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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, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, D4, C23, C32, C12, D6, C2×C6, C2×C8, D8, C2×D4, C3⋊S3, C3×C6, C3×C6, C24, D12, C2×C12, C22×S3, C2×D8, C3×C12, C2×C3⋊S3, C62, D24, C2×C24, C2×D12, C3×C24, C12⋊S3, C12⋊S3, C6×C12, C22×C3⋊S3, C2×D24, C325D8, C6×C24, C2×C12⋊S3, C2×C325D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊S3, D12, C22×S3, C2×D8, C2×C3⋊S3, D24, C2×D12, C12⋊S3, C22×C3⋊S3, C2×D24, C325D8, 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)(9 109)(10 110)(11 111)(12 112)(13 105)(14 106)(15 107)(16 108)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 92)(26 93)(27 94)(28 95)(29 96)(30 89)(31 90)(32 91)(33 72)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(49 137)(50 138)(51 139)(52 140)(53 141)(54 142)(55 143)(56 144)(73 120)(74 113)(75 114)(76 115)(77 116)(78 117)(79 118)(80 119)(81 131)(82 132)(83 133)(84 134)(85 135)(86 136)(87 129)(88 130)(97 128)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)
(1 52 29)(2 53 30)(3 54 31)(4 55 32)(5 56 25)(6 49 26)(7 50 27)(8 51 28)(9 115 124)(10 116 125)(11 117 126)(12 118 127)(13 119 128)(14 120 121)(15 113 122)(16 114 123)(17 66 87)(18 67 88)(19 68 81)(20 69 82)(21 70 83)(22 71 84)(23 72 85)(24 65 86)(33 135 63)(34 136 64)(35 129 57)(36 130 58)(37 131 59)(38 132 60)(39 133 61)(40 134 62)(41 137 93)(42 138 94)(43 139 95)(44 140 96)(45 141 89)(46 142 90)(47 143 91)(48 144 92)(73 98 106)(74 99 107)(75 100 108)(76 101 109)(77 102 110)(78 103 111)(79 104 112)(80 97 105)
(1 113 18)(2 114 19)(3 115 20)(4 116 21)(5 117 22)(6 118 23)(7 119 24)(8 120 17)(9 82 31)(10 83 32)(11 84 25)(12 85 26)(13 86 27)(14 87 28)(15 88 29)(16 81 30)(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 79 63)(42 80 64)(43 73 57)(44 74 58)(45 75 59)(46 76 60)(47 77 61)(48 78 62)(49 127 72)(50 128 65)(51 121 66)(52 122 67)(53 123 68)(54 124 69)(55 125 70)(56 126 71)(89 108 131)(90 109 132)(91 110 133)(92 111 134)(93 112 135)(94 105 136)(95 106 129)(96 107 130)
(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 33)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 74)(18 73)(19 80)(20 79)(21 78)(22 77)(23 76)(24 75)(25 143)(26 142)(27 141)(28 140)(29 139)(30 138)(31 137)(32 144)(49 90)(50 89)(51 96)(52 95)(53 94)(54 93)(55 92)(56 91)(57 113)(58 120)(59 119)(60 118)(61 117)(62 116)(63 115)(64 114)(65 108)(66 107)(67 106)(68 105)(69 112)(70 111)(71 110)(72 109)(81 97)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(121 130)(122 129)(123 136)(124 135)(125 134)(126 133)(127 132)(128 131)

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

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

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

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