Copied to
clipboard

## G = C22×C9⋊D4order 288 = 25·32

### Direct product of C22 and C9⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C22×C9⋊D4
 Chief series C1 — C3 — C9 — C18 — D18 — C22×D9 — C23×D9 — C22×C9⋊D4
 Lower central C9 — C18 — C22×C9⋊D4
 Upper central C1 — C23 — C24

Generators and relations for C22×C9⋊D4
G = < a,b,c,d,e | a2=b2=c9=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1384 in 354 conjugacy classes, 132 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×11], C22 [×28], S3 [×4], C6, C6 [×6], C6 [×4], C2×C4 [×6], D4 [×16], C23, C23 [×6], C23 [×14], C9, Dic3 [×4], D6 [×16], C2×C6 [×11], C2×C6 [×12], C22×C4, C2×D4 [×12], C24, C24, D9 [×4], C18, C18 [×6], C18 [×4], C2×Dic3 [×6], C3⋊D4 [×16], C22×S3 [×10], C22×C6, C22×C6 [×6], C22×C6 [×4], C22×D4, Dic9 [×4], D18 [×4], D18 [×12], C2×C18 [×11], C2×C18 [×12], C22×Dic3, C2×C3⋊D4 [×12], S3×C23, C23×C6, C2×Dic9 [×6], C9⋊D4 [×16], C22×D9 [×6], C22×D9 [×4], C22×C18, C22×C18 [×6], C22×C18 [×4], C22×C3⋊D4, C22×Dic9, C2×C9⋊D4 [×12], C23×D9, C23×C18, C22×C9⋊D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D9, C3⋊D4 [×4], C22×S3 [×7], C22×D4, D18 [×7], C2×C3⋊D4 [×6], S3×C23, C9⋊D4 [×4], C22×D9 [×7], C22×C3⋊D4, C2×C9⋊D4 [×6], C23×D9, C22×C9⋊D4

Smallest permutation representation of C22×C9⋊D4
On 144 points
Generators in S144
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 82)(8 83)(9 84)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)
(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)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(91 127)(92 128)(93 129)(94 130)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)
(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 94 13 103)(2 93 14 102)(3 92 15 101)(4 91 16 100)(5 99 17 108)(6 98 18 107)(7 97 10 106)(8 96 11 105)(9 95 12 104)(19 88 28 79)(20 87 29 78)(21 86 30 77)(22 85 31 76)(23 84 32 75)(24 83 33 74)(25 82 34 73)(26 90 35 81)(27 89 36 80)(37 133 46 142)(38 132 47 141)(39 131 48 140)(40 130 49 139)(41 129 50 138)(42 128 51 137)(43 127 52 136)(44 135 53 144)(45 134 54 143)(55 124 64 115)(56 123 65 114)(57 122 66 113)(58 121 67 112)(59 120 68 111)(60 119 69 110)(61 118 70 109)(62 126 71 117)(63 125 72 116)
(1 40)(2 39)(3 38)(4 37)(5 45)(6 44)(7 43)(8 42)(9 41)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 54)(18 53)(19 70)(20 69)(21 68)(22 67)(23 66)(24 65)(25 64)(26 72)(27 71)(28 61)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 63)(36 62)(73 115)(74 114)(75 113)(76 112)(77 111)(78 110)(79 109)(80 117)(81 116)(82 124)(83 123)(84 122)(85 121)(86 120)(87 119)(88 118)(89 126)(90 125)(91 142)(92 141)(93 140)(94 139)(95 138)(96 137)(97 136)(98 144)(99 143)(100 133)(101 132)(102 131)(103 130)(104 129)(105 128)(106 127)(107 135)(108 134)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 6A ··· 6O 9A 9B 9C 18A ··· 18AS order 1 2 ··· 2 2 2 2 2 2 2 2 2 3 4 4 4 4 6 ··· 6 9 9 9 18 ··· 18 size 1 1 ··· 1 2 2 2 2 18 18 18 18 2 18 18 18 18 2 ··· 2 2 2 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 D9 C3⋊D4 D18 C9⋊D4 kernel C22×C9⋊D4 C22×Dic9 C2×C9⋊D4 C23×D9 C23×C18 C23×C6 C2×C18 C22×C6 C24 C2×C6 C23 C22 # reps 1 1 12 1 1 1 4 7 3 8 21 24

Matrix representation of C22×C9⋊D4 in GL4(𝔽37) generated by

 1 0 0 0 0 36 0 0 0 0 36 0 0 0 0 36
,
 36 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 11 17 0 0 20 31
,
 36 0 0 0 0 36 0 0 0 0 7 14 0 0 7 30
,
 1 0 0 0 0 36 0 0 0 0 1 0 0 0 36 36
G:=sub<GL(4,GF(37))| [1,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,11,20,0,0,17,31],[36,0,0,0,0,36,0,0,0,0,7,7,0,0,14,30],[1,0,0,0,0,36,0,0,0,0,1,36,0,0,0,36] >;

C22×C9⋊D4 in GAP, Magma, Sage, TeX

C_2^2\times C_9\rtimes D_4
% in TeX

G:=Group("C2^2xC9:D4");
// GroupNames label

G:=SmallGroup(288,366);
// by ID

G=gap.SmallGroup(288,366);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,675,6725,292,9414]);
// Polycyclic

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

׿
×
𝔽