Abstract group 972.42: $C_9:C_{18}.S_3$

• Label: 972.42
• Order: \( 2^{2} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $31$
• Trans deg.: $108$
• Rank: $2$

Group information

Description:	$C_9:C_{18}.S_3$
Order:	\(972\)\(\medspace = 2^{2} \cdot 3^{5} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3.\He_3.C_6^2$, of order \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \)
Composition factors:	$C_2$ x 2, $C_3$ x 5
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	1	116	54	116	126	108	126	324	972
Conjugacy classes	1	1	5	2	5	8	4	8	12	46
Divisions	1	1	3	1	3	3	1	3	1	17
Autjugacy classes	1	1	5	1	5	4	2	4	2	25

Dimension	1	2	3	4	6	18	36
Irr. complex chars.	12	6	24	0	2	2	0	46
Irr. rational chars.	2	5	0	3	2	4	1	17

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$108$
Rank:	$2$
Inequivalent generating pairs:	$144$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	36	36
Arbitrary	18	20	20

Constructions

Presentation:	${\langle a, b, c, d \mid a^{4}=c^{9}=d^{9}=[a,c]=1, b^{3}=d^{6}, b^{a}=b^{2}d^{3}, d^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $31$ $\langle(2,5)(3,9)(4,12)(6,13)(7,8)(10,11)(14,18)(15,22)(16,21)(17,26)(19,20)(23,24) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 54 \\ 54 & 1 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 73 & 12 \\ 36 & 37 \end{array}\right), \left(\begin{array}{rr} 80 & 3 \\ 27 & 82 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 73 \end{array}\right), \left(\begin{array}{rr} 49 & 100 \\ 24 & 49 \end{array}\right), \left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/108\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^2.\He_3)$ $\,\rtimes\,$ $C_4$	$((C_3\times C_{18}).S_3)$ $\,\rtimes\,$ $C_3$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_9:C_{18})$ . $S_3$	$(C_6.\He_3)$ . $C_6$	$C_6$ . $(\He_3.C_6)$	$(C_3.\He_3)$ . $C_{12}$	all 12

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

Homology

Abelianization:	$C_{12} \simeq C_{4} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 371 subgroups in 47 conjugacy classes, 16 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_9:C_9.S_3$
Commutator:	$G' \simeq$ $C_3.\He_3$	$G/G' \simeq$ $C_{12}$
Frattini:	$\Phi \simeq$ $C_3\times C_{18}$	$G/\Phi \simeq$ $C_3\times S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_3^2.\He_3$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_9:C_{18}.S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $\He_3.C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2.\He_3$

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 972: $C_9:C_{18}.S_3$

Order 486: $C_2\times C_3^2.\He_3$

Order 324: $C_9:C_{36}$, $(C_3\times C_{18}).S_3$

Order 243: $C_3^2.\He_3$

Order 162: $C_6.\He_3$, $\He_3:C_6$, $C_9:C_{18}$

Order 108: $C_9:C_{12}$, $C_3:C_{36}$, $C_9:C_{12}$

Order 81: $\He_3:C_3$, $C_9:C_9$, $C_3.\He_3$

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

Order 36: $C_9:C_4$ x 2, $C_3:C_{12}$, $C_{36}$

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

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

Order 12: $C_{12}$, $C_3:C_4$

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

Order 6: $C_6$ x 3

Order 4: $C_4$

Order 3: $C_3$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 972: $C_9:C_{18}.S_3$

Order 486: $C_2\times C_3^2.\He_3$

Order 324: $C_9:C_{36}$, $(C_3\times C_{18}).S_3$

Order 243: $C_3^2.\He_3$

Order 162: $C_6.\He_3$, $\He_3:C_6$, $C_9:C_{18}$

Order 108: $C_9:C_{12}$, $C_3:C_{36}$, $C_9:C_{12}$

Order 81: $\He_3:C_3$, $C_9:C_9$, $C_3.\He_3$

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

Order 36: $C_9:C_4$ x 2, $C_3:C_{12}$, $C_{36}$

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

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

Order 12: $C_{12}$, $C_3:C_4$

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

Order 6: $C_6$ x 3

Order 4: $C_4$

Order 3: $C_3$ x 3

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 972: $C_9:C_{18}.S_3$ ($C_1$)

Order 486: $C_2\times C_3^2.\He_3$ ($C_2$)

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

Order 243: $C_3^2.\He_3$ ($C_4$)

Order 162: $C_6.\He_3$ ($C_6$), $C_9:C_{18}$ ($S_3$)

Order 81: $C_9:C_9$ ($C_3:C_4$), $C_3.\He_3$ ($C_{12}$)

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

Order 27: $C_3\times C_9$ ($C_3:C_{12}$)

Order 18: $C_3\times C_6$ ($C_3^2:C_6$)

Order 9: $C_3^2$ ($C_3^2:C_{12}$)

Order 6: $C_6$ ($\He_3.C_6$)

Order 3: $C_3$ ($\He_3.C_{12}$)

Order 2: $C_2$ ($C_9:C_9.S_3$)

Order 1: $C_1$ ($C_9:C_{18}.S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 972: $C_9:C_{18}.S_3$ ($C_1$)

Order 486: $C_2\times C_3^2.\He_3$ ($C_2$)

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

Order 243: $C_3^2.\He_3$ ($C_4$)

Order 162: $C_6.\He_3$ ($C_6$), $C_9:C_{18}$ ($S_3$)

Order 81: $C_9:C_9$ ($C_3:C_4$), $C_3.\He_3$ ($C_{12}$)

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

Order 27: $C_3\times C_9$ ($C_3:C_{12}$)

Order 18: $C_3\times C_6$ ($C_3^2:C_6$)

Order 9: $C_3^2$ ($C_3^2:C_{12}$)

Order 6: $C_6$ ($\He_3.C_6$)

Order 3: $C_3$ ($\He_3.C_{12}$)

Order 2: $C_2$ ($C_9:C_9.S_3$)

Order 1: $C_1$ ($C_9:C_{18}.S_3$)

Series

Derived series

$C_9:C_{18}.S_3$

$\rhd$

$C_3.\He_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_9:C_{18}.S_3$

$\rhd$

$C_2\times C_3^2.\He_3$

$\rhd$

$C_3^2.\He_3$

$\rhd$

$C_9:C_9$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_9:C_{18}.S_3$

$\rhd$

$C_3.\He_3$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $46 \times 46$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $17 \times 17$ rational character table.