Abstract group 324.38: $S_3\times D_{27}$

• Label: 324.38
• Order: \( 2^{2} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3^{3} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 3^{2} \)
• Perm deg.: $30$
• Trans deg.: $54$
• Rank: $2$

Group information

Description:	$S_3\times D_{27}$
Order:	\(324\)\(\medspace = 2^{2} \cdot 3^{4} \)
Exponent:	\(54\)\(\medspace = 2 \cdot 3^{3} \)
Automorphism group:	$C_9.(C_9\times S_3)\times S_3$, of order \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \)
Composition factors:	$C_2$ x 2, $C_3$ x 4
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	9	18	27	54
Elements	1	111	8	60	18	18	54	54	324
Conjugacy classes	1	3	3	2	6	3	18	9	45
Divisions	1	3	3	2	2	1	2	1	15
Autjugacy classes	1	3	3	2	2	1	2	1	15

Dimension	1	2	4	6	12	18	36
Irr. complex chars.	4	28	13	0	0	0	0	45
Irr. rational chars.	4	4	1	2	1	2	1	15

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$54$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	4	36
Arbitrary	4	4	20

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{6}=c^{27}=[a,c]=1, b^{a}=b^{5}, c^{b}=c^{26} \rangle$
Permutation group:	Degree $30$ $\langle(29,30), (2,5)(3,9)(4,12)(6,7)(8,18)(10,11)(13,20)(14,19)(15,25)(16,24) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 53 & 13 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 26 & 27 \\ 27 & 26 \end{array}\right), \left(\begin{array}{rr} 1 & 27 \\ 27 & 28 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/54\Z)$
Direct product:	$S_3$ $\, \times\, $ $D_{27}$
Semidirect product:	$C_{27}$ $\,\rtimes\,$ $D_6$	$C_3$ $\,\rtimes\,$ $D_{54}$	$(S_3\times C_{27})$ $\,\rtimes\,$ $C_2$	$(C_3\times D_{27})$ $\,\rtimes\,$ $C_2$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_9$ . $S_3^2$	$C_3^2$ . $D_{18}$	$(S_3\times C_9)$ . $S_3$	$(C_3\times S_3)$ . $D_9$	all 6

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 508 subgroups in 46 conjugacy classes, 16 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_3\times D_{27}$
Commutator:	$G' \simeq$ $C_3\times C_{27}$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_9$	$G/\Phi \simeq$ $S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_{27}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $S_3\times D_{27}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2$	$G/\operatorname{soc} \simeq$ $D_{18}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_{27}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 324: $S_3\times D_{27}$

Order 162: $S_3\times C_{27}$, $C_3\times D_{27}$, $C_3:D_{27}$

Order 108: $D_{54}$, $S_3\times D_9$

Order 81: $C_3\times C_{27}$

Order 54: $D_{27}$ x 3, $C_3:D_9$, $S_3\times C_9$, $C_3\times D_9$, $C_{54}$

Order 36: $D_{18}$, $S_3^2$

Order 27: $C_{27}$ x 2, $C_3\times C_9$

Order 18: $D_9$ x 3, $C_3\times S_3$ x 2, $C_3:S_3$, $C_{18}$

Order 12: $D_6$ x 2

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

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

Order 4: $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 324: $S_3\times D_{27}$

Order 162: $S_3\times C_{27}$, $C_3\times D_{27}$, $C_3:D_{27}$

Order 108: $D_{54}$, $S_3\times D_9$

Order 81: $C_3\times C_{27}$

Order 54: $D_{27}$ x 3, $C_3:D_9$, $S_3\times C_9$, $C_3\times D_9$, $C_{54}$

Order 36: $D_{18}$, $S_3^2$

Order 27: $C_{27}$ x 2, $C_3\times C_9$

Order 18: $D_9$ x 3, $C_3\times S_3$ x 2, $C_3:S_3$, $C_{18}$

Order 12: $D_6$ x 2

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

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

Order 4: $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 324: $S_3\times D_{27}$ ($C_1$)

Order 162: $S_3\times C_{27}$ ($C_2$), $C_3\times D_{27}$ ($C_2$), $C_3:D_{27}$ ($C_2$)

Order 81: $C_3\times C_{27}$ ($C_2^2$)

Order 54: $S_3\times C_9$ ($S_3$), $D_{27}$ ($S_3$)

Order 27: $C_3\times C_9$ ($D_6$), $C_{27}$ ($D_6$)

Order 18: $C_3\times S_3$ ($D_9$)

Order 9: $C_3^2$ ($D_{18}$), $C_9$ ($S_3^2$)

Order 6: $S_3$ ($D_{27}$)

Order 3: $C_3$ ($D_{54}$), $C_3$ ($S_3\times D_9$)

Order 1: $C_1$ ($S_3\times D_{27}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 324: $S_3\times D_{27}$ ($C_1$)

Order 162: $S_3\times C_{27}$ ($C_2$), $C_3\times D_{27}$ ($C_2$), $C_3:D_{27}$ ($C_2$)

Order 81: $C_3\times C_{27}$ ($C_2^2$)

Order 54: $S_3\times C_9$ ($S_3$), $D_{27}$ ($S_3$)

Order 27: $C_3\times C_9$ ($D_6$), $C_{27}$ ($D_6$)

Order 18: $C_3\times S_3$ ($D_9$)

Order 9: $C_3^2$ ($D_{18}$), $C_9$ ($S_3^2$)

Order 6: $S_3$ ($D_{27}$)

Order 3: $C_3$ ($D_{54}$), $C_3$ ($S_3\times D_9$)

Order 1: $C_1$ ($S_3\times D_{27}$)

Series

Derived series

$S_3\times D_{27}$

$\rhd$

$C_3\times C_{27}$

$\rhd$

$C_1$

Chief series

$S_3\times D_{27}$

$\rhd$

$S_3\times C_{27}$

$\rhd$

$C_3\times C_{27}$

$\rhd$

$C_{27}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$S_3\times D_{27}$

$\rhd$

$C_3\times C_{27}$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 18 larger groups in the database.

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

Character theory

Complex character table

See the $45 \times 45$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	3A	3B	3C	6A	6B	9A	9B	18A	27A	27B	54A
	Size	1	3	27	81	2	2	4	6	54	6	12	18	18	36	54
	2 P	1A	1A	1A	1A	3A	3B	3C	3A	3B	9A	9B	9A	27A	27B	27A
	3 P	1A	2A	2B	2C	1A	1A	1A	2A	2B	3A	3A	6A	9A	9A	18A
324.38.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
324.38.1b		$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$
324.38.1c		$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$
324.38.1d		$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$
324.38.2a		$2$	$0$	$2$	$0$	$2$	$-1$	$-1$	$0$	$-1$	$2$	$-1$	$0$	$2$	$-1$	$0$
324.38.2b		$2$	$2$	$0$	$0$	$2$	$2$	$2$	$2$	$0$	$2$	$2$	$2$	$-1$	$-1$	$-1$
324.38.2c		$2$	$-2$	$0$	$0$	$2$	$2$	$2$	$-2$	$0$	$2$	$2$	$-2$	$-1$	$-1$	$1$
324.38.2d		$2$	$0$	$-2$	$0$	$2$	$-1$	$-1$	$0$	$1$	$2$	$-1$	$0$	$2$	$-1$	$0$
324.38.2e		$6$	$6$	$0$	$0$	$6$	$6$	$6$	$6$	$0$	$-3$	$-3$	$-3$	$0$	$0$	$0$
324.38.2f		$6$	$-6$	$0$	$0$	$6$	$6$	$6$	$-6$	$0$	$-3$	$-3$	$3$	$0$	$0$	$0$
324.38.2g		$18$	$18$	$0$	$0$	$-9$	$18$	$-9$	$-9$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
324.38.2h		$18$	$-18$	$0$	$0$	$-9$	$18$	$-9$	$9$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
324.38.4a		$4$	$0$	$0$	$0$	$4$	$-2$	$-2$	$0$	$0$	$4$	$-2$	$0$	$-2$	$1$	$0$
324.38.4b		$12$	$0$	$0$	$0$	$12$	$-6$	$-6$	$0$	$0$	$-6$	$3$	$0$	$0$	$0$	$0$
324.38.4c		$36$	$0$	$0$	$0$	$-18$	$-18$	$9$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$