direct product, non-abelian, soluble
Aliases: C2×He3⋊2C8, He3⋊5(C2×C8), (C2×He3)⋊2C8, He3⋊3C4.2C4, C6.3(C32⋊2C8), C22.2(He3⋊C4), (C22×He3).1C4, He3⋊3C4.9C22, C2.3(C2×He3⋊C4), C3.(C2×C32⋊2C8), C6.17(C2×C32⋊C4), (C2×C6).6(C32⋊C4), (C2×He3).5(C2×C4), (C2×He3⋊3C4).3C2, SmallGroup(432,277)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C2×He3⋊2C8 |
Generators and relations for C2×He3⋊2C8
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=bcd, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 281 in 65 conjugacy classes, 21 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C32, Dic3, C12, C2×C6, C2×C6, C2×C8, C3×C6, C24, C2×Dic3, C2×C12, He3, C3×Dic3, C62, C2×C24, C2×He3, C2×He3, C6×Dic3, He3⋊3C4, C22×He3, He3⋊2C8, C2×He3⋊3C4, C2×He3⋊2C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C2×C8, C32⋊C4, C32⋊2C8, C2×C32⋊C4, He3⋊C4, C2×C32⋊2C8, He3⋊2C8, C2×He3⋊C4, C2×He3⋊2C8
►On 144 points
Generators in S144
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 65)(9 84)(10 85)(11 86)(12 87)(13 88)(14 81)(15 82)(16 83)(17 92)(18 93)(19 94)(20 95)(21 96)(22 89)(23 90)(24 91)(25 54)(26 55)(27 56)(28 49)(29 50)(30 51)(31 52)(32 53)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(57 108)(58 109)(59 110)(60 111)(61 112)(62 105)(63 106)(64 107)(73 124)(74 125)(75 126)(76 127)(77 128)(78 121)(79 122)(80 123)(97 130)(98 131)(99 132)(100 133)(101 134)(102 135)(103 136)(104 129)
(1 15 31)(2 76 46)(4 130 64)(5 11 27)(6 80 42)(8 134 60)(10 120 78)(12 17 132)(14 116 74)(16 21 136)(18 59 73)(19 30 44)(22 63 77)(23 26 48)(28 58 114)(32 62 118)(33 98 137)(34 49 109)(35 100 139)(36 125 81)(37 102 141)(38 53 105)(39 104 143)(40 121 85)(41 113 131)(43 115 133)(45 117 135)(47 119 129)(51 140 94)(52 66 82)(55 144 90)(56 70 86)(65 101 111)(67 127 142)(69 97 107)(71 123 138)(83 96 103)(87 92 99)(89 106 128)(93 110 124)
(1 15 31)(2 16 32)(3 9 25)(4 10 26)(5 11 27)(6 12 28)(7 13 29)(8 14 30)(17 58 80)(18 59 73)(19 60 74)(20 61 75)(21 62 76)(22 63 77)(23 64 78)(24 57 79)(33 137 98)(34 138 99)(35 139 100)(36 140 101)(37 141 102)(38 142 103)(39 143 104)(40 144 97)(41 131 113)(42 132 114)(43 133 115)(44 134 116)(45 135 117)(46 136 118)(47 129 119)(48 130 120)(49 71 87)(50 72 88)(51 65 81)(52 66 82)(53 67 83)(54 68 84)(55 69 85)(56 70 86)(89 106 128)(90 107 121)(91 108 122)(92 109 123)(93 110 124)(94 111 125)(95 112 126)(96 105 127)
(1 20 45)(2 136 21)(3 47 22)(4 23 130)(5 24 41)(6 132 17)(7 43 18)(8 19 134)(9 129 63)(10 64 120)(11 57 131)(12 114 58)(13 133 59)(14 60 116)(15 61 135)(16 118 62)(25 119 77)(26 78 48)(27 79 113)(28 42 80)(29 115 73)(30 74 44)(31 75 117)(32 46 76)(33 56 122)(34 109 87)(35 124 50)(36 81 111)(37 52 126)(38 105 83)(39 128 54)(40 85 107)(49 138 123)(51 125 140)(53 142 127)(55 121 144)(65 94 101)(66 95 141)(67 103 96)(68 143 89)(69 90 97)(70 91 137)(71 99 92)(72 139 93)(82 112 102)(84 104 106)(86 108 98)(88 100 110)
(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)| (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,65)(9,84)(10,85)(11,86)(12,87)(13,88)(14,81)(15,82)(16,83)(17,92)(18,93)(19,94)(20,95)(21,96)(22,89)(23,90)(24,91)(25,54)(26,55)(27,56)(28,49)(29,50)(30,51)(31,52)(32,53)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(57,108)(58,109)(59,110)(60,111)(61,112)(62,105)(63,106)(64,107)(73,124)(74,125)(75,126)(76,127)(77,128)(78,121)(79,122)(80,123)(97,130)(98,131)(99,132)(100,133)(101,134)(102,135)(103,136)(104,129), (1,15,31)(2,76,46)(4,130,64)(5,11,27)(6,80,42)(8,134,60)(10,120,78)(12,17,132)(14,116,74)(16,21,136)(18,59,73)(19,30,44)(22,63,77)(23,26,48)(28,58,114)(32,62,118)(33,98,137)(34,49,109)(35,100,139)(36,125,81)(37,102,141)(38,53,105)(39,104,143)(40,121,85)(41,113,131)(43,115,133)(45,117,135)(47,119,129)(51,140,94)(52,66,82)(55,144,90)(56,70,86)(65,101,111)(67,127,142)(69,97,107)(71,123,138)(83,96,103)(87,92,99)(89,106,128)(93,110,124), (1,15,31)(2,16,32)(3,9,25)(4,10,26)(5,11,27)(6,12,28)(7,13,29)(8,14,30)(17,58,80)(18,59,73)(19,60,74)(20,61,75)(21,62,76)(22,63,77)(23,64,78)(24,57,79)(33,137,98)(34,138,99)(35,139,100)(36,140,101)(37,141,102)(38,142,103)(39,143,104)(40,144,97)(41,131,113)(42,132,114)(43,133,115)(44,134,116)(45,135,117)(46,136,118)(47,129,119)(48,130,120)(49,71,87)(50,72,88)(51,65,81)(52,66,82)(53,67,83)(54,68,84)(55,69,85)(56,70,86)(89,106,128)(90,107,121)(91,108,122)(92,109,123)(93,110,124)(94,111,125)(95,112,126)(96,105,127), (1,20,45)(2,136,21)(3,47,22)(4,23,130)(5,24,41)(6,132,17)(7,43,18)(8,19,134)(9,129,63)(10,64,120)(11,57,131)(12,114,58)(13,133,59)(14,60,116)(15,61,135)(16,118,62)(25,119,77)(26,78,48)(27,79,113)(28,42,80)(29,115,73)(30,74,44)(31,75,117)(32,46,76)(33,56,122)(34,109,87)(35,124,50)(36,81,111)(37,52,126)(38,105,83)(39,128,54)(40,85,107)(49,138,123)(51,125,140)(53,142,127)(55,121,144)(65,94,101)(66,95,141)(67,103,96)(68,143,89)(69,90,97)(70,91,137)(71,99,92)(72,139,93)(82,112,102)(84,104,106)(86,108,98)(88,100,110), (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( (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,65)(9,84)(10,85)(11,86)(12,87)(13,88)(14,81)(15,82)(16,83)(17,92)(18,93)(19,94)(20,95)(21,96)(22,89)(23,90)(24,91)(25,54)(26,55)(27,56)(28,49)(29,50)(30,51)(31,52)(32,53)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(57,108)(58,109)(59,110)(60,111)(61,112)(62,105)(63,106)(64,107)(73,124)(74,125)(75,126)(76,127)(77,128)(78,121)(79,122)(80,123)(97,130)(98,131)(99,132)(100,133)(101,134)(102,135)(103,136)(104,129), (1,15,31)(2,76,46)(4,130,64)(5,11,27)(6,80,42)(8,134,60)(10,120,78)(12,17,132)(14,116,74)(16,21,136)(18,59,73)(19,30,44)(22,63,77)(23,26,48)(28,58,114)(32,62,118)(33,98,137)(34,49,109)(35,100,139)(36,125,81)(37,102,141)(38,53,105)(39,104,143)(40,121,85)(41,113,131)(43,115,133)(45,117,135)(47,119,129)(51,140,94)(52,66,82)(55,144,90)(56,70,86)(65,101,111)(67,127,142)(69,97,107)(71,123,138)(83,96,103)(87,92,99)(89,106,128)(93,110,124), (1,15,31)(2,16,32)(3,9,25)(4,10,26)(5,11,27)(6,12,28)(7,13,29)(8,14,30)(17,58,80)(18,59,73)(19,60,74)(20,61,75)(21,62,76)(22,63,77)(23,64,78)(24,57,79)(33,137,98)(34,138,99)(35,139,100)(36,140,101)(37,141,102)(38,142,103)(39,143,104)(40,144,97)(41,131,113)(42,132,114)(43,133,115)(44,134,116)(45,135,117)(46,136,118)(47,129,119)(48,130,120)(49,71,87)(50,72,88)(51,65,81)(52,66,82)(53,67,83)(54,68,84)(55,69,85)(56,70,86)(89,106,128)(90,107,121)(91,108,122)(92,109,123)(93,110,124)(94,111,125)(95,112,126)(96,105,127), (1,20,45)(2,136,21)(3,47,22)(4,23,130)(5,24,41)(6,132,17)(7,43,18)(8,19,134)(9,129,63)(10,64,120)(11,57,131)(12,114,58)(13,133,59)(14,60,116)(15,61,135)(16,118,62)(25,119,77)(26,78,48)(27,79,113)(28,42,80)(29,115,73)(30,74,44)(31,75,117)(32,46,76)(33,56,122)(34,109,87)(35,124,50)(36,81,111)(37,52,126)(38,105,83)(39,128,54)(40,85,107)(49,138,123)(51,125,140)(53,142,127)(55,121,144)(65,94,101)(66,95,141)(67,103,96)(68,143,89)(69,90,97)(70,91,137)(71,99,92)(72,139,93)(82,112,102)(84,104,106)(86,108,98)(88,100,110), (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([[(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,65),(9,84),(10,85),(11,86),(12,87),(13,88),(14,81),(15,82),(16,83),(17,92),(18,93),(19,94),(20,95),(21,96),(22,89),(23,90),(24,91),(25,54),(26,55),(27,56),(28,49),(29,50),(30,51),(31,52),(32,53),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,137),(42,138),(43,139),(44,140),(45,141),(46,142),(47,143),(48,144),(57,108),(58,109),(59,110),(60,111),(61,112),(62,105),(63,106),(64,107),(73,124),(74,125),(75,126),(76,127),(77,128),(78,121),(79,122),(80,123),(97,130),(98,131),(99,132),(100,133),(101,134),(102,135),(103,136),(104,129)], [(1,15,31),(2,76,46),(4,130,64),(5,11,27),(6,80,42),(8,134,60),(10,120,78),(12,17,132),(14,116,74),(16,21,136),(18,59,73),(19,30,44),(22,63,77),(23,26,48),(28,58,114),(32,62,118),(33,98,137),(34,49,109),(35,100,139),(36,125,81),(37,102,141),(38,53,105),(39,104,143),(40,121,85),(41,113,131),(43,115,133),(45,117,135),(47,119,129),(51,140,94),(52,66,82),(55,144,90),(56,70,86),(65,101,111),(67,127,142),(69,97,107),(71,123,138),(83,96,103),(87,92,99),(89,106,128),(93,110,124)], [(1,15,31),(2,16,32),(3,9,25),(4,10,26),(5,11,27),(6,12,28),(7,13,29),(8,14,30),(17,58,80),(18,59,73),(19,60,74),(20,61,75),(21,62,76),(22,63,77),(23,64,78),(24,57,79),(33,137,98),(34,138,99),(35,139,100),(36,140,101),(37,141,102),(38,142,103),(39,143,104),(40,144,97),(41,131,113),(42,132,114),(43,133,115),(44,134,116),(45,135,117),(46,136,118),(47,129,119),(48,130,120),(49,71,87),(50,72,88),(51,65,81),(52,66,82),(53,67,83),(54,68,84),(55,69,85),(56,70,86),(89,106,128),(90,107,121),(91,108,122),(92,109,123),(93,110,124),(94,111,125),(95,112,126),(96,105,127)], [(1,20,45),(2,136,21),(3,47,22),(4,23,130),(5,24,41),(6,132,17),(7,43,18),(8,19,134),(9,129,63),(10,64,120),(11,57,131),(12,114,58),(13,133,59),(14,60,116),(15,61,135),(16,118,62),(25,119,77),(26,78,48),(27,79,113),(28,42,80),(29,115,73),(30,74,44),(31,75,117),(32,46,76),(33,56,122),(34,109,87),(35,124,50),(36,81,111),(37,52,126),(38,105,83),(39,128,54),(40,85,107),(49,138,123),(51,125,140),(53,142,127),(55,121,144),(65,94,101),(66,95,141),(67,103,96),(68,143,89),(69,90,97),(70,91,137),(71,99,92),(72,139,93),(82,112,102),(84,104,106),(86,108,98),(88,100,110)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6L | 8A | ··· | 8H | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 12 | 12 | 9 | 9 | 9 | 9 | 1 | ··· | 1 | 12 | ··· | 12 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | ··· | 9 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | He3⋊C4 | He3⋊2C8 | C2×He3⋊C4 | C32⋊C4 | C32⋊2C8 | C2×C32⋊C4 |
| kernel | C2×He3⋊2C8 | He3⋊2C8 | C2×He3⋊3C4 | He3⋊3C4 | C22×He3 | C2×He3 | C22 | C2 | C2 | C2×C6 | C6 | C6 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 8 | 16 | 8 | 2 | 4 | 2 |
Matrix representation of C2×He3⋊2C8 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 9 | 9 |
| 0 | 9 | 72 | 9 |
| 0 | 72 | 72 | 65 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,8,0,0,0,0,1,0,0,0,0,64],[1,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,0,1,0,0,0,0,1,0,1,0,0],[1,0,0,0,0,72,9,72,0,9,72,72,0,9,9,65] >;
C2×He3⋊2C8 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3\rtimes_2C_8 % in TeX
G:=Group("C2xHe3:2C8"); // GroupNames label
G:=SmallGroup(432,277);
// by ID
G=gap.SmallGroup(432,277);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,58,3924,298,5381,2539,537]);
// 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*c*d,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations