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G = C2×C325D8order 288 = 25·32

Direct product of C2 and C325D8

direct product, metabelian, supersoluble, monomial

Aliases: C2×C325D8, C61D24, C2423D6, C12.49D12, C62.91D4, (C3×C6)⋊5D8, (C2×C24)⋊5S3, (C6×C24)⋊7C2, C32(C2×D24), C3210(C2×D8), C6.57(C2×D12), (C2×C6).39D12, (C3×C24)⋊24C22, (C2×C12).385D6, (C3×C12).124D4, C4.7(C12⋊S3), C12⋊S314C22, C12.190(C22×S3), (C6×C12).301C22, (C3×C12).152C23, C22.13(C12⋊S3), C87(C2×C3⋊S3), (C2×C8)⋊3(C3⋊S3), (C2×C12⋊S3)⋊7C2, (C3×C6).197(C2×D4), C4.27(C22×C3⋊S3), C2.12(C2×C12⋊S3), (C2×C4).80(C2×C3⋊S3), SmallGroup(288,760)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C2×C325D8
C1C3C32C3×C6C3×C12C12⋊S3C2×C12⋊S3 — C2×C325D8
C32C3×C6C3×C12 — C2×C325D8
C1C22C2×C4C2×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 2A2B2C2D2E2F2G3A3B3C3D4A4B6A···6L8A8B8C8D12A···12P24A···24AF
order122222223333446···6888812···1224···24
size1111363636362222222···222222···22···2

78 irreducible representations

dim1111222222222
type+++++++++++++
imageC1C2C2C2S3D4D4D6D6D8D12D12D24
kernelC2×C325D8C325D8C6×C24C2×C12⋊S3C2×C24C3×C12C62C24C2×C12C3×C6C12C2×C6C6
# reps14124118448832

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

72000
07200
0010
0001
,
0100
727200
0010
0001
,
1000
0100
00721
00720
,
23500
681800
00505
006855
,
506800
182300
006855
00505
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