^2:C_{18}$ ($C_6$) x 2, $C_3^3:C_6$ ($C_6$) x 2, $C_3^2:C_{18}$ ($C_6$) x 2, $C_6\times \He_3$ ($C_6$), $C_3^2\times C_{18}$ ($C_6$), $C_3^2\times C_{18}$ ($S_3$)

Order 108: $C_3^2:D_6$ ($C_9$) x 3, $C_3^2:D_6$ ($C_3^2$)

Order 81: $C_3^2:C_9$ ($C_2\times C_6$) x 2, $C_3\times \He_3$ ($C_2\times C_6$), $C_3^2\times C_9$ ($C_2\times C_6$), $C_3^2\times C_9$ ($D_6$)

Order 54: $C_3^2:C_6$ ($C_{18}$) x 6, $C_3\times C_{18}$ ($C_3\times S_3$) x 3, $C_2\times \He_3$ ($C_{18}$) x 3, $C_3^2:C_6$ ($C_3\times C_6$) x 2, $C_3^2\times C_6$ ($C_3\times C_6$), $C_3^2\times C_6$ ($C_3\times S_3$)

Order 36: $C_6:S_3$ ($C_3\times C_9$)

Order 27: $\He_3$ ($C_2\times C_{18}$) x 3, $C_3\times C_9$ ($C_6\times S_3$) x 3, $C_3^3$ ($C_6^2$), $C_3^3$ ($C_6\times S_3$)

Order 18: $C_3\times C_6$ ($S_3\times C_9$) x 3, $C_3:S_3$ ($C_3\times C_{18}$) x 2, $C_3\times C_6$ ($C_3\times C_{18}$), $C_3\times C_6$ ($S_3\times C_3^2$), $C_{18}$ ($C_3^2:C_6$)

Order 9: $C_3^2$ ($S_3\times C_{18}$) x 3, $C_3^2$ ($C_3^2\times D_6$), $C_3^2$ ($C_6\times C_{18}$), $C_9$ ($C_3^2:D_6$)

Order 6: $C_6$ ($C_3^3:C_6$), $C_6$ ($C_3^2:C_{18}$)

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

Order 2: $C_2$ ($\He_3:C_{18}$)

Order 1: $C_1$ ($C_3^3.C_6^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 972: $C_3^3.C_6^2$ ($C_1$)

Order 486: $\He_3:C_{18}$ ($C_2$), $C_{18}\times \He_3$ ($C_2$)

Order 324: $C_2\times C_3^2:C_{18}$ ($C_3$), $C_3:S_3\times C_{18}$ ($C_3$), $C_3^2:C_6^2$ ($C_3$)

Order 243: $C_9\times \He_3$ ($C_2^2$)

Order 162: $C_6\times \He_3$ ($C_6$), $C_3^2\times C_{18}$ ($C_6$), $C_3^2\times C_{18}$ ($S_3$), $C_3^2:C_{18}$ ($C_6$), $C_3^2:C_{18}$ ($C_6$), $C_3^3:C_6$ ($C_6$), $C_3^2:C_{18}$ ($C_6$)

Order 108: $C_3^2:D_6$ ($C_3^2$), $C_3^2:D_6$ ($C_9$)

Order 81: $C_3^2:C_9$ ($C_2\times C_6$), $C_3\times \He_3$ ($C_2\times C_6$), $C_3^2\times C_9$ ($C_2\times C_6$), $C_3^2\times C_9$ ($D_6$)

Order 54: $C_3\times C_{18}$ ($C_3\times S_3$) x 2, $C_3^2:C_6$ ($C_{18}$), $C_3^2\times C_6$ ($C_3\times C_6$), $C_3^2\times C_6$ ($C_3\times S_3$), $C_3^2:C_6$ ($C_3\times C_6$), $C_2\times \He_3$ ($C_{18}$)

Order 36: $C_6:S_3$ ($C_3\times C_9$)

Order 27: $C_3\times C_9$ ($C_6\times S_3$) x 2, $C_3^3$ ($C_6^2$), $C_3^3$ ($C_6\times S_3$), $\He_3$ ($C_2\times C_{18}$)

Order 18: $C_3\times C_6$ ($C_3\times C_{18}$), $C_3\times C_6$ ($S_3\times C_9$), $C_3\times C_6$ ($S_3\times C_3^2$), $C_3:S_3$ ($C_3\times C_{18}$), $C_{18}$ ($C_3^2:C_6$)

Order 9: $C_3^2$ ($C_3^2\times D_6$), $C_3^2$ ($C_6\times C_{18}$), $C_3^2$ ($S_3\times C_{18}$), $C_9$ ($C_3^2:D_6$)

Order 6: $C_6$ ($C_3^3:C_6$), $C_6$ ($C_3^2:C_{18}$)

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

Order 2: $C_2$ ($\He_3:C_{18}$)

Order 1: $C_1$ ($C_3^3.C_6^2$)

Series

Derived series

$C_3^3.C_6^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_3^3.C_6^2$

$\rhd$

$C_{18}\times \He_3$

$\rhd$

$C_3^2\times C_{18}$

$\rhd$

$C_3\times C_{18}$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^3.C_6^2$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_{18}$

Supergroups

This group is a maximal subgroup of 21 larger groups in the database.

This group is a maximal quotient of 7 larger groups in the database.

Character theory

Complex character table

See the $180 \times 180$ 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 $54 \times 54$ rational character table.