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## G = C2×C32⋊5D8order 288 = 25·32

### Direct product of C2 and C32⋊5D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C2×C32⋊5D8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C12⋊S3 — C2×C12⋊S3 — C2×C32⋊5D8
 Lower central C32 — C3×C6 — C3×C12 — C2×C32⋊5D8
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×C325D8
G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1364 in 228 conjugacy classes, 77 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×4], C4 [×2], C22, C22 [×8], S3 [×16], C6 [×12], C8 [×2], C2×C4, D4 [×6], C23 [×2], C32, C12 [×8], D6 [×32], C2×C6 [×4], C2×C8, D8 [×4], C2×D4 [×2], C3⋊S3 [×4], C3×C6, C3×C6 [×2], C24 [×8], D12 [×24], C2×C12 [×4], C22×S3 [×8], C2×D8, C3×C12 [×2], C2×C3⋊S3 [×8], C62, D24 [×16], C2×C24 [×4], C2×D12 [×8], C3×C24 [×2], C12⋊S3 [×4], C12⋊S3 [×2], C6×C12, C22×C3⋊S3 [×2], C2×D24 [×4], C325D8 [×4], C6×C24, C2×C12⋊S3 [×2], C2×C325D8
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], D8 [×2], C2×D4, C3⋊S3, D12 [×8], C22×S3 [×4], C2×D8, C2×C3⋊S3 [×3], D24 [×8], C2×D12 [×4], C12⋊S3 [×2], C22×C3⋊S3, C2×D24 [×4], C325D8 [×2], C2×C12⋊S3, C2×C325D8

Smallest permutation representation of C2×C325D8
On 144 points
Generators in S144
(1 44)(2 45)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 57)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 89)(24 90)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 68)(34 69)(35 70)(36 71)(37 72)(38 65)(39 66)(40 67)(73 114)(74 115)(75 116)(76 117)(77 118)(78 119)(79 120)(80 113)(81 112)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(97 128)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(129 144)(130 137)(131 138)(132 139)(133 140)(134 141)(135 142)(136 143)
(1 133 22)(2 134 23)(3 135 24)(4 136 17)(5 129 18)(6 130 19)(7 131 20)(8 132 21)(9 124 86)(10 125 87)(11 126 88)(12 127 81)(13 128 82)(14 121 83)(15 122 84)(16 123 85)(25 79 39)(26 80 40)(27 73 33)(28 74 34)(29 75 35)(30 76 36)(31 77 37)(32 78 38)(41 137 93)(42 138 94)(43 139 95)(44 140 96)(45 141 89)(46 142 90)(47 143 91)(48 144 92)(49 114 68)(50 115 69)(51 116 70)(52 117 71)(53 118 72)(54 119 65)(55 120 66)(56 113 67)(57 100 108)(58 101 109)(59 102 110)(60 103 111)(61 104 112)(62 97 105)(63 98 106)(64 99 107)
(1 15 117)(2 16 118)(3 9 119)(4 10 120)(5 11 113)(6 12 114)(7 13 115)(8 14 116)(17 87 55)(18 88 56)(19 81 49)(20 82 50)(21 83 51)(22 84 52)(23 85 53)(24 86 54)(25 91 110)(26 92 111)(27 93 112)(28 94 105)(29 95 106)(30 96 107)(31 89 108)(32 90 109)(33 137 104)(34 138 97)(35 139 98)(36 140 99)(37 141 100)(38 142 101)(39 143 102)(40 144 103)(41 61 73)(42 62 74)(43 63 75)(44 64 76)(45 57 77)(46 58 78)(47 59 79)(48 60 80)(65 135 124)(66 136 125)(67 129 126)(68 130 127)(69 131 128)(70 132 121)(71 133 122)(72 134 123)
(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 43)(2 42)(3 41)(4 48)(5 47)(6 46)(7 45)(8 44)(9 73)(10 80)(11 79)(12 78)(13 77)(14 76)(15 75)(16 74)(17 144)(18 143)(19 142)(20 141)(21 140)(22 139)(23 138)(24 137)(25 126)(26 125)(27 124)(28 123)(29 122)(30 121)(31 128)(32 127)(33 86)(34 85)(35 84)(36 83)(37 82)(38 81)(39 88)(40 87)(49 101)(50 100)(51 99)(52 98)(53 97)(54 104)(55 103)(56 102)(57 115)(58 114)(59 113)(60 120)(61 119)(62 118)(63 117)(64 116)(65 112)(66 111)(67 110)(68 109)(69 108)(70 107)(71 106)(72 105)(89 131)(90 130)(91 129)(92 136)(93 135)(94 134)(95 133)(96 132)

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 6A ··· 6L 8A 8B 8C 8D 12A ··· 12P 24A ··· 24AF order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 6 ··· 6 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 1 1 36 36 36 36 2 2 2 2 2 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D4 D6 D6 D8 D12 D12 D24 kernel C2×C32⋊5D8 C32⋊5D8 C6×C24 C2×C12⋊S3 C2×C24 C3×C12 C62 C24 C2×C12 C3×C6 C12 C2×C6 C6 # reps 1 4 1 2 4 1 1 8 4 4 8 8 32

Matrix representation of C2×C325D8 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 72 72 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 72 1 0 0 72 0
,
 23 5 0 0 68 18 0 0 0 0 50 5 0 0 68 55
,
 50 68 0 0 18 23 0 0 0 0 68 55 0 0 50 5
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,1,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,72,0,0,1,0],[23,68,0,0,5,18,0,0,0,0,50,68,0,0,5,55],[50,18,0,0,68,23,0,0,0,0,68,50,0,0,55,5] >;

C2×C325D8 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_5D_8
% in TeX

G:=Group("C2xC3^2:5D8");
// GroupNames label

G:=SmallGroup(288,760);
// by ID

G=gap.SmallGroup(288,760);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,142,675,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=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,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽