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## G = C22×C9⋊S3order 216 = 23·33

### Direct product of C22 and C9⋊S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×C9⋊S3
G = < a,b,c,d,e | a2=b2=c9=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 886 in 160 conjugacy classes, 61 normal (9 characteristic)
C1, C2 [×3], C2 [×4], C3, C3 [×3], C22, C22 [×6], S3 [×16], C6 [×12], C23, C9 [×3], C32, D6 [×24], C2×C6, C2×C6 [×3], D9 [×12], C18 [×9], C3⋊S3 [×4], C3×C6 [×3], C22×S3 [×4], C3×C9, D18 [×18], C2×C18 [×3], C2×C3⋊S3 [×6], C62, C9⋊S3 [×4], C3×C18 [×3], C22×D9 [×3], C22×C3⋊S3, C2×C9⋊S3 [×6], C6×C18, C22×C9⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], C23, D6 [×12], D9 [×3], C3⋊S3, C22×S3 [×4], D18 [×9], C2×C3⋊S3 [×3], C9⋊S3, C22×D9 [×3], C22×C3⋊S3, C2×C9⋊S3 [×3], C22×C9⋊S3

Smallest permutation representation of C22×C9⋊S3
On 108 points
Generators in S108
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 73)(8 74)(9 75)(10 59)(11 60)(12 61)(13 62)(14 63)(15 55)(16 56)(17 57)(18 58)(19 71)(20 72)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 46)(8 47)(9 48)(10 86)(11 87)(12 88)(13 89)(14 90)(15 82)(16 83)(17 84)(18 85)(19 98)(20 99)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)
(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)
(2 9)(3 8)(4 7)(5 6)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 27)(17 26)(18 25)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(36 45)(46 52)(47 51)(48 50)(53 54)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 72)(73 79)(74 78)(75 77)(80 81)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(90 99)(100 106)(101 105)(102 104)(107 108)

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

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

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

C22×C9⋊S3 is a maximal subgroup of   C6.18D36  C6.11D36  D18⋊D6  C22×S3×D9
C22×C9⋊S3 is a maximal quotient of   C36.70D6  C36.27D6  C36.29D6

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 6A ··· 6L 9A ··· 9I 18A ··· 18AA order 1 2 2 2 2 2 2 2 3 3 3 3 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 27 27 27 27 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 S3 D6 D6 D9 D18 kernel C22×C9⋊S3 C2×C9⋊S3 C6×C18 C2×C18 C62 C18 C3×C6 C2×C6 C6 # reps 1 6 1 3 1 9 3 9 27

Matrix representation of C22×C9⋊S3 in GL5(𝔽19)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 0 0 18
,
 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 0 0 18
,
 1 0 0 0 0 0 12 17 0 0 0 2 14 0 0 0 0 0 1 3 0 0 0 18 17
,
 1 0 0 0 0 0 0 1 0 0 0 18 18 0 0 0 0 0 17 16 0 0 0 1 1
,
 18 0 0 0 0 0 14 12 0 0 0 17 5 0 0 0 0 0 1 0 0 0 0 18 18

G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18],[18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,12,2,0,0,0,17,14,0,0,0,0,0,1,18,0,0,0,3,17],[1,0,0,0,0,0,0,18,0,0,0,1,18,0,0,0,0,0,17,1,0,0,0,16,1],[18,0,0,0,0,0,14,17,0,0,0,12,5,0,0,0,0,0,1,18,0,0,0,0,18] >;

C22×C9⋊S3 in GAP, Magma, Sage, TeX

C_2^2\times C_9\rtimes S_3
% in TeX

G:=Group("C2^2xC9:S3");
// GroupNames label

G:=SmallGroup(216,112);
// by ID

G=gap.SmallGroup(216,112);
# by ID

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

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

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