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## G = C2×C32.3S4order 432 = 24·33

### Direct product of C2 and C32.3S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C3.A4 — C2×C32.3S4
 Chief series C1 — C22 — C2×C6 — C62 — C3×C3.A4 — C32.3S4 — C2×C32.3S4
 Lower central C3×C3.A4 — C2×C32.3S4
 Upper central C1 — C2

Generators and relations for C2×C32.3S4
G = < a,b,c,d,e | a2=b6=c6=e2=1, d3=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b4c3, ebe=b2c3, dcd-1=b3c, ece=b3c2, ede=c4d2 >

Subgroups: 1540 in 178 conjugacy classes, 35 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C9, C32, Dic3, D6, C2×C6, C2×C6, C2×C6, C2×D4, D9, C18, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22×C6, C3×C9, C3.A4, D18, C3⋊Dic3, C2×C3⋊S3, C62, C62, C2×C3⋊D4, C9⋊S3, C3×C18, C3.S4, C2×C3.A4, C2×C3⋊Dic3, C327D4, C22×C3⋊S3, C2×C62, C3×C3.A4, C2×C9⋊S3, C2×C3.S4, C2×C327D4, C32.3S4, C6×C3.A4, C2×C32.3S4
Quotients: C1, C2, C22, S3, D6, D9, C3⋊S3, S4, D18, C2×C3⋊S3, C2×S4, C9⋊S3, C3.S4, C3⋊S4, C2×C9⋊S3, C2×C3.S4, C2×C3⋊S4, C32.3S4, C2×C32.3S4

Smallest permutation representation of C2×C32.3S4
On 54 points
Generators in S54
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 46)(8 47)(9 48)(10 24)(11 25)(12 26)(13 27)(14 19)(15 20)(16 21)(17 22)(18 23)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 37)(35 38)(36 39)
(1 33 14 49 45 19)(2 34 15 50 37 20)(3 38 16)(4 36 17 52 39 22)(5 28 18 53 40 23)(6 41 10)(7 30 11 46 42 25)(8 31 12 47 43 26)(9 44 13)(21 51 35)(24 54 29)(27 48 32)
(1 46 4 49 7 52)(2 8 5)(3 48 6 51 9 54)(10 21 13 24 16 27)(11 22 14 25 17 19)(12 18 15)(20 26 23)(28 34 31)(29 38 32 41 35 44)(30 39 33 42 36 45)(37 43 40)(47 53 50)
(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 9)(3 8)(4 7)(5 6)(10 40)(11 39)(12 38)(13 37)(14 45)(15 44)(16 43)(17 42)(18 41)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(25 36)(26 35)(27 34)(46 52)(47 51)(48 50)(53 54)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E ··· 6L 9A ··· 9I 18A ··· 18I order 1 2 2 2 2 2 3 3 3 3 4 4 6 6 6 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 3 3 54 54 2 2 2 2 54 54 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 S3 D6 D6 D9 D18 S4 C2×S4 C3.S4 C3⋊S4 C2×C3.S4 C2×C3⋊S4 kernel C2×C32.3S4 C32.3S4 C6×C3.A4 C2×C3.A4 C2×C62 C3.A4 C62 C22×C6 C2×C6 C3×C6 C32 C6 C6 C3 C3 # reps 1 2 1 3 1 3 1 9 9 2 2 3 1 3 1

Matrix representation of C2×C32.3S4 in GL7(𝔽37)

 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36
,
 36 36 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 36 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 1 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 36
,
 36 36 0 0 0 0 0 1 0 0 0 0 0 0 0 0 26 31 0 0 0 0 0 6 20 0 0 0 0 0 0 0 1 0 35 0 0 0 0 0 0 36 0 0 0 0 0 1 36
,
 1 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0

G:=sub<GL(7,GF(37))| [36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[36,1,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,36,0,0,0,0,0,36,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,36,36,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36],[36,1,0,0,0,0,0,36,0,0,0,0,0,0,0,0,26,6,0,0,0,0,0,31,20,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,35,36,36],[1,36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0] >;

C2×C32.3S4 in GAP, Magma, Sage, TeX

C_2\times C_3^2._3S_4
% in TeX

G:=Group("C2xC3^2.3S4");
// GroupNames label

G:=SmallGroup(432,537);
// by ID

G=gap.SmallGroup(432,537);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,926,394,675,2524,9077,2287,5298,3989]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=c^6=e^2=1,d^3=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^4*c^3,e*b*e=b^2*c^3,d*c*d^-1=b^3*c,e*c*e=b^3*c^2,e*d*e=c^4*d^2>;
// generators/relations

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