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## G = C2×C9.S4order 432 = 24·33

### Direct product of C2 and C9.S4

Aliases: C2×C9.S4, C23⋊D27, C22⋊D54, C18.4S4, C9.(C2×S4), (C2×C18).D6, (C2×C6).D18, C9.A4⋊C22, C6.4(C3.S4), (C22×C6).3D9, (C22×C18).3S3, (C2×C9.A4)⋊C2, C3.(C2×C3.S4), SmallGroup(432,224)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C9.A4 — C2×C9.S4
 Chief series C1 — C22 — C2×C6 — C2×C18 — C9.A4 — C9.S4 — C2×C9.S4
 Lower central C9.A4 — C2×C9.S4
 Upper central C1 — C2

Generators and relations for C2×C9.S4
G = < a,b,c,d,e,f | a2=b9=c2=d2=f2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=b-1e2 >

Subgroups: 814 in 79 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C9, Dic3, D6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C27, Dic9, D18, C2×C18, C2×C18, C2×C3⋊D4, D27, C54, C2×Dic9, C9⋊D4, C22×D9, C22×C18, C9.A4, D54, C2×C9⋊D4, C9.S4, C2×C9.A4, C2×C9.S4
Quotients: C1, C2, C22, S3, D6, D9, S4, D18, C2×S4, D27, C3.S4, D54, C2×C3.S4, C9.S4, C2×C9.S4

Smallest permutation representation of C2×C9.S4
On 54 points
Generators in S54
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)
(1 4 7 10 13 16 19 22 25)(2 5 8 11 14 17 20 23 26)(3 6 9 12 15 18 21 24 27)(28 31 34 37 40 43 46 49 52)(29 32 35 38 41 44 47 50 53)(30 33 36 39 42 45 48 51 54)
(1 45)(2 46)(4 48)(5 49)(7 51)(8 52)(10 54)(11 28)(13 30)(14 31)(16 33)(17 34)(19 36)(20 37)(22 39)(23 40)(25 42)(26 43)
(2 46)(3 47)(5 49)(6 50)(8 52)(9 53)(11 28)(12 29)(14 31)(15 32)(17 34)(18 35)(20 37)(21 38)(23 40)(24 41)(26 43)(27 44)
(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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)
(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(28 35)(29 34)(30 33)(31 32)(36 54)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)

G:=sub<Sym(54)| (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44), (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54), (1,45)(2,46)(4,48)(5,49)(7,51)(8,52)(10,54)(11,28)(13,30)(14,31)(16,33)(17,34)(19,36)(20,37)(22,39)(23,40)(25,42)(26,43), (2,46)(3,47)(5,49)(6,50)(8,52)(9,53)(11,28)(12,29)(14,31)(15,32)(17,34)(18,35)(20,37)(21,38)(23,40)(24,41)(26,43)(27,44), (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)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,35)(29,34)(30,33)(31,32)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)>;

G:=Group( (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44), (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54), (1,45)(2,46)(4,48)(5,49)(7,51)(8,52)(10,54)(11,28)(13,30)(14,31)(16,33)(17,34)(19,36)(20,37)(22,39)(23,40)(25,42)(26,43), (2,46)(3,47)(5,49)(6,50)(8,52)(9,53)(11,28)(12,29)(14,31)(15,32)(17,34)(18,35)(20,37)(21,38)(23,40)(24,41)(26,43)(27,44), (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)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,35)(29,34)(30,33)(31,32)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46) );

G=PermutationGroup([[(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44)], [(1,4,7,10,13,16,19,22,25),(2,5,8,11,14,17,20,23,26),(3,6,9,12,15,18,21,24,27),(28,31,34,37,40,43,46,49,52),(29,32,35,38,41,44,47,50,53),(30,33,36,39,42,45,48,51,54)], [(1,45),(2,46),(4,48),(5,49),(7,51),(8,52),(10,54),(11,28),(13,30),(14,31),(16,33),(17,34),(19,36),(20,37),(22,39),(23,40),(25,42),(26,43)], [(2,46),(3,47),(5,49),(6,50),(8,52),(9,53),(11,28),(12,29),(14,31),(15,32),(17,34),(18,35),(20,37),(21,38),(23,40),(24,41),(26,43),(27,44)], [(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),(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)], [(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(28,35),(29,34),(30,33),(31,32),(36,54),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46)]])

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 6A 6B 6C 9A 9B 9C 18A 18B 18C 18D ··· 18I 27A ··· 27I 54A ··· 54I order 1 2 2 2 2 2 3 4 4 6 6 6 9 9 9 18 18 18 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 3 3 54 54 2 54 54 2 6 6 2 2 2 2 2 2 6 ··· 6 8 ··· 8 8 ··· 8

42 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 3 3 6 6 6 6 type + + + + + + + + + + + + + + + image C1 C2 C2 S3 D6 D9 D18 D27 D54 S4 C2×S4 C3.S4 C2×C3.S4 C9.S4 C2×C9.S4 kernel C2×C9.S4 C9.S4 C2×C9.A4 C22×C18 C2×C18 C22×C6 C2×C6 C23 C22 C18 C9 C6 C3 C2 C1 # reps 1 2 1 1 1 3 3 9 9 2 2 1 1 3 3

Matrix representation of C2×C9.S4 in GL5(𝔽109)

 108 0 0 0 0 0 108 0 0 0 0 0 108 0 0 0 0 0 108 0 0 0 0 0 108
,
 5 54 0 0 0 14 86 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 108 0 0 0 0 0 1 0 0 0 0 108 108
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 108 0 0 0 108 0 108
,
 82 34 0 0 0 29 24 0 0 0 0 0 0 1 0 0 0 108 108 107 0 0 0 0 1
,
 106 20 0 0 0 65 3 0 0 0 0 0 1 0 0 0 0 108 108 107 0 0 0 0 1

G:=sub<GL(5,GF(109))| [108,0,0,0,0,0,108,0,0,0,0,0,108,0,0,0,0,0,108,0,0,0,0,0,108],[5,14,0,0,0,54,86,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,108,0,0,0,0,0,1,108,0,0,0,0,108],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,108,0,0,0,108,0,0,0,0,0,108],[82,29,0,0,0,34,24,0,0,0,0,0,0,108,0,0,0,1,108,0,0,0,0,107,1],[106,65,0,0,0,20,3,0,0,0,0,0,1,108,0,0,0,0,108,0,0,0,0,107,1] >;

C2×C9.S4 in GAP, Magma, Sage, TeX

C_2\times C_9.S_4
% in TeX

G:=Group("C2xC9.S4");
// GroupNames label

G:=SmallGroup(432,224);
// by ID

G=gap.SmallGroup(432,224);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,422,331,1683,192,2524,9077,2287,5298,3989]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^9=c^2=d^2=f^2=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=b^-1*e^2>;
// generators/relations

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