Abstract group 324.72: $(C_3\times C_{18}):S_3$

• Label: 324.72
• Order: \( 2^{2} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $29$
• Trans deg.: $54$
• Rank: $2$

Group information

Description:	$(C_3\times C_{18}):S_3$
Order:	\(324\)\(\medspace = 2^{2} \cdot 3^{4} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_6\times C_3^2:D_6$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
Composition factors:	$C_2$ x 2, $C_3$ x 4
Derived length:	$3$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	19	62	98	18	126	324
Conjugacy classes	1	3	6	10	6	18	44
Divisions	1	3	4	6	1	3	18
Autjugacy classes	1	2	4	5	1	2	15

Dimension	1	2	3	4	6	18
Irr. complex chars.	12	6	24	0	2	0	44
Irr. rational chars.	4	6	0	2	2	4	18

Minimal presentations

Permutation degree:	$29$
Transitive degree:	$54$
Rank:	$2$
Inequivalent generating pairs:	$36$

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:	$\langle a, b, c, d \mid a^{2}=b^{3}=c^{3}=d^{18}=[a,d]=[c,d]=1, b^{a}=b^{2}, c^{a}=c^{2}d^{12}, c^{b}=cd^{12}, d^{b}=c^{2}d \rangle$
Permutation group:	Degree $29$ $\langle(3,10)(7,17)(11,20)(12,21)(13,23)(14,24)(18,25)(19,26)(22,27)(28,29), (28,29) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(\He_3.C_6)$
Semidirect product:	$(C_3\times C_9)$ $\,\rtimes\,$ $D_6$	$(C_3\times C_{18})$ $\,\rtimes\,$ $S_3$	$(C_3^2:S_3)$ $\,\rtimes\,$ $C_6$ (2)	$(C_3^2:D_6)$ $\,\rtimes\,$ $C_3$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^2$ . $(C_6\times S_3)$	$C_3$ . $(C_3^2:D_6)$	$C_6$ . $(C_3^2:C_6)$	$(C_2\times \He_3)$ . $C_6$	all 6

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

Homology

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

Subgroups

There are 268 subgroups in 56 conjugacy classes, 18 normal (14 characteristic).

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

Special subgroups

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

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 324: $(C_3\times C_{18}):S_3$

Order 162: $\He_3.C_6$ x 2, $\He_3:C_6$

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

Order 81: $\He_3:C_3$

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

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

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

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

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

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

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

Order 4: $C_2^2$

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 324: $(C_3\times C_{18}):S_3$

Order 162: $\He_3:C_6$, $\He_3.C_6$

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

Order 81: $\He_3:C_3$

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

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

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

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

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

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

Order 6: $C_6$ x 5, $S_3$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 4

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 324: $(C_3\times C_{18}):S_3$ ($C_1$)

Order 162: $\He_3.C_6$ ($C_2$) x 2, $\He_3:C_6$ ($C_2$)

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

Order 81: $\He_3:C_3$ ($C_2^2$)

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

Order 27: $\He_3$ ($C_2\times C_6$), $C_3\times C_9$ ($D_6$)

Order 18: $C_3\times C_6$ ($C_3\times S_3$)

Order 9: $C_3^2$ ($C_6\times S_3$)

Order 6: $C_6$ ($C_3^2:C_6$)

Order 3: $C_3$ ($C_3^2:D_6$)

Order 2: $C_2$ ($\He_3.C_6$)

Order 1: $C_1$ ($(C_3\times C_{18}):S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 324: $(C_3\times C_{18}):S_3$ ($C_1$)

Order 162: $\He_3:C_6$ ($C_2$), $\He_3.C_6$ ($C_2$)

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

Order 81: $\He_3:C_3$ ($C_2^2$)

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

Order 27: $\He_3$ ($C_2\times C_6$), $C_3\times C_9$ ($D_6$)

Order 18: $C_3\times C_6$ ($C_3\times S_3$)

Order 9: $C_3^2$ ($C_6\times S_3$)

Order 6: $C_6$ ($C_3^2:C_6$)

Order 3: $C_3$ ($C_3^2:D_6$)

Order 2: $C_2$ ($\He_3.C_6$)

Order 1: $C_1$ ($(C_3\times C_{18}):S_3$)

Series

Derived series

$(C_3\times C_{18}):S_3$

$\rhd$

$\He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$(C_3\times C_{18}):S_3$

$\rhd$

$\He_3.C_6$

$\rhd$

$\He_3:C_3$

$\rhd$

$\He_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_3\times C_{18}):S_3$

$\rhd$

$\He_3$

Upper central series

$C_1$

$\lhd$

$C_6$

Supergroups

This group is a maximal subgroup of 23 larger groups in the database.

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

Character theory

Complex character table

See the $44 \times 44$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $18 \times 18$ rational character table.