Abstract group 648.617: $C_9:C_6\times A_4$

• Label: 648.617
• Order: \( 2^{3} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{2} \)
• Perm deg.: $15$
• Trans deg.: $54$
• Rank: $3$

Group information

Description:	$C_9:C_6\times A_4$
Order:	\(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_6^2.C_3^3.C_2^2$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \)
Composition factors:	$C_2$ x 3, $C_3$ x 4
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	7	80	128	162	270	648
Conjugacy classes	1	3	14	22	18	30	88
Divisions	1	3	7	11	9	15	46
Autjugacy classes	1	3	6	12	2	4	28

Dimension	1	2	3	6	9	18
Irr. complex chars.	54	0	30	0	4	0	88
Irr. rational chars.	2	26	2	14	0	2	46

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$54$
Rank:	$3$
Inequivalent generating triples:	$32760$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	18	18
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{6}=b^{6}=c^{18}=1, b^{a}=b^{4}c^{9}, c^{a}=b^{3}c, c^{b}=c^{13} \rangle$
Permutation group:	Degree $15$ $\langle(10,11)(12,13)(14,15), (10,12,14)(11,13,15), (2,4,7)(5,9,8), (1,2,5,3,4,8,6,7,9), (1,3,6)(2,4,7)(5,8,9), (12,13)(14,15), (10,11)(12,13)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 35 & 18 \\ 18 & 17 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 18 & 19 \end{array}\right), \left(\begin{array}{rr} 1 & 27 \\ 27 & 10 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 20 \\ 24 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Direct product:	$C_2$ $\, \times\, $ $A_4$ $\, \times\, $ $(C_9:C_3)$
Semidirect product:	$(C_9\times A_4)$ $\,\rtimes\,$ $C_6$	$(C_9:A_4)$ $\,\rtimes\,$ $C_6$	$C_9$ $\,\rtimes\,$ $(C_6\times A_4)$	$(A_4\times C_{18})$ $\,\rtimes\,$ $C_3$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^2\times A_4)$ . $C_6$	$(C_6\times A_4)$ . $C_3^2$ (2)	$C_3^2$ . $(C_6\times A_4)$	$C_6$ . $(C_3^2\times A_4)$	all 12

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

Homology

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

Subgroups

There are 560 subgroups in 184 conjugacy classes, 78 normal (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_3^2\times A_4$
Commutator:	$G' \simeq$ $C_2\times C_6$	$G/G' \simeq$ $C_3^2\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_6^2:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_6.C_6^2$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_9:C_6\times A_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_3^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9:C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

Normal subgroups

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Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 648: $C_9:C_6\times A_4$

Order 324: $C_6^2.C_3^2$

Order 216: $C_{18}:A_4$ x 6, $A_4\times C_{18}$ x 3, $C_6^2.C_6$ x 2, $C_6^2:C_6$, $C_6.C_6^2$

Order 162: $C_{18}:C_3^2$

Order 108: $C_9:A_4$ x 6, $C_{18}:C_6$ x 3, $C_9\times A_4$ x 3, $C_3^2.A_4$ x 2, $C_3^2\times A_4$

Order 81: $C_9:C_3^2$

Order 72: $C_6\times A_4$ x 6, $C_2^2:C_{18}$ x 6, $C_2^2\times C_{18}$ x 3, $C_2\times C_6^2$

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

Order 36: $C_2\times C_{18}$ x 9, $C_2^2:C_9$ x 6, $C_3\times A_4$ x 6, $C_6^2$ x 3

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

Order 24: $C_2\times A_4$ x 5, $C_2^2\times C_6$ x 2

Order 18: $C_{18}$ x 15, $C_3\times C_6$ x 9

Order 12: $C_2\times C_6$ x 6, $A_4$ x 5

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

Order 8: $C_2^3$

Order 6: $C_6$ x 11

Order 4: $C_2^2$ x 3

Order 3: $C_3$ x 7

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 648: $C_9:C_6\times A_4$

Order 324: $C_6^2.C_3^2$

Order 216: $C_6^2:C_6$, $C_6.C_6^2$, $C_6^2.C_6$, $C_{18}:A_4$, $A_4\times C_{18}$

Order 162: $C_{18}:C_3^2$

Order 108: $C_{18}:C_6$ x 3, $C_3^2\times A_4$, $C_3^2.A_4$, $C_9:A_4$, $C_9\times A_4$

Order 81: $C_9:C_3^2$

Order 72: $C_6\times A_4$ x 3, $C_2\times C_6^2$, $C_2^2\times C_{18}$, $C_2^2:C_{18}$

Order 54: $C_9:C_6$ x 5, $C_3\times C_{18}$, $C_3^2\times C_6$

Order 36: $C_2\times C_{18}$ x 3, $C_6^2$ x 3, $C_3\times A_4$ x 3, $C_2^2:C_9$

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

Order 24: $C_2^2\times C_6$ x 2, $C_2\times A_4$ x 2

Order 18: $C_3\times C_6$ x 6, $C_{18}$ x 4

Order 12: $C_2\times C_6$ x 6, $A_4$ x 2

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

Order 8: $C_2^3$

Order 6: $C_6$ x 8

Order 4: $C_2^2$ x 3

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 648: $C_9:C_6\times A_4$ ($C_1$)

Order 324: $C_6^2.C_3^2$ ($C_2$)

Order 216: $C_{18}:A_4$ ($C_3$) x 6, $A_4\times C_{18}$ ($C_3$) x 3, $C_6^2.C_6$ ($C_3$) x 2, $C_6^2:C_6$ ($C_3$), $C_6.C_6^2$ ($C_3$)

Order 108: $C_9:A_4$ ($C_6$) x 6, $C_9\times A_4$ ($C_6$) x 3, $C_3^2.A_4$ ($C_6$) x 2, $C_3^2\times A_4$ ($C_6$), $C_{18}:C_6$ ($C_6$)

Order 72: $C_2^2:C_{18}$ ($C_3^2$) x 6, $C_6\times A_4$ ($C_3^2$) x 3, $C_2^2\times C_{18}$ ($C_3^2$) x 3, $C_2\times C_6^2$ ($C_3^2$)

Order 54: $C_9:C_6$ ($A_4$)

Order 36: $C_2^2:C_9$ ($C_3\times C_6$) x 6, $C_2\times C_{18}$ ($C_3\times C_6$) x 3, $C_3\times A_4$ ($C_3\times C_6$) x 3, $C_6^2$ ($C_3\times C_6$)

Order 27: $C_9:C_3$ ($C_2\times A_4$)

Order 24: $C_2\times A_4$ ($C_9:C_3$) x 3, $C_2^2\times C_6$ ($C_3^3$)

Order 18: $C_{18}$ ($C_3\times A_4$) x 3, $C_3\times C_6$ ($C_3\times A_4$)

Order 12: $A_4$ ($C_9:C_6$) x 3, $C_2\times C_6$ ($C_3^2\times C_6$)

Order 9: $C_9$ ($C_6\times A_4$) x 3, $C_3^2$ ($C_6\times A_4$)

Order 8: $C_2^3$ ($C_9:C_3^2$)

Order 6: $C_6$ ($C_3^2\times A_4$)

Order 4: $C_2^2$ ($C_{18}:C_3^2$)

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

Order 2: $C_2$ ($C_6^2.C_3^2$)

Order 1: $C_1$ ($C_9:C_6\times A_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 648: $C_9:C_6\times A_4$ ($C_1$)

Order 324: $C_6^2.C_3^2$ ($C_2$)

Order 216: $C_6^2:C_6$ ($C_3$), $C_6.C_6^2$ ($C_3$), $C_6^2.C_6$ ($C_3$), $C_{18}:A_4$ ($C_3$), $A_4\times C_{18}$ ($C_3$)

Order 108: $C_3^2\times A_4$ ($C_6$), $C_{18}:C_6$ ($C_6$), $C_3^2.A_4$ ($C_6$), $C_9:A_4$ ($C_6$), $C_9\times A_4$ ($C_6$)

Order 72: $C_6\times A_4$ ($C_3^2$) x 2, $C_2\times C_6^2$ ($C_3^2$), $C_2^2\times C_{18}$ ($C_3^2$), $C_2^2:C_{18}$ ($C_3^2$)

Order 54: $C_9:C_6$ ($A_4$)

Order 36: $C_3\times A_4$ ($C_3\times C_6$) x 2, $C_2\times C_{18}$ ($C_3\times C_6$), $C_2^2:C_9$ ($C_3\times C_6$), $C_6^2$ ($C_3\times C_6$)

Order 27: $C_9:C_3$ ($C_2\times A_4$)

Order 24: $C_2^2\times C_6$ ($C_3^3$), $C_2\times A_4$ ($C_9:C_3$)

Order 18: $C_3\times C_6$ ($C_3\times A_4$), $C_{18}$ ($C_3\times A_4$)

Order 12: $C_2\times C_6$ ($C_3^2\times C_6$), $A_4$ ($C_9:C_6$)

Order 9: $C_3^2$ ($C_6\times A_4$), $C_9$ ($C_6\times A_4$)

Order 8: $C_2^3$ ($C_9:C_3^2$)

Order 6: $C_6$ ($C_3^2\times A_4$)

Order 4: $C_2^2$ ($C_{18}:C_3^2$)

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

Order 2: $C_2$ ($C_6^2.C_3^2$)

Order 1: $C_1$ ($C_9:C_6\times A_4$)

Series

Derived series

$C_9:C_6\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$C_9:C_6\times A_4$

$\rhd$

$C_6^2.C_3^2$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_9:C_6\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_9:C_6$

Supergroups

This group is a maximal subgroup of 19 larger groups in the database.

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

Character theory

Complex character table

See the $88 \times 88$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $46 \times 46$ rational character table.