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## G = C32×D4⋊2S3order 432 = 24·33

### Direct product of C32 and D4⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C32×D4⋊2S3
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — S3×C3×C6 — S3×C3×C12 — C32×D4⋊2S3
 Lower central C3 — C6 — C32×D4⋊2S3
 Upper central C1 — C3×C6 — D4×C32

Generators and relations for C32×D42S3
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=c-1, ce=ec, cf=fc, de=ed, fdf=c2d, fef=e-1 >

Subgroups: 592 in 304 conjugacy classes, 138 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, D4, Q8, C32, C32, C32, Dic3, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×D4, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C62, C62, D42S3, C3×C4○D4, S3×C32, C32×C6, C32×C6, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, C6×C12, D4×C32, D4×C32, D4×C32, Q8×C32, C32×Dic3, C32×Dic3, C32×C12, S3×C3×C6, C3×C62, C3×D42S3, C32×C4○D4, C32×Dic6, S3×C3×C12, Dic3×C3×C6, C32×C3⋊D4, D4×C33, C32×D42S3
Quotients: C1, C2, C3, C22, S3, C6, C23, C32, D6, C2×C6, C4○D4, C3×S3, C3×C6, C22×S3, C22×C6, S3×C6, C62, D42S3, C3×C4○D4, S3×C32, S3×C2×C6, C2×C62, S3×C3×C6, C3×D42S3, C32×C4○D4, S3×C62, C32×D42S3

Smallest permutation representation of C32×D42S3
On 72 points
Generators in S72
(1 29 14)(2 30 15)(3 31 16)(4 32 13)(5 10 33)(6 11 34)(7 12 35)(8 9 36)(17 22 67)(18 23 68)(19 24 65)(20 21 66)(25 39 58)(26 40 59)(27 37 60)(28 38 57)(41 52 53)(42 49 54)(43 50 55)(44 51 56)(45 61 70)(46 62 71)(47 63 72)(48 64 69)
(1 26 35)(2 27 36)(3 28 33)(4 25 34)(5 31 38)(6 32 39)(7 29 40)(8 30 37)(9 15 60)(10 16 57)(11 13 58)(12 14 59)(17 47 49)(18 48 50)(19 45 51)(20 46 52)(21 62 53)(22 63 54)(23 64 55)(24 61 56)(41 66 71)(42 67 72)(43 68 69)(44 65 70)
(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)(5 8)(6 7)(9 10)(11 12)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)(25 26)(27 28)(29 32)(30 31)(33 36)(34 35)(37 38)(39 40)(41 42)(43 44)(45 48)(46 47)(49 52)(50 51)(53 54)(55 56)(57 60)(58 59)(61 64)(62 63)(65 68)(66 67)(69 70)(71 72)
(1 14 29)(2 15 30)(3 16 31)(4 13 32)(5 33 10)(6 34 11)(7 35 12)(8 36 9)(17 22 67)(18 23 68)(19 24 65)(20 21 66)(25 58 39)(26 59 40)(27 60 37)(28 57 38)(41 52 53)(42 49 54)(43 50 55)(44 51 56)(45 61 70)(46 62 71)(47 63 72)(48 64 69)
(1 41)(2 42)(3 43)(4 44)(5 48)(6 45)(7 46)(8 47)(9 63)(10 64)(11 61)(12 62)(13 56)(14 53)(15 54)(16 55)(17 37)(18 38)(19 39)(20 40)(21 59)(22 60)(23 57)(24 58)(25 65)(26 66)(27 67)(28 68)(29 52)(30 49)(31 50)(32 51)(33 69)(34 70)(35 71)(36 72)

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

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

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

135 conjugacy classes

 class 1 2A 2B 2C 2D 3A ··· 3H 3I ··· 3Q 4A 4B 4C 4D 4E 6A ··· 6H 6I ··· 6AG 6AH ··· 6AY 6AZ ··· 6BG 12A ··· 12H 12I ··· 12X 12Y ··· 12AG 12AH ··· 12AW order 1 2 2 2 2 3 ··· 3 3 ··· 3 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 2 2 6 1 ··· 1 2 ··· 2 2 3 3 6 6 1 ··· 1 2 ··· 2 4 ··· 4 6 ··· 6 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 C3×C4○D4 D4⋊2S3 C3×D4⋊2S3 kernel C32×D4⋊2S3 C32×Dic6 S3×C3×C12 Dic3×C3×C6 C32×C3⋊D4 D4×C33 C3×D4⋊2S3 C3×Dic6 S3×C12 C6×Dic3 C3×C3⋊D4 D4×C32 D4×C32 C3×C12 C62 C33 C3×D4 C12 C2×C6 C32 C32 C3 # reps 1 1 1 2 2 1 8 8 8 16 16 8 1 1 2 2 8 8 16 16 1 8

Matrix representation of C32×D42S3 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 3
,
 12 0 0 0 0 12 0 0 0 0 1 1 0 0 11 12
,
 12 0 0 0 0 12 0 0 0 0 1 1 0 0 0 12
,
 9 0 0 0 9 3 0 0 0 0 1 0 0 0 0 1
,
 1 8 0 0 0 12 0 0 0 0 8 8 0 0 10 5
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[12,0,0,0,0,12,0,0,0,0,1,11,0,0,1,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,1,12],[9,9,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,8,12,0,0,0,0,8,10,0,0,8,5] >;

C32×D42S3 in GAP, Magma, Sage, TeX

C_3^2\times D_4\rtimes_2S_3
% in TeX

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

G:=SmallGroup(432,705);
// by ID

G=gap.SmallGroup(432,705);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,512,1598,807,14118]);
// 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=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=c^2*d,f*e*f=e^-1>;
// generators/relations

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