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## G = C22×C12⋊S3order 288 = 25·32

### Direct product of C22 and C12⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C22×C12⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C23×C3⋊S3 — C22×C12⋊S3
 Lower central C32 — C3×C6 — C22×C12⋊S3
 Upper central C1 — C23 — C22×C4

Generators and relations for C22×C12⋊S3
G = < a,b,c,d,e | a2=b2=c12=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: 3044 in 708 conjugacy classes, 213 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×8], C3 [×4], C4 [×4], C22 [×7], C22 [×32], S3 [×32], C6 [×28], C2×C4 [×6], D4 [×16], C23, C23 [×20], C32, C12 [×16], D6 [×128], C2×C6 [×28], C22×C4, C2×D4 [×12], C24 [×2], C3⋊S3 [×8], C3×C6, C3×C6 [×6], D12 [×64], C2×C12 [×24], C22×S3 [×80], C22×C6 [×4], C22×D4, C3×C12 [×4], C2×C3⋊S3 [×8], C2×C3⋊S3 [×24], C62 [×7], C2×D12 [×48], C22×C12 [×4], S3×C23 [×8], C12⋊S3 [×16], C6×C12 [×6], C22×C3⋊S3 [×12], C22×C3⋊S3 [×8], C2×C62, C22×D12 [×4], C2×C12⋊S3 [×12], C2×C6×C12, C23×C3⋊S3 [×2], C22×C12⋊S3
Quotients: C1, C2 [×15], C22 [×35], S3 [×4], D4 [×4], C23 [×15], D6 [×28], C2×D4 [×6], C24, C3⋊S3, D12 [×16], C22×S3 [×28], C22×D4, C2×C3⋊S3 [×7], C2×D12 [×24], S3×C23 [×4], C12⋊S3 [×4], C22×C3⋊S3 [×7], C22×D12 [×4], C2×C12⋊S3 [×6], C23×C3⋊S3, C22×C12⋊S3

Smallest permutation representation of C22×C12⋊S3
On 144 points
Generators in S144
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 135)(14 136)(15 137)(16 138)(17 139)(18 140)(19 141)(20 142)(21 143)(22 144)(23 133)(24 134)(25 108)(26 97)(27 98)(28 99)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(61 109)(62 110)(63 111)(64 112)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(73 130)(74 131)(75 132)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 127)(83 128)(84 129)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 124)(26 125)(27 126)(28 127)(29 128)(30 129)(31 130)(32 131)(33 132)(34 121)(35 122)(36 123)(37 139)(38 140)(39 141)(40 142)(41 143)(42 144)(43 133)(44 134)(45 135)(46 136)(47 137)(48 138)(73 102)(74 103)(75 104)(76 105)(77 106)(78 107)(79 108)(80 97)(81 98)(82 99)(83 100)(84 101)(85 110)(86 111)(87 112)(88 113)(89 114)(90 115)(91 116)(92 117)(93 118)(94 119)(95 120)(96 109)
(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)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)
(1 48 108)(2 37 97)(3 38 98)(4 39 99)(5 40 100)(6 41 101)(7 42 102)(8 43 103)(9 44 104)(10 45 105)(11 46 106)(12 47 107)(13 121 113)(14 122 114)(15 123 115)(16 124 116)(17 125 117)(18 126 118)(19 127 119)(20 128 120)(21 129 109)(22 130 110)(23 131 111)(24 132 112)(25 91 54)(26 92 55)(27 93 56)(28 94 57)(29 95 58)(30 96 59)(31 85 60)(32 86 49)(33 87 50)(34 88 51)(35 89 52)(36 90 53)(61 143 84)(62 144 73)(63 133 74)(64 134 75)(65 135 76)(66 136 77)(67 137 78)(68 138 79)(69 139 80)(70 140 81)(71 141 82)(72 142 83)
(1 94)(2 93)(3 92)(4 91)(5 90)(6 89)(7 88)(8 87)(9 86)(10 85)(11 96)(12 95)(13 73)(14 84)(15 83)(16 82)(17 81)(18 80)(19 79)(20 78)(21 77)(22 76)(23 75)(24 74)(25 39)(26 38)(27 37)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(49 104)(50 103)(51 102)(52 101)(53 100)(54 99)(55 98)(56 97)(57 108)(58 107)(59 106)(60 105)(61 114)(62 113)(63 112)(64 111)(65 110)(66 109)(67 120)(68 119)(69 118)(70 117)(71 116)(72 115)(121 144)(122 143)(123 142)(124 141)(125 140)(126 139)(127 138)(128 137)(129 136)(130 135)(131 134)(132 133)

G:=sub<Sym(144)| (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,141)(20,142)(21,143)(22,144)(23,133)(24,134)(25,108)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,130)(74,131)(75,132)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129), (1,68)(2,69)(3,70)(4,71)(5,72)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,124)(26,125)(27,126)(28,127)(29,128)(30,129)(31,130)(32,131)(33,132)(34,121)(35,122)(36,123)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(43,133)(44,134)(45,135)(46,136)(47,137)(48,138)(73,102)(74,103)(75,104)(76,105)(77,106)(78,107)(79,108)(80,97)(81,98)(82,99)(83,100)(84,101)(85,110)(86,111)(87,112)(88,113)(89,114)(90,115)(91,116)(92,117)(93,118)(94,119)(95,120)(96,109), (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)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,48,108)(2,37,97)(3,38,98)(4,39,99)(5,40,100)(6,41,101)(7,42,102)(8,43,103)(9,44,104)(10,45,105)(11,46,106)(12,47,107)(13,121,113)(14,122,114)(15,123,115)(16,124,116)(17,125,117)(18,126,118)(19,127,119)(20,128,120)(21,129,109)(22,130,110)(23,131,111)(24,132,112)(25,91,54)(26,92,55)(27,93,56)(28,94,57)(29,95,58)(30,96,59)(31,85,60)(32,86,49)(33,87,50)(34,88,51)(35,89,52)(36,90,53)(61,143,84)(62,144,73)(63,133,74)(64,134,75)(65,135,76)(66,136,77)(67,137,78)(68,138,79)(69,139,80)(70,140,81)(71,141,82)(72,142,83), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,96)(12,95)(13,73)(14,84)(15,83)(16,82)(17,81)(18,80)(19,79)(20,78)(21,77)(22,76)(23,75)(24,74)(25,39)(26,38)(27,37)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,108)(58,107)(59,106)(60,105)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,120)(68,119)(69,118)(70,117)(71,116)(72,115)(121,144)(122,143)(123,142)(124,141)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)>;

G:=Group( (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,141)(20,142)(21,143)(22,144)(23,133)(24,134)(25,108)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,130)(74,131)(75,132)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129), (1,68)(2,69)(3,70)(4,71)(5,72)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,124)(26,125)(27,126)(28,127)(29,128)(30,129)(31,130)(32,131)(33,132)(34,121)(35,122)(36,123)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(43,133)(44,134)(45,135)(46,136)(47,137)(48,138)(73,102)(74,103)(75,104)(76,105)(77,106)(78,107)(79,108)(80,97)(81,98)(82,99)(83,100)(84,101)(85,110)(86,111)(87,112)(88,113)(89,114)(90,115)(91,116)(92,117)(93,118)(94,119)(95,120)(96,109), (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)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,48,108)(2,37,97)(3,38,98)(4,39,99)(5,40,100)(6,41,101)(7,42,102)(8,43,103)(9,44,104)(10,45,105)(11,46,106)(12,47,107)(13,121,113)(14,122,114)(15,123,115)(16,124,116)(17,125,117)(18,126,118)(19,127,119)(20,128,120)(21,129,109)(22,130,110)(23,131,111)(24,132,112)(25,91,54)(26,92,55)(27,93,56)(28,94,57)(29,95,58)(30,96,59)(31,85,60)(32,86,49)(33,87,50)(34,88,51)(35,89,52)(36,90,53)(61,143,84)(62,144,73)(63,133,74)(64,134,75)(65,135,76)(66,136,77)(67,137,78)(68,138,79)(69,139,80)(70,140,81)(71,141,82)(72,142,83), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,96)(12,95)(13,73)(14,84)(15,83)(16,82)(17,81)(18,80)(19,79)(20,78)(21,77)(22,76)(23,75)(24,74)(25,39)(26,38)(27,37)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,108)(58,107)(59,106)(60,105)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,120)(68,119)(69,118)(70,117)(71,116)(72,115)(121,144)(122,143)(123,142)(124,141)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133) );

G=PermutationGroup([(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,135),(14,136),(15,137),(16,138),(17,139),(18,140),(19,141),(20,142),(21,143),(22,144),(23,133),(24,134),(25,108),(26,97),(27,98),(28,99),(29,100),(30,101),(31,102),(32,103),(33,104),(34,105),(35,106),(36,107),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54),(61,109),(62,110),(63,111),(64,112),(65,113),(66,114),(67,115),(68,116),(69,117),(70,118),(71,119),(72,120),(73,130),(74,131),(75,132),(76,121),(77,122),(78,123),(79,124),(80,125),(81,126),(82,127),(83,128),(84,129)], [(1,68),(2,69),(3,70),(4,71),(5,72),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,49),(24,50),(25,124),(26,125),(27,126),(28,127),(29,128),(30,129),(31,130),(32,131),(33,132),(34,121),(35,122),(36,123),(37,139),(38,140),(39,141),(40,142),(41,143),(42,144),(43,133),(44,134),(45,135),(46,136),(47,137),(48,138),(73,102),(74,103),(75,104),(76,105),(77,106),(78,107),(79,108),(80,97),(81,98),(82,99),(83,100),(84,101),(85,110),(86,111),(87,112),(88,113),(89,114),(90,115),(91,116),(92,117),(93,118),(94,119),(95,120),(96,109)], [(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),(109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144)], [(1,48,108),(2,37,97),(3,38,98),(4,39,99),(5,40,100),(6,41,101),(7,42,102),(8,43,103),(9,44,104),(10,45,105),(11,46,106),(12,47,107),(13,121,113),(14,122,114),(15,123,115),(16,124,116),(17,125,117),(18,126,118),(19,127,119),(20,128,120),(21,129,109),(22,130,110),(23,131,111),(24,132,112),(25,91,54),(26,92,55),(27,93,56),(28,94,57),(29,95,58),(30,96,59),(31,85,60),(32,86,49),(33,87,50),(34,88,51),(35,89,52),(36,90,53),(61,143,84),(62,144,73),(63,133,74),(64,134,75),(65,135,76),(66,136,77),(67,137,78),(68,138,79),(69,139,80),(70,140,81),(71,141,82),(72,142,83)], [(1,94),(2,93),(3,92),(4,91),(5,90),(6,89),(7,88),(8,87),(9,86),(10,85),(11,96),(12,95),(13,73),(14,84),(15,83),(16,82),(17,81),(18,80),(19,79),(20,78),(21,77),(22,76),(23,75),(24,74),(25,39),(26,38),(27,37),(28,48),(29,47),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(49,104),(50,103),(51,102),(52,101),(53,100),(54,99),(55,98),(56,97),(57,108),(58,107),(59,106),(60,105),(61,114),(62,113),(63,112),(64,111),(65,110),(66,109),(67,120),(68,119),(69,118),(70,117),(71,116),(72,115),(121,144),(122,143),(123,142),(124,141),(125,140),(126,139),(127,138),(128,137),(129,136),(130,135),(131,134),(132,133)])

84 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6AB 12A ··· 12AF order 1 2 ··· 2 2 ··· 2 3 3 3 3 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 18 ··· 18 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D6 D12 kernel C22×C12⋊S3 C2×C12⋊S3 C2×C6×C12 C23×C3⋊S3 C22×C12 C62 C2×C12 C22×C6 C2×C6 # reps 1 12 1 2 4 4 24 4 32

Matrix representation of C22×C12⋊S3 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 10 6 0 0 0 0 7 3 0 0 0 0 0 0 0 1 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 6 10 0 0 0 0 3 7 0 0 0 0 0 0 12 0 0 0 0 0 1 1

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,6,3,0,0,0,0,10,7,0,0,0,0,0,0,12,1,0,0,0,0,0,1] >;

C22×C12⋊S3 in GAP, Magma, Sage, TeX

C_2^2\times C_{12}\rtimes S_3
% in TeX

G:=Group("C2^2xC12:S3");
// GroupNames label

G:=SmallGroup(288,1005);
// by ID

G=gap.SmallGroup(288,1005);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,675,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^12=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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