Abstract group 108.24: $S_3\times C_{18}$

• Label: 108.24
• Order: \( 2^{2} \cdot 3^{3} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$S_3\times C_{18}$
Order:	\(108\)\(\medspace = 2^{2} \cdot 3^{3} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_6\times D_6$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Composition factors:	$C_2$ x 2, $C_3$ x 3
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	7	8	20	18	54	108
Conjugacy classes	1	3	5	9	12	24	54
Divisions	1	3	3	5	2	4	18
Autjugacy classes	1	2	3	4	2	3	15

Dimension	1	2	4	6	12
Irr. complex chars.	36	18	0	0	0	54
Irr. rational chars.	4	6	2	4	2	18

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$36$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	12
Arbitrary	2	4	8

Constructions

Presentation:	$\langle a, b \mid a^{18}=b^{6}=1, b^{a}=b^{5} \rangle$
Permutation group:	Degree $14$ $\langle(2,3), (4,5), (6,7,9,8,10,12,11,13,14), (6,8,11)(7,10,13)(9,12,14), (1,2,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 2 & 0 \\ 0 & 2 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 12 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{19})$
Transitive group:	36T63				more information
Direct product:	$C_2$ $\, \times\, $ $C_9$ $\, \times\, $ $S_3$
Semidirect product:	$C_6$ $\,\rtimes\,$ $C_{18}$	$(C_3\times C_{18})$ $\,\rtimes\,$ $C_2$	$C_3$ $\,\rtimes\,$ $(C_2\times C_{18})$	$(C_3\times C_9)$ $\,\rtimes\,$ $C_2^2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6\times S_3)$ . $C_3$	$(C_3\times S_3)$ . $C_6$ (2)	$C_6$ . $(C_3\times S_3)$	$C_3$ . $(C_6\times S_3)$	all 6
Aut. group:	$\Aut(C_2\times C_{38})$	$\Aut(C_2\times C_{54})$	$\Aut(S_3\times C_{19})$	$\Aut(S_3\times C_{27})$

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

Homology

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

Subgroups

There are 56 subgroups in 34 conjugacy classes, 21 normal (15 characteristic).

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

Special subgroups

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

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 108: $S_3\times C_{18}$

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

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

Order 27: $C_3\times C_9$

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

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

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

Order 6: $C_6$ x 5, $S_3$ 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 108: $S_3\times C_{18}$

Order 54: $C_3\times C_{18}$, $S_3\times C_9$

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

Order 27: $C_3\times C_9$

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

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

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

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

Order 4: $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 108: $S_3\times C_{18}$ ($C_1$)

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

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

Order 27: $C_3\times C_9$ ($C_2^2$)

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

Order 12: $D_6$ ($C_9$)

Order 9: $C_3^2$ ($C_2\times C_6$), $C_9$ ($D_6$)

Order 6: $S_3$ ($C_{18}$) x 2, $C_6$ ($C_3\times S_3$), $C_6$ ($C_{18}$)

Order 3: $C_3$ ($C_2\times C_{18}$), $C_3$ ($C_6\times S_3$)

Order 2: $C_2$ ($S_3\times C_9$)

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 108: $S_3\times C_{18}$ ($C_1$)

Order 54: $C_3\times C_{18}$ ($C_2$), $S_3\times C_9$ ($C_2$)

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

Order 27: $C_3\times C_9$ ($C_2^2$)

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

Order 12: $D_6$ ($C_9$)

Order 9: $C_3^2$ ($C_2\times C_6$), $C_9$ ($D_6$)

Order 6: $C_6$ ($C_3\times S_3$), $C_6$ ($C_{18}$), $S_3$ ($C_{18}$)

Order 3: $C_3$ ($C_2\times C_{18}$), $C_3$ ($C_6\times S_3$)

Order 2: $C_2$ ($S_3\times C_9$)

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

Series

Derived series

$S_3\times C_{18}$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$S_3\times C_{18}$

$\rhd$

$C_3\times C_{18}$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$S_3\times C_{18}$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_{18}$

Supergroups

This group is a maximal subgroup of 53 larger groups in the database.

This group is a maximal quotient of 43 larger groups in the database.

Character theory

Complex character table

See the $54 \times 54$ 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.