Abstract group 116964.i: $C_{19}^2:(C_9\times D_{18})$

• Label: 116964.i
• Order: \( 2^{2} \cdot 3^{4} \cdot 19^{2} \)
• Exponent: \( 2 \cdot 3^{2} \cdot 19 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{4} \cdot 19^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $38$
• Trans deg.: $38$
• Rank: $2$

Group information

Description:	$C_{19}^2:(C_9\times D_{18})$
Order:	\(116964\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 19^{2} \)
Exponent:	\(342\)\(\medspace = 2 \cdot 3^{2} \cdot 19 \)
Automorphism group:	$F_{19}\wr C_2$, of order \(233928\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 19^{2} \)
Composition factors:	$C_2$ x 2, $C_3$ x 4, $C_{19}$ x 2
Derived length:	$3$

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

Group statistics

Order	1	2	3	6	9	18	19	38	57	171
Elements	1	703	1520	15884	21888	64980	360	6156	1368	4104	116964
Conjugacy classes	1	3	5	9	39	51	3	2	2	6	121
Divisions	1	3	3	5	7	9	3	2	1	1	35
Autjugacy classes	1	2	5	7	39	45	2	1	2	6	110

Dimension	1	2	4	6	12	36	72	162	216
Irr. complex chars.	36	72	0	0	0	9	0	4	0	121
Irr. rational chars.	4	6	2	6	10	1	1	4	1	35

Minimal presentations

Permutation degree:	$38$
Transitive degree:	$38$
Rank:	$2$
Inequivalent generating pairs:	$12960$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d \mid a^{18}=b^{18}=c^{19}=d^{19}=[c,d]=1, b^{a}=b^{17}c^{18}d^{6}, c^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $38$ $\langle(1,7,15,13,4,11,14,18,17,3,16,8,10,19,12,9,5,6)(21,28,27,38,31,32)(22,36,34,37,23,25) \!\cdots\! \rangle$
Transitive group:	38T46				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{19}:F_{19})$ $\,\rtimes\,$ $D_9$	$(C_{19}^2:C_9)$ $\,\rtimes\,$ $D_{18}$	$C_{19}^2$ $\,\rtimes\,$ $(C_9\times D_{18})$	$(C_{19}:D_{19})$ $\,\rtimes\,$ $(C_9\times D_9)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{19}^2:C_6)$ . $(S_3\times C_9)$	$(C_{19}^2:(C_3\times D_9))$ . $C_6$ (2)	$(C_{19}^2:(C_3\times C_9))$ . $D_6$	$(C_{19}^2:C_6)$ . $(C_3\times D_9)$	all 12
Aut. group:	$\Aut(C_{19}^2:D_9)$	$\Aut(C_{19}^2:(C_3\times D_9))$	$\Aut(C_{19}^2:(C_9\times D_9))$

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 57244 subgroups in 194 conjugacy classes, 28 normal (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_{19}^2:(C_9\times D_{18})$
Commutator:	$G' \simeq$ $C_{19}^2:C_9$	$G/G' \simeq$ $C_2\times C_{18}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{19}^2:(C_9\times D_{18})$
Fitting:	$\operatorname{Fit} \simeq$ $C_{19}^2$	$G/\operatorname{Fit} \simeq$ $C_9\times D_{18}$
Radical:	$R \simeq$ $C_{19}^2:(C_9\times D_{18})$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{19}^2$	$G/\operatorname{soc} \simeq$ $C_9\times D_{18}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9^2$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $C_{19}^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 116964: $C_{19}^2:(C_9\times D_{18})$

Order 58482: $C_{19}^2:(C_9\times D_9)$ x 2, $C_{19}^2:(C_9\times C_{18})$

Order 38988: $C_{19}^2:(S_3\times C_{18})$, $C_{19}^2:(C_3\times D_{18})$

Order 29241: $C_{19}^2:C_9^2$

Order 19494: $C_{19}^2:(C_3\times C_{18})$ x 2, $C_{19}^2:(S_3\times C_9)$ x 2, $C_{19}^2:(C_3\times D_9)$ x 2, $C_{19}^2:(C_3\times C_{18})$

Order 12996: $C_{19}^2:(C_6\times S_3)$, $C_{19}^2:D_{18}$, $C_{19}^2:(C_2\times C_{18})$

Order 9747: $C_{19}^2:(C_3\times C_9)$, $C_{19}^2:(C_3\times C_9)$, $C_{19}^2:(C_3\times C_9)$

Order 6498: $C_{19}^2:C_{18}$ x 2, $C_{19}^2:(C_3\times S_3)$ x 2, $C_{19}^2:D_9$ x 2, $C_{19}^2:C_{18}$, $C_{19}^2:C_{18}$, $C_{19}^2:C_{18}$, $C_{19}^2:C_{18}$, $C_{19}^2:(C_3\times C_6)$, $C_{19}^2:C_{18}$, $C_{19}^2:C_{18}$, $C_{19}:F_{19}$

Order 4332: $C_{19}^2:(C_2\times C_6)$, $C_{19}^2:D_6$

Order 3249: $C_{19}^2:C_9$, $(C_{19}:C_3)^2$, $C_{19}^2:C_9$, $C_{19}^2:C_9$, $C_{19}^2:C_9$, $C_{19}^2:C_9$, $C_{19}^2:C_9$, $C_{19}\times C_{19}:C_3.C_3$

Order 3078: $C_9\times F_{19}$

Order 2166: $C_{19}^2:C_6$ x 2, $C_{19}^2:S_3$ x 2, $C_{19}^2:C_6$, $C_{19}^2:C_6$, $C_{19}^2:C_6$

Order 1539: $C_{171}:C_9$

Order 1444: $D_{19}^2$

Order 1083: $C_{19}^2:C_3$, $C_{19}^2:C_3$, $C_{19}:C_{57}$

Order 1026: $C_3\times F_{19}$ x 3, $C_{171}:C_6$

Order 722: $C_{19}\times D_{19}$ x 2, $C_{19}:D_{19}$

Order 684: $C_2\times F_{19}$ x 2

Order 513: $C_{57}:C_9$ x 3, $C_{171}:C_3$

Order 361: $C_{19}^2$

Order 342: $F_{19}$ x 13, $C_{19}:C_{18}$ x 2, $C_{19}:C_{18}$ x 2, $C_{57}:C_6$, $C_9\times D_{19}$

Order 324: $C_9\times D_{18}$

Order 228: $C_{38}:C_6$ x 2

Order 171: $C_{19}:C_9$ x 11, $C_{19}:C_9$ x 2, $C_{57}:C_3$, $C_{171}$

Order 162: $C_9\times D_9$ x 2, $C_9\times C_{18}$

Order 114: $C_{19}:C_6$ x 7, $C_{19}:C_6$ x 2, $C_3\times D_{19}$

Order 108: $S_3\times C_{18}$, $C_3\times D_{18}$

Order 81: $C_9^2$

Order 76: $D_{38}$ x 2

Order 57: $C_{19}:C_3$ x 5, $C_{57}$

Order 54: $C_3\times C_{18}$ x 3, $S_3\times C_9$ x 2, $C_3\times D_9$ x 2

Order 38: $D_{19}$ x 5, $C_{38}$ x 2

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

Order 27: $C_3\times C_9$ x 3

Order 19: $C_{19}$ x 3

Order 18: $C_{18}$ x 9, $C_3\times S_3$ x 2, $D_9$ x 2, $C_3\times C_6$

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

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

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 116964: $C_{19}^2:(C_9\times D_{18})$

Order 58482: $C_{19}^2:(C_9\times C_{18})$, $C_{19}^2:(C_9\times D_9)$

Order 38988: $C_{19}^2:(S_3\times C_{18})$, $C_{19}^2:(C_3\times D_{18})$

Order 29241: $C_{19}^2:C_9^2$

Order 19494: $C_{19}^2:(C_3\times C_{18})$ x 2, $C_{19}^2:(C_3\times C_{18})$, $C_{19}^2:(S_3\times C_9)$, $C_{19}^2:(C_3\times D_9)$

Order 12996: $C_{19}^2:(C_6\times S_3)$, $C_{19}^2:D_{18}$, $C_{19}^2:(C_2\times C_{18})$

Order 9747: $C_{19}^2:(C_3\times C_9)$, $C_{19}^2:(C_3\times C_9)$, $C_{19}^2:(C_3\times C_9)$

Order 6498: $C_{19}^2:C_{18}$, $C_{19}^2:C_{18}$, $C_{19}^2:C_{18}$, $C_{19}^2:C_{18}$, $C_{19}^2:(C_3\times C_6)$, $C_{19}^2:C_{18}$, $C_{19}^2:C_{18}$, $C_{19}^2:C_{18}$, $C_{19}^2:(C_3\times S_3)$, $C_{19}^2:D_9$, $C_{19}:F_{19}$

Order 4332: $C_{19}^2:(C_2\times C_6)$, $C_{19}^2:D_6$

Order 3249: $C_{19}^2:C_9$, $(C_{19}:C_3)^2$, $C_{19}^2:C_9$, $C_{19}^2:C_9$, $C_{19}^2:C_9$, $C_{19}^2:C_9$, $C_{19}^2:C_9$, $C_{19}\times C_{19}:C_3.C_3$

Order 3078: $C_9\times F_{19}$

Order 2166: $C_{19}^2:C_6$, $C_{19}^2:C_6$, $C_{19}^2:C_6$, $C_{19}^2:S_3$, $C_{19}^2:C_6$

Order 1539: $C_{171}:C_9$

Order 1444: $D_{19}^2$

Order 1083: $C_{19}^2:C_3$, $C_{19}^2:C_3$, $C_{19}:C_{57}$

Order 1026: $C_3\times F_{19}$ x 3, $C_{171}:C_6$

Order 722: $C_{19}:D_{19}$, $C_{19}\times D_{19}$

Order 684: $C_2\times F_{19}$

Order 513: $C_{57}:C_9$ x 3, $C_{171}:C_3$

Order 361: $C_{19}^2$

Order 342: $F_{19}$ x 11, $C_{19}:C_{18}$ x 2, $C_{57}:C_6$, $C_{19}:C_{18}$, $C_9\times D_{19}$

Order 324: $C_9\times D_{18}$

Order 228: $C_{38}:C_6$

Order 171: $C_{19}:C_9$ x 10, $C_{19}:C_9$ x 2, $C_{57}:C_3$, $C_{171}$

Order 162: $C_9\times D_9$, $C_9\times C_{18}$

Order 114: $C_{19}:C_6$ x 5, $C_3\times D_{19}$, $C_{19}:C_6$

Order 108: $S_3\times C_{18}$, $C_3\times D_{18}$

Order 81: $C_9^2$

Order 76: $D_{38}$

Order 57: $C_{19}:C_3$ x 4, $C_{57}$

Order 54: $C_3\times C_{18}$ x 3, $S_3\times C_9$, $C_3\times D_9$

Order 38: $D_{19}$ x 3, $C_{38}$

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

Order 27: $C_3\times C_9$ x 3

Order 19: $C_{19}$ x 2

Order 18: $C_{18}$ x 8, $C_3\times C_6$, $C_3\times S_3$, $D_9$

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

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

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 116964: $C_{19}^2:(C_9\times D_{18})$ ($C_1$)

Order 58482: $C_{19}^2:(C_9\times D_9)$ ($C_2$) x 2, $C_{19}^2:(C_9\times C_{18})$ ($C_2$)

Order 38988: $C_{19}^2:(C_3\times D_{18})$ ($C_3$)

Order 29241: $C_{19}^2:C_9^2$ ($C_2^2$)

Order 19494: $C_{19}^2:(C_3\times D_9)$ ($C_6$) x 2, $C_{19}^2:(C_3\times C_{18})$ ($C_6$), $C_{19}^2:(C_3\times C_{18})$ ($S_3$)

Order 12996: $C_{19}^2:D_{18}$ ($C_9$)

Order 9747: $C_{19}^2:(C_3\times C_9)$ ($C_2\times C_6$), $C_{19}^2:(C_3\times C_9)$ ($D_6$)

Order 6498: $C_{19}^2:D_9$ ($C_{18}$) x 2, $C_{19}^2:C_{18}$ ($C_{18}$), $C_{19}^2:(C_3\times C_6)$ ($C_3\times S_3$), $C_{19}:F_{19}$ ($D_9$)

Order 3249: $C_{19}^2:C_9$ ($D_{18}$), $(C_{19}:C_3)^2$ ($C_6\times S_3$), $C_{19}^2:C_9$ ($C_2\times C_{18}$)

Order 2166: $C_{19}^2:C_6$ ($C_3\times D_9$), $C_{19}^2:C_6$ ($S_3\times C_9$)

Order 1083: $C_{19}^2:C_3$ ($S_3\times C_{18}$), $C_{19}^2:C_3$ ($C_3\times D_{18}$)

Order 722: $C_{19}:D_{19}$ ($C_9\times D_9$)

Order 361: $C_{19}^2$ ($C_9\times D_{18}$)

Order 1: $C_1$ ($C_{19}^2:(C_9\times D_{18})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 116964: $C_{19}^2:(C_9\times D_{18})$ ($C_1$)

Order 58482: $C_{19}^2:(C_9\times C_{18})$ ($C_2$), $C_{19}^2:(C_9\times D_9)$ ($C_2$)

Order 38988: $C_{19}^2:(C_3\times D_{18})$ ($C_3$)

Order 29241: $C_{19}^2:C_9^2$ ($C_2^2$)

Order 19494: $C_{19}^2:(C_3\times C_{18})$ ($C_6$), $C_{19}^2:(C_3\times C_{18})$ ($S_3$), $C_{19}^2:(C_3\times D_9)$ ($C_6$)

Order 12996: $C_{19}^2:D_{18}$ ($C_9$)

Order 9747: $C_{19}^2:(C_3\times C_9)$ ($C_2\times C_6$), $C_{19}^2:(C_3\times C_9)$ ($D_6$)

Order 6498: $C_{19}^2:C_{18}$ ($C_{18}$), $C_{19}^2:(C_3\times C_6)$ ($C_3\times S_3$), $C_{19}^2:D_9$ ($C_{18}$), $C_{19}:F_{19}$ ($D_9$)

Order 3249: $C_{19}^2:C_9$ ($D_{18}$), $(C_{19}:C_3)^2$ ($C_6\times S_3$), $C_{19}^2:C_9$ ($C_2\times C_{18}$)

Order 2166: $C_{19}^2:C_6$ ($C_3\times D_9$), $C_{19}^2:C_6$ ($S_3\times C_9$)

Order 1083: $C_{19}^2:C_3$ ($S_3\times C_{18}$), $C_{19}^2:C_3$ ($C_3\times D_{18}$)

Order 722: $C_{19}:D_{19}$ ($C_9\times D_9$)

Order 361: $C_{19}^2$ ($C_9\times D_{18}$)

Order 1: $C_1$ ($C_{19}^2:(C_9\times D_{18})$)

Series

Derived series

$C_{19}^2:(C_9\times D_{18})$

$\rhd$

$C_{19}^2:C_9$

$\rhd$

$C_{19}^2$

$\rhd$

$C_1$

Chief series

$C_{19}^2:(C_9\times D_{18})$

$\rhd$

$C_{19}^2:(C_9\times C_{18})$

$\rhd$

$C_{19}^2:(C_3\times C_{18})$

$\rhd$

$C_{19}^2:C_{18}$

$\rhd$

$C_{19}^2:C_9$

$\rhd$

$C_{19}^2:C_3$

$\rhd$

$C_{19}^2$

$\rhd$

$C_1$

Lower central series

$C_{19}^2:(C_9\times D_{18})$

$\rhd$

$C_{19}^2:C_9$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $121 \times 121$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $35 \times 35$ rational character table.