direct product, metabelian, nilpotent (class 2), monomial
Aliases: C2×C8×He3, C62.9C12, C12.38C62, (C6×C24)⋊C3, (C3×C24)⋊8C6, (C3×C6)⋊4C24, C6.4(C3×C24), C3.1(C6×C24), C4.3(C4×He3), C24.18(C3×C6), (C6×C12).18C6, C6.14(C6×C12), C32⋊8(C2×C24), (C3×C12).13C12, C12.15(C3×C12), (C4×He3).12C4, (C2×C24).1C32, C4.5(C22×He3), C22.2(C4×He3), (C22×He3).8C4, (C4×He3).54C22, C2.2(C2×C4×He3), (C2×C4×He3).17C2, (C2×C4).5(C2×He3), (C2×C6).15(C3×C12), (C3×C6).32(C2×C12), (C3×C12).66(C2×C6), (C2×C12).30(C3×C6), (C2×He3).38(C2×C4), SmallGroup(432,210)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C8×He3
G = < a,b,c,d,e | a2=b8=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 209 in 121 conjugacy classes, 77 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C32, C12, C12, C2×C6, C2×C6, C2×C8, C3×C6, C24, C24, C2×C12, C2×C12, He3, C3×C12, C62, C2×C24, C2×C24, C2×He3, C2×He3, C3×C24, C6×C12, C4×He3, C22×He3, C6×C24, C8×He3, C2×C4×He3, C2×C8×He3
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C32, C12, C2×C6, C2×C8, C3×C6, C24, C2×C12, He3, C3×C12, C62, C2×C24, C2×He3, C3×C24, C6×C12, C4×He3, C22×He3, C6×C24, C8×He3, C2×C4×He3, C2×C8×He3
►On 144 points
Generators in S144
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 52)(26 53)(27 54)(28 55)(29 56)(30 49)(31 50)(32 51)(33 96)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 111)(66 112)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)(73 126)(74 127)(75 128)(76 121)(77 122)(78 123)(79 124)(80 125)(81 134)(82 135)(83 136)(84 129)(85 130)(86 131)(87 132)(88 133)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 113)
(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 86 143)(2 87 144)(3 88 137)(4 81 138)(5 82 139)(6 83 140)(7 84 141)(8 85 142)(9 132 64)(10 133 57)(11 134 58)(12 135 59)(13 136 60)(14 129 61)(15 130 62)(16 131 63)(17 49 72)(18 50 65)(19 51 66)(20 52 67)(21 53 68)(22 54 69)(23 55 70)(24 56 71)(25 105 47)(26 106 48)(27 107 41)(28 108 42)(29 109 43)(30 110 44)(31 111 45)(32 112 46)(33 113 76)(34 114 77)(35 115 78)(36 116 79)(37 117 80)(38 118 73)(39 119 74)(40 120 75)(89 97 122)(90 98 123)(91 99 124)(92 100 125)(93 101 126)(94 102 127)(95 103 128)(96 104 121)
(1 35 26)(2 36 27)(3 37 28)(4 38 29)(5 39 30)(6 40 31)(7 33 32)(8 34 25)(9 91 54)(10 92 55)(11 93 56)(12 94 49)(13 95 50)(14 96 51)(15 89 52)(16 90 53)(17 59 127)(18 60 128)(19 61 121)(20 62 122)(21 63 123)(22 64 124)(23 57 125)(24 58 126)(41 144 79)(42 137 80)(43 138 73)(44 139 74)(45 140 75)(46 141 76)(47 142 77)(48 143 78)(65 136 103)(66 129 104)(67 130 97)(68 131 98)(69 132 99)(70 133 100)(71 134 101)(72 135 102)(81 118 109)(82 119 110)(83 120 111)(84 113 112)(85 114 105)(86 115 106)(87 116 107)(88 117 108)
(1 143 106)(2 144 107)(3 137 108)(4 138 109)(5 139 110)(6 140 111)(7 141 112)(8 142 105)(9 64 69)(10 57 70)(11 58 71)(12 59 72)(13 60 65)(14 61 66)(15 62 67)(16 63 68)(17 102 49)(18 103 50)(19 104 51)(20 97 52)(21 98 53)(22 99 54)(23 100 55)(24 101 56)(25 47 114)(26 48 115)(27 41 116)(28 42 117)(29 43 118)(30 44 119)(31 45 120)(32 46 113)(33 76 84)(34 77 85)(35 78 86)(36 79 87)(37 80 88)(38 73 81)(39 74 82)(40 75 83)(89 122 130)(90 123 131)(91 124 132)(92 125 133)(93 126 134)(94 127 135)(95 128 136)(96 121 129)
G:=sub<Sym(144)| (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,52)(26,53)(27,54)(28,55)(29,56)(30,49)(31,50)(32,51)(33,96)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,111)(66,112)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,126)(74,127)(75,128)(76,121)(77,122)(78,123)(79,124)(80,125)(81,134)(82,135)(83,136)(84,129)(85,130)(86,131)(87,132)(88,133)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,113), (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), (1,86,143)(2,87,144)(3,88,137)(4,81,138)(5,82,139)(6,83,140)(7,84,141)(8,85,142)(9,132,64)(10,133,57)(11,134,58)(12,135,59)(13,136,60)(14,129,61)(15,130,62)(16,131,63)(17,49,72)(18,50,65)(19,51,66)(20,52,67)(21,53,68)(22,54,69)(23,55,70)(24,56,71)(25,105,47)(26,106,48)(27,107,41)(28,108,42)(29,109,43)(30,110,44)(31,111,45)(32,112,46)(33,113,76)(34,114,77)(35,115,78)(36,116,79)(37,117,80)(38,118,73)(39,119,74)(40,120,75)(89,97,122)(90,98,123)(91,99,124)(92,100,125)(93,101,126)(94,102,127)(95,103,128)(96,104,121), (1,35,26)(2,36,27)(3,37,28)(4,38,29)(5,39,30)(6,40,31)(7,33,32)(8,34,25)(9,91,54)(10,92,55)(11,93,56)(12,94,49)(13,95,50)(14,96,51)(15,89,52)(16,90,53)(17,59,127)(18,60,128)(19,61,121)(20,62,122)(21,63,123)(22,64,124)(23,57,125)(24,58,126)(41,144,79)(42,137,80)(43,138,73)(44,139,74)(45,140,75)(46,141,76)(47,142,77)(48,143,78)(65,136,103)(66,129,104)(67,130,97)(68,131,98)(69,132,99)(70,133,100)(71,134,101)(72,135,102)(81,118,109)(82,119,110)(83,120,111)(84,113,112)(85,114,105)(86,115,106)(87,116,107)(88,117,108), (1,143,106)(2,144,107)(3,137,108)(4,138,109)(5,139,110)(6,140,111)(7,141,112)(8,142,105)(9,64,69)(10,57,70)(11,58,71)(12,59,72)(13,60,65)(14,61,66)(15,62,67)(16,63,68)(17,102,49)(18,103,50)(19,104,51)(20,97,52)(21,98,53)(22,99,54)(23,100,55)(24,101,56)(25,47,114)(26,48,115)(27,41,116)(28,42,117)(29,43,118)(30,44,119)(31,45,120)(32,46,113)(33,76,84)(34,77,85)(35,78,86)(36,79,87)(37,80,88)(38,73,81)(39,74,82)(40,75,83)(89,122,130)(90,123,131)(91,124,132)(92,125,133)(93,126,134)(94,127,135)(95,128,136)(96,121,129)>;
G:=Group( (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,52)(26,53)(27,54)(28,55)(29,56)(30,49)(31,50)(32,51)(33,96)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,111)(66,112)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,126)(74,127)(75,128)(76,121)(77,122)(78,123)(79,124)(80,125)(81,134)(82,135)(83,136)(84,129)(85,130)(86,131)(87,132)(88,133)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,113), (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), (1,86,143)(2,87,144)(3,88,137)(4,81,138)(5,82,139)(6,83,140)(7,84,141)(8,85,142)(9,132,64)(10,133,57)(11,134,58)(12,135,59)(13,136,60)(14,129,61)(15,130,62)(16,131,63)(17,49,72)(18,50,65)(19,51,66)(20,52,67)(21,53,68)(22,54,69)(23,55,70)(24,56,71)(25,105,47)(26,106,48)(27,107,41)(28,108,42)(29,109,43)(30,110,44)(31,111,45)(32,112,46)(33,113,76)(34,114,77)(35,115,78)(36,116,79)(37,117,80)(38,118,73)(39,119,74)(40,120,75)(89,97,122)(90,98,123)(91,99,124)(92,100,125)(93,101,126)(94,102,127)(95,103,128)(96,104,121), (1,35,26)(2,36,27)(3,37,28)(4,38,29)(5,39,30)(6,40,31)(7,33,32)(8,34,25)(9,91,54)(10,92,55)(11,93,56)(12,94,49)(13,95,50)(14,96,51)(15,89,52)(16,90,53)(17,59,127)(18,60,128)(19,61,121)(20,62,122)(21,63,123)(22,64,124)(23,57,125)(24,58,126)(41,144,79)(42,137,80)(43,138,73)(44,139,74)(45,140,75)(46,141,76)(47,142,77)(48,143,78)(65,136,103)(66,129,104)(67,130,97)(68,131,98)(69,132,99)(70,133,100)(71,134,101)(72,135,102)(81,118,109)(82,119,110)(83,120,111)(84,113,112)(85,114,105)(86,115,106)(87,116,107)(88,117,108), (1,143,106)(2,144,107)(3,137,108)(4,138,109)(5,139,110)(6,140,111)(7,141,112)(8,142,105)(9,64,69)(10,57,70)(11,58,71)(12,59,72)(13,60,65)(14,61,66)(15,62,67)(16,63,68)(17,102,49)(18,103,50)(19,104,51)(20,97,52)(21,98,53)(22,99,54)(23,100,55)(24,101,56)(25,47,114)(26,48,115)(27,41,116)(28,42,117)(29,43,118)(30,44,119)(31,45,120)(32,46,113)(33,76,84)(34,77,85)(35,78,86)(36,79,87)(37,80,88)(38,73,81)(39,74,82)(40,75,83)(89,122,130)(90,123,131)(91,124,132)(92,125,133)(93,126,134)(94,127,135)(95,128,136)(96,121,129) );
G=PermutationGroup([[(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(17,44),(18,45),(19,46),(20,47),(21,48),(22,41),(23,42),(24,43),(25,52),(26,53),(27,54),(28,55),(29,56),(30,49),(31,50),(32,51),(33,96),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,111),(66,112),(67,105),(68,106),(69,107),(70,108),(71,109),(72,110),(73,126),(74,127),(75,128),(76,121),(77,122),(78,123),(79,124),(80,125),(81,134),(82,135),(83,136),(84,129),(85,130),(86,131),(87,132),(88,133),(97,114),(98,115),(99,116),(100,117),(101,118),(102,119),(103,120),(104,113)], [(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)], [(1,86,143),(2,87,144),(3,88,137),(4,81,138),(5,82,139),(6,83,140),(7,84,141),(8,85,142),(9,132,64),(10,133,57),(11,134,58),(12,135,59),(13,136,60),(14,129,61),(15,130,62),(16,131,63),(17,49,72),(18,50,65),(19,51,66),(20,52,67),(21,53,68),(22,54,69),(23,55,70),(24,56,71),(25,105,47),(26,106,48),(27,107,41),(28,108,42),(29,109,43),(30,110,44),(31,111,45),(32,112,46),(33,113,76),(34,114,77),(35,115,78),(36,116,79),(37,117,80),(38,118,73),(39,119,74),(40,120,75),(89,97,122),(90,98,123),(91,99,124),(92,100,125),(93,101,126),(94,102,127),(95,103,128),(96,104,121)], [(1,35,26),(2,36,27),(3,37,28),(4,38,29),(5,39,30),(6,40,31),(7,33,32),(8,34,25),(9,91,54),(10,92,55),(11,93,56),(12,94,49),(13,95,50),(14,96,51),(15,89,52),(16,90,53),(17,59,127),(18,60,128),(19,61,121),(20,62,122),(21,63,123),(22,64,124),(23,57,125),(24,58,126),(41,144,79),(42,137,80),(43,138,73),(44,139,74),(45,140,75),(46,141,76),(47,142,77),(48,143,78),(65,136,103),(66,129,104),(67,130,97),(68,131,98),(69,132,99),(70,133,100),(71,134,101),(72,135,102),(81,118,109),(82,119,110),(83,120,111),(84,113,112),(85,114,105),(86,115,106),(87,116,107),(88,117,108)], [(1,143,106),(2,144,107),(3,137,108),(4,138,109),(5,139,110),(6,140,111),(7,141,112),(8,142,105),(9,64,69),(10,57,70),(11,58,71),(12,59,72),(13,60,65),(14,61,66),(15,62,67),(16,63,68),(17,102,49),(18,103,50),(19,104,51),(20,97,52),(21,98,53),(22,99,54),(23,100,55),(24,101,56),(25,47,114),(26,48,115),(27,41,116),(28,42,117),(29,43,118),(30,44,119),(31,45,120),(32,46,113),(33,76,84),(34,77,85),(35,78,86),(36,79,87),(37,80,88),(38,73,81),(39,74,82),(40,75,83),(89,122,130),(90,123,131),(91,124,132),(92,125,133),(93,126,134),(94,127,135),(95,128,136),(96,121,129)]])
176 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3J | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6AD | 8A | ··· | 8H | 12A | ··· | 12H | 12I | ··· | 12AN | 24A | ··· | 24P | 24Q | ··· | 24CB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 |
176 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | He3 | C2×He3 | C2×He3 | C4×He3 | C4×He3 | C8×He3 |
| kernel | C2×C8×He3 | C8×He3 | C2×C4×He3 | C6×C24 | C4×He3 | C22×He3 | C3×C24 | C6×C12 | C2×He3 | C3×C12 | C62 | C3×C6 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 16 | 8 | 8 | 16 | 16 | 64 | 2 | 4 | 2 | 4 | 4 | 16 |
Matrix representation of C2×C8×He3 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 10 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 72 | 72 | 7 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 8 | 0 | 0 | 0 |
| 0 | 65 | 65 | 56 |
| 0 | 1 | 0 | 0 |
| 0 | 72 | 8 | 8 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[10,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[1,0,0,0,0,0,72,0,0,1,72,0,0,0,7,1],[1,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,0,65,1,72,0,65,0,8,0,56,0,8] >;
C2×C8×He3 in GAP, Magma, Sage, TeX
C_2\times C_8\times {\rm He}_3 % in TeX
G:=Group("C2xC8xHe3"); // GroupNames label
G:=SmallGroup(432,210);
// by ID
G=gap.SmallGroup(432,210);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,252,605,242]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^3=d^3=e^3=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,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations