direct product, metabelian, supersoluble, monomial
Aliases: C2×C32⋊5D8, C6⋊1D24, C24⋊23D6, C12.49D12, C62.91D4, (C3×C6)⋊5D8, (C2×C24)⋊5S3, (C6×C24)⋊7C2, C3⋊2(C2×D24), C32⋊10(C2×D8), C6.57(C2×D12), (C2×C6).39D12, (C3×C24)⋊24C22, (C2×C12).385D6, (C3×C12).124D4, C4.7(C12⋊S3), C12⋊S3⋊14C22, C12.190(C22×S3), (C6×C12).301C22, (C3×C12).152C23, C22.13(C12⋊S3), C8⋊7(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
Generators and relations for C2×C32⋊5D8
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, C32⋊5D8, C6×C24, C2×C12⋊S3, C2×C32⋊5D8
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, C32⋊5D8, C2×C12⋊S3, C2×C32⋊5D8
►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 | 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×C32⋊5D8 ►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×C32⋊5D8 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