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## G = S3×C9⋊D4order 432 = 24·33

### Direct product of S3 and C9⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — S3×C9⋊D4
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — S3×C18 — C2×S3×D9 — S3×C9⋊D4
 Lower central C3×C9 — C3×C18 — S3×C9⋊D4
 Upper central C1 — C2 — C22

Generators and relations for S3×C9⋊D4
G = < a,b,c,d,e | a3=b2=c9=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1160 in 178 conjugacy classes, 47 normal (41 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, S3, C6, C6, C2×C4, D4, C23, C9, C9, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C22×S3, C22×C6, C3×C9, Dic9, Dic9, D18, D18, C2×C18, C2×C18, C3×Dic3, C3⋊Dic3, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, S3×D4, C2×C3⋊D4, C3×D9, S3×C9, S3×C9, C9⋊S3, C3×C18, C3×C18, C2×Dic9, C9⋊D4, C9⋊D4, C22×D9, C22×C18, S3×Dic3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C327D4, C2×S32, S3×C2×C6, C3×Dic9, C9⋊Dic3, S3×D9, C6×D9, S3×C18, S3×C18, C2×C9⋊S3, C6×C18, C2×C9⋊D4, S3×C3⋊D4, S3×Dic9, D6⋊D9, C9⋊D12, C3×C9⋊D4, C6.D18, C2×S3×D9, S3×C2×C18, S3×C9⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, D18, S32, S3×D4, C2×C3⋊D4, C9⋊D4, C22×D9, C2×S32, S3×D9, C2×C9⋊D4, S3×C3⋊D4, C2×S3×D9, S3×C9⋊D4

Smallest permutation representation of S3×C9⋊D4
On 72 points
Generators in S72
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 37)(8 38)(9 39)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)
(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 32 13 23)(2 31 14 22)(3 30 15 21)(4 29 16 20)(5 28 17 19)(6 36 18 27)(7 35 10 26)(8 34 11 25)(9 33 12 24)(37 71 46 62)(38 70 47 61)(39 69 48 60)(40 68 49 59)(41 67 50 58)(42 66 51 57)(43 65 52 56)(44 64 53 55)(45 72 54 63)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 27)(7 26)(8 25)(9 24)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 36)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 63)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 72)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 9A 9B 9C 9D 9E 9F 12 18A ··· 18I 18J ··· 18R 18S ··· 18AD order 1 2 2 2 2 2 2 2 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 9 9 9 9 9 9 12 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 2 3 3 6 18 54 2 2 4 18 54 2 2 2 2 4 4 4 4 6 6 6 6 36 2 2 2 4 4 4 36 2 ··· 2 4 ··· 4 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D6 D6 D9 C3⋊D4 D18 D18 C9⋊D4 S32 S3×D4 C2×S32 S3×D9 S3×C3⋊D4 C2×S3×D9 S3×C9⋊D4 kernel S3×C9⋊D4 S3×Dic9 D6⋊D9 C9⋊D12 C3×C9⋊D4 C6.D18 C2×S3×D9 S3×C2×C18 C9⋊D4 S3×C2×C6 S3×C9 Dic9 D18 C2×C18 S3×C6 C62 C22×S3 C3×S3 D6 C2×C6 S3 C2×C6 C9 C6 C22 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 3 4 6 3 12 1 1 1 3 2 3 6

Matrix representation of S3×C9⋊D4 in GL6(𝔽37)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 1 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 10 0 0 0 0 0 0 26 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 0 0 0 0 0 0 16
,
 0 1 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[10,0,0,0,0,0,0,26,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,16],[0,36,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

S3×C9⋊D4 in GAP, Magma, Sage, TeX

S_3\times C_9\rtimes D_4
% in TeX

G:=Group("S3xC9:D4");
// GroupNames label

G:=SmallGroup(432,313);
// by ID

G=gap.SmallGroup(432,313);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^9=d^4=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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