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## G = C4×C9⋊S3order 216 = 23·33

### Direct product of C4 and C9⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C4×C9⋊S3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C2×C9⋊S3 — C4×C9⋊S3
 Lower central C3×C9 — C4×C9⋊S3
 Upper central C1 — C4

Generators and relations for C4×C9⋊S3
G = < a,b,c,d | a4=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 410 in 80 conjugacy classes, 35 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, C9, C32, Dic3, C12, C12, D6, D9, C18, C3⋊S3, C3×C6, C4×S3, C3×C9, Dic9, C36, D18, C3⋊Dic3, C3×C12, C2×C3⋊S3, C9⋊S3, C3×C18, C4×D9, C4×C3⋊S3, C9⋊Dic3, C3×C36, C2×C9⋊S3, C4×C9⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, D9, C3⋊S3, C4×S3, D18, C2×C3⋊S3, C9⋊S3, C4×D9, C4×C3⋊S3, C2×C9⋊S3, C4×C9⋊S3

Smallest permutation representation of C4×C9⋊S3
On 108 points
Generators in S108
(1 103 49 76)(2 104 50 77)(3 105 51 78)(4 106 52 79)(5 107 53 80)(6 108 54 81)(7 100 46 73)(8 101 47 74)(9 102 48 75)(10 59 86 32)(11 60 87 33)(12 61 88 34)(13 62 89 35)(14 63 90 36)(15 55 82 28)(16 56 83 29)(17 57 84 30)(18 58 85 31)(19 71 98 44)(20 72 99 45)(21 64 91 37)(22 65 92 38)(23 66 93 39)(24 67 94 40)(25 68 95 41)(26 69 96 42)(27 70 97 43)
(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)(73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)
(1 35 37)(2 36 38)(3 28 39)(4 29 40)(5 30 41)(6 31 42)(7 32 43)(8 33 44)(9 34 45)(10 27 100)(11 19 101)(12 20 102)(13 21 103)(14 22 104)(15 23 105)(16 24 106)(17 25 107)(18 26 108)(46 59 70)(47 60 71)(48 61 72)(49 62 64)(50 63 65)(51 55 66)(52 56 67)(53 57 68)(54 58 69)(73 86 97)(74 87 98)(75 88 99)(76 89 91)(77 90 92)(78 82 93)(79 83 94)(80 84 95)(81 85 96)
(1 49)(2 48)(3 47)(4 46)(5 54)(6 53)(7 52)(8 51)(9 50)(10 94)(11 93)(12 92)(13 91)(14 99)(15 98)(16 97)(17 96)(18 95)(19 82)(20 90)(21 89)(22 88)(23 87)(24 86)(25 85)(26 84)(27 83)(28 71)(29 70)(30 69)(31 68)(32 67)(33 66)(34 65)(35 64)(36 72)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 63)(73 106)(74 105)(75 104)(76 103)(77 102)(78 101)(79 100)(80 108)(81 107)

G:=sub<Sym(108)| (1,103,49,76)(2,104,50,77)(3,105,51,78)(4,106,52,79)(5,107,53,80)(6,108,54,81)(7,100,46,73)(8,101,47,74)(9,102,48,75)(10,59,86,32)(11,60,87,33)(12,61,88,34)(13,62,89,35)(14,63,90,36)(15,55,82,28)(16,56,83,29)(17,57,84,30)(18,58,85,31)(19,71,98,44)(20,72,99,45)(21,64,91,37)(22,65,92,38)(23,66,93,39)(24,67,94,40)(25,68,95,41)(26,69,96,42)(27,70,97,43), (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (1,35,37)(2,36,38)(3,28,39)(4,29,40)(5,30,41)(6,31,42)(7,32,43)(8,33,44)(9,34,45)(10,27,100)(11,19,101)(12,20,102)(13,21,103)(14,22,104)(15,23,105)(16,24,106)(17,25,107)(18,26,108)(46,59,70)(47,60,71)(48,61,72)(49,62,64)(50,63,65)(51,55,66)(52,56,67)(53,57,68)(54,58,69)(73,86,97)(74,87,98)(75,88,99)(76,89,91)(77,90,92)(78,82,93)(79,83,94)(80,84,95)(81,85,96), (1,49)(2,48)(3,47)(4,46)(5,54)(6,53)(7,52)(8,51)(9,50)(10,94)(11,93)(12,92)(13,91)(14,99)(15,98)(16,97)(17,96)(18,95)(19,82)(20,90)(21,89)(22,88)(23,87)(24,86)(25,85)(26,84)(27,83)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,72)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(73,106)(74,105)(75,104)(76,103)(77,102)(78,101)(79,100)(80,108)(81,107)>;

G:=Group( (1,103,49,76)(2,104,50,77)(3,105,51,78)(4,106,52,79)(5,107,53,80)(6,108,54,81)(7,100,46,73)(8,101,47,74)(9,102,48,75)(10,59,86,32)(11,60,87,33)(12,61,88,34)(13,62,89,35)(14,63,90,36)(15,55,82,28)(16,56,83,29)(17,57,84,30)(18,58,85,31)(19,71,98,44)(20,72,99,45)(21,64,91,37)(22,65,92,38)(23,66,93,39)(24,67,94,40)(25,68,95,41)(26,69,96,42)(27,70,97,43), (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (1,35,37)(2,36,38)(3,28,39)(4,29,40)(5,30,41)(6,31,42)(7,32,43)(8,33,44)(9,34,45)(10,27,100)(11,19,101)(12,20,102)(13,21,103)(14,22,104)(15,23,105)(16,24,106)(17,25,107)(18,26,108)(46,59,70)(47,60,71)(48,61,72)(49,62,64)(50,63,65)(51,55,66)(52,56,67)(53,57,68)(54,58,69)(73,86,97)(74,87,98)(75,88,99)(76,89,91)(77,90,92)(78,82,93)(79,83,94)(80,84,95)(81,85,96), (1,49)(2,48)(3,47)(4,46)(5,54)(6,53)(7,52)(8,51)(9,50)(10,94)(11,93)(12,92)(13,91)(14,99)(15,98)(16,97)(17,96)(18,95)(19,82)(20,90)(21,89)(22,88)(23,87)(24,86)(25,85)(26,84)(27,83)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,72)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(73,106)(74,105)(75,104)(76,103)(77,102)(78,101)(79,100)(80,108)(81,107) );

G=PermutationGroup([[(1,103,49,76),(2,104,50,77),(3,105,51,78),(4,106,52,79),(5,107,53,80),(6,108,54,81),(7,100,46,73),(8,101,47,74),(9,102,48,75),(10,59,86,32),(11,60,87,33),(12,61,88,34),(13,62,89,35),(14,63,90,36),(15,55,82,28),(16,56,83,29),(17,57,84,30),(18,58,85,31),(19,71,98,44),(20,72,99,45),(21,64,91,37),(22,65,92,38),(23,66,93,39),(24,67,94,40),(25,68,95,41),(26,69,96,42),(27,70,97,43)], [(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),(73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108)], [(1,35,37),(2,36,38),(3,28,39),(4,29,40),(5,30,41),(6,31,42),(7,32,43),(8,33,44),(9,34,45),(10,27,100),(11,19,101),(12,20,102),(13,21,103),(14,22,104),(15,23,105),(16,24,106),(17,25,107),(18,26,108),(46,59,70),(47,60,71),(48,61,72),(49,62,64),(50,63,65),(51,55,66),(52,56,67),(53,57,68),(54,58,69),(73,86,97),(74,87,98),(75,88,99),(76,89,91),(77,90,92),(78,82,93),(79,83,94),(80,84,95),(81,85,96)], [(1,49),(2,48),(3,47),(4,46),(5,54),(6,53),(7,52),(8,51),(9,50),(10,94),(11,93),(12,92),(13,91),(14,99),(15,98),(16,97),(17,96),(18,95),(19,82),(20,90),(21,89),(22,88),(23,87),(24,86),(25,85),(26,84),(27,83),(28,71),(29,70),(30,69),(31,68),(32,67),(33,66),(34,65),(35,64),(36,72),(37,62),(38,61),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,63),(73,106),(74,105),(75,104),(76,103),(77,102),(78,101),(79,100),(80,108),(81,107)]])

C4×C9⋊S3 is a maximal subgroup of
C36.38D6  C36.40D6  C72⋊S3  D18.D6  Dic18⋊S3  D12⋊D9  D6.D18  C4×S3×D9  C36⋊D6  C36.70D6  C36.27D6  C36.29D6
C4×C9⋊S3 is a maximal quotient of
C72⋊S3  C6.Dic18  C6.11D36

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 9A ··· 9I 12A ··· 12H 18A ··· 18I 36A ··· 36R order 1 2 2 2 3 3 3 3 4 4 4 4 6 6 6 6 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 27 27 2 2 2 2 1 1 27 27 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C4 S3 S3 D6 D6 D9 C4×S3 C4×S3 D18 C4×D9 kernel C4×C9⋊S3 C9⋊Dic3 C3×C36 C2×C9⋊S3 C9⋊S3 C36 C3×C12 C18 C3×C6 C12 C9 C32 C6 C3 # reps 1 1 1 1 4 3 1 3 1 9 6 2 9 18

Matrix representation of C4×C9⋊S3 in GL4(𝔽37) generated by

 31 0 0 0 0 31 0 0 0 0 31 0 0 0 0 31
,
 6 11 0 0 26 17 0 0 0 0 6 11 0 0 26 17
,
 36 1 0 0 36 0 0 0 0 0 0 36 0 0 1 36
,
 26 17 0 0 6 11 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(37))| [31,0,0,0,0,31,0,0,0,0,31,0,0,0,0,31],[6,26,0,0,11,17,0,0,0,0,6,26,0,0,11,17],[36,36,0,0,1,0,0,0,0,0,0,1,0,0,36,36],[26,6,0,0,17,11,0,0,0,0,0,1,0,0,1,0] >;

C4×C9⋊S3 in GAP, Magma, Sage, TeX

C_4\times C_9\rtimes S_3
% in TeX

G:=Group("C4xC9:S3");
// GroupNames label

G:=SmallGroup(216,64);
// by ID

G=gap.SmallGroup(216,64);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,31,2115,453,1444,5189]);
// Polycyclic

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

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