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## G = S3×C32⋊7D4order 432 = 24·33

### Direct product of S3 and C32⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — S3×C32⋊7D4
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C2×S3×C3⋊S3 — S3×C32⋊7D4
 Lower central C33 — C32×C6 — S3×C32⋊7D4
 Upper central C1 — C2 — C22

Generators and relations for S3×C327D4
G = < a,b,c,d,e,f | a3=b2=c3=d3=e4=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=fcf=c-1, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 2336 in 388 conjugacy classes, 80 normal (32 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, S3×D4, C2×C3⋊D4, S3×C32, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, C32×C6, S3×Dic3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C2×C3⋊Dic3, C327D4, C327D4, C2×S32, S3×C2×C6, C22×C3⋊S3, C2×C62, C3×C3⋊Dic3, C335C4, S3×C3⋊S3, S3×C3×C6, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, C3×C62, S3×C3⋊D4, C2×C327D4, S3×C3⋊Dic3, C336D4, C337D4, C3×C327D4, C3315D4, C2×S3×C3⋊S3, S3×C62, S3×C327D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C3⋊D4, C22×S3, S32, C2×C3⋊S3, S3×D4, C2×C3⋊D4, C327D4, C2×S32, C22×C3⋊S3, S3×C3⋊S3, S3×C3⋊D4, C2×C327D4, C2×S3×C3⋊S3, S3×C327D4

Smallest permutation representation of S3×C327D4
On 72 points
Generators in S72
(1 29 7)(2 30 8)(3 31 5)(4 32 6)(9 61 55)(10 62 56)(11 63 53)(12 64 54)(13 37 41)(14 38 42)(15 39 43)(16 40 44)(17 49 72)(18 50 69)(19 51 70)(20 52 71)(21 59 34)(22 60 35)(23 57 36)(24 58 33)(25 66 45)(26 67 46)(27 68 47)(28 65 48)
(1 11)(2 12)(3 9)(4 10)(5 61)(6 62)(7 63)(8 64)(13 46)(14 47)(15 48)(16 45)(17 59)(18 60)(19 57)(20 58)(21 49)(22 50)(23 51)(24 52)(25 44)(26 41)(27 42)(28 43)(29 53)(30 54)(31 55)(32 56)(33 71)(34 72)(35 69)(36 70)(37 67)(38 68)(39 65)(40 66)
(1 58 44)(2 41 59)(3 60 42)(4 43 57)(5 22 38)(6 39 23)(7 24 40)(8 37 21)(9 18 27)(10 28 19)(11 20 25)(12 26 17)(13 34 30)(14 31 35)(15 36 32)(16 29 33)(45 53 71)(46 72 54)(47 55 69)(48 70 56)(49 64 67)(50 68 61)(51 62 65)(52 66 63)
(1 16 24)(2 21 13)(3 14 22)(4 23 15)(5 42 35)(6 36 43)(7 44 33)(8 34 41)(9 47 50)(10 51 48)(11 45 52)(12 49 46)(17 67 54)(18 55 68)(19 65 56)(20 53 66)(25 71 63)(26 64 72)(27 69 61)(28 62 70)(29 40 58)(30 59 37)(31 38 60)(32 57 39)
(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 12)(10 11)(13 14)(15 16)(17 18)(19 20)(21 22)(23 24)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)(37 38)(39 40)(41 42)(43 44)(45 48)(46 47)(49 50)(51 52)(53 56)(54 55)(57 58)(59 60)(61 64)(62 63)(65 66)(67 68)(69 72)(70 71)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3E 3F 3G 3H 3I 4A 4B 6A ··· 6M 6N ··· 6Z 6AA ··· 6AP 6AQ 12 order 1 2 2 2 2 2 2 2 3 ··· 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 12 size 1 1 2 3 3 6 18 54 2 ··· 2 4 4 4 4 18 54 2 ··· 2 4 ··· 4 6 ··· 6 36 36

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D6 C3⋊D4 S32 S3×D4 C2×S32 S3×C3⋊D4 kernel S3×C32⋊7D4 S3×C3⋊Dic3 C33⋊6D4 C33⋊7D4 C3×C32⋊7D4 C33⋊15D4 C2×S3×C3⋊S3 S3×C62 C32⋊7D4 S3×C2×C6 S3×C32 C3⋊Dic3 S3×C6 C2×C3⋊S3 C62 C3×S3 C2×C6 C32 C6 C3 # reps 1 1 1 1 1 1 1 1 1 4 2 1 8 1 5 16 4 1 4 8

Matrix representation of S3×C327D4 in GL8(ℤ)

 1 0 0 0 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 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 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 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 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 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 0 0 0 1
,
 0 1 0 0 0 0 0 0 -1 0 0 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 0 0 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 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 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

S3×C327D4 in GAP, Magma, Sage, TeX

S_3\times C_3^2\rtimes_7D_4
% in TeX

G:=Group("S3xC3^2:7D4");
// GroupNames label

G:=SmallGroup(432,684);
// by ID

G=gap.SmallGroup(432,684);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,571,2028,14118]);
// Polycyclic

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

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