direct product, non-abelian, supersoluble, monomial
Aliases: C2xHe3:4C8, C62.6Dic3, He3:9(C2xC8), (C2xHe3):4C8, (C6xC12).20S3, (C3xC12).66D6, (C4xHe3).9C4, C4.3(He3:3C4), C6.5(C32:4C8), (C3xC12).11Dic3, (C22xHe3).6C4, C12.13(C3:Dic3), (C4xHe3).47C22, C22.2(He3:3C4), (C3xC6):3(C3:C8), C32:4(C2xC3:C8), C12.90(C2xC3:S3), (C2xC4xHe3).12C2, C6.17(C2xC3:Dic3), C2.1(C2xHe3:3C4), C3.2(C2xC32:4C8), (C2xC12).28(C3:S3), (C2xHe3).31(C2xC4), C4.14(C2xHe3:C2), (C3xC6).16(C2xDic3), (C2xC6).13(C3:Dic3), (C2xC4).5(He3:C2), SmallGroup(432,184)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C2xHe3:4C8 |
Generators and relations for C2xHe3:4C8
G = < a,b,c,d,e | a2=b3=c3=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=b-1, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 281 in 121 conjugacy classes, 59 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2xC4, C32, C12, C12, C2xC6, C2xC6, C2xC8, C3xC6, C3:C8, C24, C2xC12, C2xC12, He3, C3xC12, C62, C2xC3:C8, C2xC24, C2xHe3, C2xHe3, C3xC3:C8, C6xC12, C4xHe3, C22xHe3, C6xC3:C8, He3:4C8, C2xC4xHe3, C2xHe3:4C8
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, Dic3, D6, C2xC8, C3:S3, C3:C8, C2xDic3, C3:Dic3, C2xC3:S3, C2xC3:C8, He3:C2, C32:4C8, C2xC3:Dic3, He3:3C4, C2xHe3:C2, C2xC32:4C8, He3:4C8, C2xHe3:3C4, C2xHe3:4C8
►On 144 points
Generators in S144
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 89)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 105)(24 106)(25 123)(26 124)(27 125)(28 126)(29 127)(30 128)(31 121)(32 122)(41 88)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 101)(50 102)(51 103)(52 104)(53 97)(54 98)(55 99)(56 100)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 137)(65 118)(66 119)(67 120)(68 113)(69 114)(70 115)(71 116)(72 117)(73 136)(74 129)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)
(1 141 44)(2 45 142)(3 143 46)(4 47 144)(5 137 48)(6 41 138)(7 139 42)(8 43 140)(9 110 51)(10 52 111)(11 112 53)(12 54 105)(13 106 55)(14 56 107)(15 108 49)(16 50 109)(17 33 100)(18 101 34)(19 35 102)(20 103 36)(21 37 104)(22 97 38)(23 39 98)(24 99 40)(25 120 129)(26 130 113)(27 114 131)(28 132 115)(29 116 133)(30 134 117)(31 118 135)(32 136 119)(57 95 88)(58 81 96)(59 89 82)(60 83 90)(61 91 84)(62 85 92)(63 93 86)(64 87 94)(65 80 121)(66 122 73)(67 74 123)(68 124 75)(69 76 125)(70 126 77)(71 78 127)(72 128 79)
(1 106 121)(2 107 122)(3 108 123)(4 109 124)(5 110 125)(6 111 126)(7 112 127)(8 105 128)(9 76 48)(10 77 41)(11 78 42)(12 79 43)(13 80 44)(14 73 45)(15 74 46)(16 75 47)(17 32 91)(18 25 92)(19 26 93)(20 27 94)(21 28 95)(22 29 96)(23 30 89)(24 31 90)(33 136 84)(34 129 85)(35 130 86)(36 131 87)(37 132 88)(38 133 81)(39 134 82)(40 135 83)(49 67 143)(50 68 144)(51 69 137)(52 70 138)(53 71 139)(54 72 140)(55 65 141)(56 66 142)(57 104 115)(58 97 116)(59 98 117)(60 99 118)(61 100 119)(62 101 120)(63 102 113)(64 103 114)
(1 141 13)(2 14 142)(3 143 15)(4 16 144)(5 137 9)(6 10 138)(7 139 11)(8 12 140)(17 136 100)(18 101 129)(19 130 102)(20 103 131)(21 132 104)(22 97 133)(23 134 98)(24 99 135)(25 120 85)(26 86 113)(27 114 87)(28 88 115)(29 116 81)(30 82 117)(31 118 83)(32 84 119)(33 61 91)(34 92 62)(35 63 93)(36 94 64)(37 57 95)(38 96 58)(39 59 89)(40 90 60)(41 70 126)(42 127 71)(43 72 128)(44 121 65)(45 66 122)(46 123 67)(47 68 124)(48 125 69)(49 74 108)(50 109 75)(51 76 110)(52 111 77)(53 78 112)(54 105 79)(55 80 106)(56 107 73)
(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)
G:=sub<Sym(144)| (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,89)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,105)(24,106)(25,123)(26,124)(27,125)(28,126)(29,127)(30,128)(31,121)(32,122)(41,88)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,101)(50,102)(51,103)(52,104)(53,97)(54,98)(55,99)(56,100)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,137)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(73,136)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135), (1,141,44)(2,45,142)(3,143,46)(4,47,144)(5,137,48)(6,41,138)(7,139,42)(8,43,140)(9,110,51)(10,52,111)(11,112,53)(12,54,105)(13,106,55)(14,56,107)(15,108,49)(16,50,109)(17,33,100)(18,101,34)(19,35,102)(20,103,36)(21,37,104)(22,97,38)(23,39,98)(24,99,40)(25,120,129)(26,130,113)(27,114,131)(28,132,115)(29,116,133)(30,134,117)(31,118,135)(32,136,119)(57,95,88)(58,81,96)(59,89,82)(60,83,90)(61,91,84)(62,85,92)(63,93,86)(64,87,94)(65,80,121)(66,122,73)(67,74,123)(68,124,75)(69,76,125)(70,126,77)(71,78,127)(72,128,79), (1,106,121)(2,107,122)(3,108,123)(4,109,124)(5,110,125)(6,111,126)(7,112,127)(8,105,128)(9,76,48)(10,77,41)(11,78,42)(12,79,43)(13,80,44)(14,73,45)(15,74,46)(16,75,47)(17,32,91)(18,25,92)(19,26,93)(20,27,94)(21,28,95)(22,29,96)(23,30,89)(24,31,90)(33,136,84)(34,129,85)(35,130,86)(36,131,87)(37,132,88)(38,133,81)(39,134,82)(40,135,83)(49,67,143)(50,68,144)(51,69,137)(52,70,138)(53,71,139)(54,72,140)(55,65,141)(56,66,142)(57,104,115)(58,97,116)(59,98,117)(60,99,118)(61,100,119)(62,101,120)(63,102,113)(64,103,114), (1,141,13)(2,14,142)(3,143,15)(4,16,144)(5,137,9)(6,10,138)(7,139,11)(8,12,140)(17,136,100)(18,101,129)(19,130,102)(20,103,131)(21,132,104)(22,97,133)(23,134,98)(24,99,135)(25,120,85)(26,86,113)(27,114,87)(28,88,115)(29,116,81)(30,82,117)(31,118,83)(32,84,119)(33,61,91)(34,92,62)(35,63,93)(36,94,64)(37,57,95)(38,96,58)(39,59,89)(40,90,60)(41,70,126)(42,127,71)(43,72,128)(44,121,65)(45,66,122)(46,123,67)(47,68,124)(48,125,69)(49,74,108)(50,109,75)(51,76,110)(52,111,77)(53,78,112)(54,105,79)(55,80,106)(56,107,73), (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)>;
G:=Group( (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,89)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,105)(24,106)(25,123)(26,124)(27,125)(28,126)(29,127)(30,128)(31,121)(32,122)(41,88)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,101)(50,102)(51,103)(52,104)(53,97)(54,98)(55,99)(56,100)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,137)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(73,136)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135), (1,141,44)(2,45,142)(3,143,46)(4,47,144)(5,137,48)(6,41,138)(7,139,42)(8,43,140)(9,110,51)(10,52,111)(11,112,53)(12,54,105)(13,106,55)(14,56,107)(15,108,49)(16,50,109)(17,33,100)(18,101,34)(19,35,102)(20,103,36)(21,37,104)(22,97,38)(23,39,98)(24,99,40)(25,120,129)(26,130,113)(27,114,131)(28,132,115)(29,116,133)(30,134,117)(31,118,135)(32,136,119)(57,95,88)(58,81,96)(59,89,82)(60,83,90)(61,91,84)(62,85,92)(63,93,86)(64,87,94)(65,80,121)(66,122,73)(67,74,123)(68,124,75)(69,76,125)(70,126,77)(71,78,127)(72,128,79), (1,106,121)(2,107,122)(3,108,123)(4,109,124)(5,110,125)(6,111,126)(7,112,127)(8,105,128)(9,76,48)(10,77,41)(11,78,42)(12,79,43)(13,80,44)(14,73,45)(15,74,46)(16,75,47)(17,32,91)(18,25,92)(19,26,93)(20,27,94)(21,28,95)(22,29,96)(23,30,89)(24,31,90)(33,136,84)(34,129,85)(35,130,86)(36,131,87)(37,132,88)(38,133,81)(39,134,82)(40,135,83)(49,67,143)(50,68,144)(51,69,137)(52,70,138)(53,71,139)(54,72,140)(55,65,141)(56,66,142)(57,104,115)(58,97,116)(59,98,117)(60,99,118)(61,100,119)(62,101,120)(63,102,113)(64,103,114), (1,141,13)(2,14,142)(3,143,15)(4,16,144)(5,137,9)(6,10,138)(7,139,11)(8,12,140)(17,136,100)(18,101,129)(19,130,102)(20,103,131)(21,132,104)(22,97,133)(23,134,98)(24,99,135)(25,120,85)(26,86,113)(27,114,87)(28,88,115)(29,116,81)(30,82,117)(31,118,83)(32,84,119)(33,61,91)(34,92,62)(35,63,93)(36,94,64)(37,57,95)(38,96,58)(39,59,89)(40,90,60)(41,70,126)(42,127,71)(43,72,128)(44,121,65)(45,66,122)(46,123,67)(47,68,124)(48,125,69)(49,74,108)(50,109,75)(51,76,110)(52,111,77)(53,78,112)(54,105,79)(55,80,106)(56,107,73), (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) );
G=PermutationGroup([[(1,90),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,89),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,107),(18,108),(19,109),(20,110),(21,111),(22,112),(23,105),(24,106),(25,123),(26,124),(27,125),(28,126),(29,127),(30,128),(31,121),(32,122),(41,88),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,101),(50,102),(51,103),(52,104),(53,97),(54,98),(55,99),(56,100),(57,138),(58,139),(59,140),(60,141),(61,142),(62,143),(63,144),(64,137),(65,118),(66,119),(67,120),(68,113),(69,114),(70,115),(71,116),(72,117),(73,136),(74,129),(75,130),(76,131),(77,132),(78,133),(79,134),(80,135)], [(1,141,44),(2,45,142),(3,143,46),(4,47,144),(5,137,48),(6,41,138),(7,139,42),(8,43,140),(9,110,51),(10,52,111),(11,112,53),(12,54,105),(13,106,55),(14,56,107),(15,108,49),(16,50,109),(17,33,100),(18,101,34),(19,35,102),(20,103,36),(21,37,104),(22,97,38),(23,39,98),(24,99,40),(25,120,129),(26,130,113),(27,114,131),(28,132,115),(29,116,133),(30,134,117),(31,118,135),(32,136,119),(57,95,88),(58,81,96),(59,89,82),(60,83,90),(61,91,84),(62,85,92),(63,93,86),(64,87,94),(65,80,121),(66,122,73),(67,74,123),(68,124,75),(69,76,125),(70,126,77),(71,78,127),(72,128,79)], [(1,106,121),(2,107,122),(3,108,123),(4,109,124),(5,110,125),(6,111,126),(7,112,127),(8,105,128),(9,76,48),(10,77,41),(11,78,42),(12,79,43),(13,80,44),(14,73,45),(15,74,46),(16,75,47),(17,32,91),(18,25,92),(19,26,93),(20,27,94),(21,28,95),(22,29,96),(23,30,89),(24,31,90),(33,136,84),(34,129,85),(35,130,86),(36,131,87),(37,132,88),(38,133,81),(39,134,82),(40,135,83),(49,67,143),(50,68,144),(51,69,137),(52,70,138),(53,71,139),(54,72,140),(55,65,141),(56,66,142),(57,104,115),(58,97,116),(59,98,117),(60,99,118),(61,100,119),(62,101,120),(63,102,113),(64,103,114)], [(1,141,13),(2,14,142),(3,143,15),(4,16,144),(5,137,9),(6,10,138),(7,139,11),(8,12,140),(17,136,100),(18,101,129),(19,130,102),(20,103,131),(21,132,104),(22,97,133),(23,134,98),(24,99,135),(25,120,85),(26,86,113),(27,114,87),(28,88,115),(29,116,81),(30,82,117),(31,118,83),(32,84,119),(33,61,91),(34,92,62),(35,63,93),(36,94,64),(37,57,95),(38,96,58),(39,59,89),(40,90,60),(41,70,126),(42,127,71),(43,72,128),(44,121,65),(45,66,122),(46,123,67),(47,68,124),(48,125,69),(49,74,108),(50,109,75),(51,76,110),(52,111,77),(53,78,112),(54,105,79),(55,80,106),(56,107,73)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6R | 8A | ··· | 8H | 12A | ··· | 12H | 12I | ··· | 12X | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 6 | ··· | 6 | 9 | ··· | 9 | 1 | ··· | 1 | 6 | ··· | 6 | 9 | ··· | 9 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | S3 | Dic3 | D6 | Dic3 | C3:C8 | He3:C2 | He3:3C4 | C2xHe3:C2 | He3:3C4 | He3:4C8 |
| kernel | C2xHe3:4C8 | He3:4C8 | C2xC4xHe3 | C4xHe3 | C22xHe3 | C2xHe3 | C6xC12 | C3xC12 | C3xC12 | C62 | C3xC6 | C2xC4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 4 | 4 | 4 | 4 | 16 | 4 | 4 | 4 | 4 | 16 |
Matrix representation of C2xHe3:4C8 ►in GL5(F73)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 0 | 72 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 64 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 |
| 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 |
| 0 | 0 | 0 | 51 | 0 |
| 0 | 0 | 51 | 0 | 0 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[0,1,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,64,0],[0,22,0,0,0,22,0,0,0,0,0,0,0,0,51,0,0,0,51,0,0,0,51,0,0] >;
C2xHe3:4C8 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3\rtimes_4C_8 % in TeX
G:=Group("C2xHe3:4C8"); // GroupNames label
G:=SmallGroup(432,184);
// by ID
G=gap.SmallGroup(432,184);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,58,1124,4037,537]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations