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## G = C32×D4⋊S3order 432 = 24·33

### Direct product of C32 and D4⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C32×D4⋊S3
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C32×C12 — C32×D12 — C32×D4⋊S3
 Lower central C3 — C6 — C12 — C32×D4⋊S3
 Upper central C1 — C3×C6 — C3×C12 — D4×C32

Generators and relations for C32×D4⋊S3
G = < a,b,c,d,e,f | a3=b3=c4=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=fcf=c-1, ce=ec, de=ed, fdf=cd, fef=e-1 >

Subgroups: 496 in 196 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C8, D4, D4, C32, C32, C32, C12, C12, C12, D6, C2×C6, D8, C3×S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×D4, C3×D4, C33, C3×C12, C3×C12, C3×C12, S3×C6, C62, D4⋊S3, C3×D8, S3×C32, C32×C6, C32×C6, C3×C3⋊C8, C3×C24, C3×D12, D4×C32, D4×C32, D4×C32, C32×C12, S3×C3×C6, C3×C62, C3×D4⋊S3, C32×D8, C32×C3⋊C8, C32×D12, D4×C33, C32×D4⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C32, D6, C2×C6, D8, C3×S3, C3×C6, C3⋊D4, C3×D4, S3×C6, C62, D4⋊S3, C3×D8, S3×C32, C3×C3⋊D4, D4×C32, S3×C3×C6, C3×D4⋊S3, C32×D8, C32×C3⋊D4, C32×D4⋊S3

Smallest permutation representation of C32×D4⋊S3
On 72 points
Generators in S72
(1 38 28)(2 39 25)(3 40 26)(4 37 27)(5 57 51)(6 58 52)(7 59 49)(8 60 50)(9 67 48)(10 68 45)(11 65 46)(12 66 47)(13 35 30)(14 36 31)(15 33 32)(16 34 29)(17 43 22)(18 44 23)(19 41 24)(20 42 21)(53 70 64)(54 71 61)(55 72 62)(56 69 63)
(1 34 22)(2 35 23)(3 36 24)(4 33 21)(5 68 56)(6 65 53)(7 66 54)(8 67 55)(9 62 50)(10 63 51)(11 64 52)(12 61 49)(13 44 25)(14 41 26)(15 42 27)(16 43 28)(17 38 29)(18 39 30)(19 40 31)(20 37 32)(45 69 57)(46 70 58)(47 71 59)(48 72 60)
(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 55 56)(57 58 59 60)(61 62 63 64)(65 66 67 68)(69 70 71 72)
(1 4)(2 3)(6 8)(9 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)(25 26)(27 28)(29 32)(30 31)(33 34)(35 36)(37 38)(39 40)(41 44)(42 43)(46 48)(50 52)(53 55)(58 60)(62 64)(65 67)(70 72)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 10 69)(6 11 70)(7 12 71)(8 9 72)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 37 42)(34 38 43)(35 39 44)(36 40 41)(45 56 51)(46 53 52)(47 54 49)(48 55 50)(57 68 63)(58 65 64)(59 66 61)(60 67 62)
(1 46)(2 45)(3 48)(4 47)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 59)(22 58)(23 57)(24 60)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 62)(32 61)(33 71)(34 70)(35 69)(36 72)

G:=sub<Sym(72)| (1,38,28)(2,39,25)(3,40,26)(4,37,27)(5,57,51)(6,58,52)(7,59,49)(8,60,50)(9,67,48)(10,68,45)(11,65,46)(12,66,47)(13,35,30)(14,36,31)(15,33,32)(16,34,29)(17,43,22)(18,44,23)(19,41,24)(20,42,21)(53,70,64)(54,71,61)(55,72,62)(56,69,63), (1,34,22)(2,35,23)(3,36,24)(4,33,21)(5,68,56)(6,65,53)(7,66,54)(8,67,55)(9,62,50)(10,63,51)(11,64,52)(12,61,49)(13,44,25)(14,41,26)(15,42,27)(16,43,28)(17,38,29)(18,39,30)(19,40,31)(20,37,32)(45,69,57)(46,70,58)(47,71,59)(48,72,60), (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)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72), (1,4)(2,3)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,28)(29,32)(30,31)(33,34)(35,36)(37,38)(39,40)(41,44)(42,43)(46,48)(50,52)(53,55)(58,60)(62,64)(65,67)(70,72), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,10,69)(6,11,70)(7,12,71)(8,9,72)(21,32,27)(22,29,28)(23,30,25)(24,31,26)(33,37,42)(34,38,43)(35,39,44)(36,40,41)(45,56,51)(46,53,52)(47,54,49)(48,55,50)(57,68,63)(58,65,64)(59,66,61)(60,67,62), (1,46)(2,45)(3,48)(4,47)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,59)(22,58)(23,57)(24,60)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,71)(34,70)(35,69)(36,72)>;

G:=Group( (1,38,28)(2,39,25)(3,40,26)(4,37,27)(5,57,51)(6,58,52)(7,59,49)(8,60,50)(9,67,48)(10,68,45)(11,65,46)(12,66,47)(13,35,30)(14,36,31)(15,33,32)(16,34,29)(17,43,22)(18,44,23)(19,41,24)(20,42,21)(53,70,64)(54,71,61)(55,72,62)(56,69,63), (1,34,22)(2,35,23)(3,36,24)(4,33,21)(5,68,56)(6,65,53)(7,66,54)(8,67,55)(9,62,50)(10,63,51)(11,64,52)(12,61,49)(13,44,25)(14,41,26)(15,42,27)(16,43,28)(17,38,29)(18,39,30)(19,40,31)(20,37,32)(45,69,57)(46,70,58)(47,71,59)(48,72,60), (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)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72), (1,4)(2,3)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,28)(29,32)(30,31)(33,34)(35,36)(37,38)(39,40)(41,44)(42,43)(46,48)(50,52)(53,55)(58,60)(62,64)(65,67)(70,72), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,10,69)(6,11,70)(7,12,71)(8,9,72)(21,32,27)(22,29,28)(23,30,25)(24,31,26)(33,37,42)(34,38,43)(35,39,44)(36,40,41)(45,56,51)(46,53,52)(47,54,49)(48,55,50)(57,68,63)(58,65,64)(59,66,61)(60,67,62), (1,46)(2,45)(3,48)(4,47)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,59)(22,58)(23,57)(24,60)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,71)(34,70)(35,69)(36,72) );

G=PermutationGroup([[(1,38,28),(2,39,25),(3,40,26),(4,37,27),(5,57,51),(6,58,52),(7,59,49),(8,60,50),(9,67,48),(10,68,45),(11,65,46),(12,66,47),(13,35,30),(14,36,31),(15,33,32),(16,34,29),(17,43,22),(18,44,23),(19,41,24),(20,42,21),(53,70,64),(54,71,61),(55,72,62),(56,69,63)], [(1,34,22),(2,35,23),(3,36,24),(4,33,21),(5,68,56),(6,65,53),(7,66,54),(8,67,55),(9,62,50),(10,63,51),(11,64,52),(12,61,49),(13,44,25),(14,41,26),(15,42,27),(16,43,28),(17,38,29),(18,39,30),(19,40,31),(20,37,32),(45,69,57),(46,70,58),(47,71,59),(48,72,60)], [(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),(53,54,55,56),(57,58,59,60),(61,62,63,64),(65,66,67,68),(69,70,71,72)], [(1,4),(2,3),(6,8),(9,11),(13,14),(15,16),(17,20),(18,19),(21,22),(23,24),(25,26),(27,28),(29,32),(30,31),(33,34),(35,36),(37,38),(39,40),(41,44),(42,43),(46,48),(50,52),(53,55),(58,60),(62,64),(65,67),(70,72)], [(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,10,69),(6,11,70),(7,12,71),(8,9,72),(21,32,27),(22,29,28),(23,30,25),(24,31,26),(33,37,42),(34,38,43),(35,39,44),(36,40,41),(45,56,51),(46,53,52),(47,54,49),(48,55,50),(57,68,63),(58,65,64),(59,66,61),(60,67,62)], [(1,46),(2,45),(3,48),(4,47),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,56),(14,55),(15,54),(16,53),(17,52),(18,51),(19,50),(20,49),(21,59),(22,58),(23,57),(24,60),(25,68),(26,67),(27,66),(28,65),(29,64),(30,63),(31,62),(32,61),(33,71),(34,70),(35,69),(36,72)]])

108 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Q 4 6A ··· 6H 6I ··· 6Q 6R ··· 6AQ 6AR ··· 6AY 8A 8B 12A ··· 12H 12I ··· 12Q 24A ··· 24P order 1 2 2 2 3 ··· 3 3 ··· 3 4 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 4 12 1 ··· 1 2 ··· 2 2 1 ··· 1 2 ··· 2 4 ··· 4 12 ··· 12 6 6 2 ··· 2 4 ··· 4 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D8 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×D8 C3×C3⋊D4 D4⋊S3 C3×D4⋊S3 kernel C32×D4⋊S3 C32×C3⋊C8 C32×D12 D4×C33 C3×D4⋊S3 C3×C3⋊C8 C3×D12 D4×C32 D4×C32 C32×C6 C3×C12 C33 C3×D4 C3×C6 C3×C6 C12 C32 C6 C32 C3 # reps 1 1 1 1 8 8 8 8 1 1 1 2 8 2 8 8 16 16 1 8

Matrix representation of C32×D4⋊S3 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 64 0 0 0 0 64
,
 1 0 0 0 0 1 0 0 0 0 64 0 0 0 0 64
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 64 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 57 16 0 0 16 16
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[64,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,57,16,0,0,16,16] >;

C32×D4⋊S3 in GAP, Magma, Sage, TeX

C_3^2\times D_4\rtimes S_3
% in TeX

G:=Group("C3^2xD4:S3");
// GroupNames label

G:=SmallGroup(432,475);
// by ID

G=gap.SmallGroup(432,475);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,533,3784,1901,102,14118]);
// Polycyclic

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

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