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G = C9×C3⋊D4  order 216 = 23·33

Direct product of C9 and C3⋊D4

direct product, metabelian, supersoluble, monomial

Aliases: C9×C3⋊D4, D6⋊2C18, Dic3⋊C18, C18.23D6, C62.11C6, (C3×C9)⋊7D4, C3⋊2(D4×C9), (C2×C6)⋊4C18, (C2×C18)⋊3S3, (C6×C18)⋊1C2, (S3×C18)⋊2C2, (S3×C6).2C6, C2.5(S3×C18), C6.33(S3×C6), C6.5(C2×C18), C22⋊3(S3×C9), (C9×Dic3)⋊4C2, C32.3(C3×D4), (C3×Dic3).3C6, (C3×C18).12C22, (C3×C3⋊D4).C3, (C2×C6).9(C3×S3), C3.4(C3×C3⋊D4), (C3×C6).22(C2×C6), SmallGroup(216,58)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C9×C3⋊D4
Chief series
C1 — C3 — C32 — C3×C6 — C3×C18 — S3×C18 — C9×C3⋊D4
Lower central
C3 — C6 — C9×C3⋊D4
Upper central
C1 — C18 — C2×C18

Generators and relations for C9×C3⋊D4
 G = < a,b,c,d | a9=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 110 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, D4, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C18, C18, C3×S3, C3×C6, C3×C6, C3⋊D4, C3×D4, C3×C9, C36, C2×C18, C2×C18, C3×Dic3, S3×C6, C62, S3×C9, C3×C18, C3×C18, D4×C9, C3×C3⋊D4, C9×Dic3, S3×C18, C6×C18, C9×C3⋊D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C9, D6, C2×C6, C18, C3×S3, C3⋊D4, C3×D4, C2×C18, S3×C6, S3×C9, D4×C9, C3×C3⋊D4, S3×C18, C9×C3⋊D4

Smallest permutation representation of C9×C3⋊D4
►On 36 points
Generators in S36
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36)

(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)

(1 14 23 34)(2 15 24 35)(3 16 25 36)(4 17 26 28)(5 18 27 29)(6 10 19 30)(7 11 20 31)(8 12 21 32)(9 13 22 33)

(1 34)(2 35)(3 36)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)
 
G:=sub<Sym(36)| (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36), (1,14,23,34)(2,15,24,35)(3,16,25,36)(4,17,26,28)(5,18,27,29)(6,10,19,30)(7,11,20,31)(8,12,21,32)(9,13,22,33), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36), (1,14,23,34)(2,15,24,35)(3,16,25,36)(4,17,26,28)(5,18,27,29)(6,10,19,30)(7,11,20,31)(8,12,21,32)(9,13,22,33), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36)], [(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36)], [(1,14,23,34),(2,15,24,35),(3,16,25,36),(4,17,26,28),(5,18,27,29),(6,10,19,30),(7,11,20,31),(8,12,21,32),(9,13,22,33)], [(1,34),(2,35),(3,36),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,25),(17,26),(18,27)]])
 

C9×C3⋊D4 is a maximal subgroup of   Dic3.D18  D18.4D6  D18⋊D6  S3×D4×C9

81 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 6A6B6C···6M6N6O9A···9F9G···9L12A12B18A···18F18G···18AD18AE···18AJ36A···36F
order1222333334666···6669···99···9121218···1818···1818···1836···36
size1126112226112···2661···12···2661···12···26···66···6

81 irreducible representations

dim111111111111222222222222
type+++++++
imageC1C2C2C2C3C6C6C6C9C18C18C18S3D4D6C3×S3C3⋊D4C3×D4S3×C6S3×C9D4×C9C3×C3⋊D4S3×C18C9×C3⋊D4
kernelC9×C3⋊D4C9×Dic3S3×C18C6×C18C3×C3⋊D4C3×Dic3S3×C6C62C3⋊D4Dic3D6C2×C6C2×C18C3×C9C18C2×C6C9C32C6C22C3C3C2C1
# reps1111222266661112222664612

Matrix representation of C9×C3⋊D4 ►in GL2(𝔽19) generated by

170
017
,
92
29
,
01
180
,
180
01
G:=sub<GL(2,GF(19))| [17,0,0,17],[9,2,2,9],[0,18,1,0],[18,0,0,1] >;
 

C9×C3⋊D4 in GAP, Magma, Sage, TeX 

C_9\times C_3\rtimes D_4
 
% in TeX
 
G:=Group("C9xC3:D4");
 
// GroupNames label
 
G:=SmallGroup(216,58);
 
// by ID
 
G=gap.SmallGroup(216,58);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,122,5189]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^9=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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