non-abelian, supersoluble, monomial
Aliases: He3⋊2C42, C62.1D6, C3.2Dic32, C32⋊(C4×Dic3), C32⋊C12⋊2C4, He3⋊3C4⋊3C4, C3⋊Dic3⋊1Dic3, C6.13(S3×Dic3), C6.14(C6.D6), C22.3(C32⋊D6), (C22×He3).1C22, (C2×C6).47S32, (C3×C6).5(C4×S3), (C2×C3⋊Dic3).1S3, (C3×C6).3(C2×Dic3), C2.2(C6.S32), C2.2(He3⋊(C2×C4)), (C2×C32⋊C12).5C2, (C2×He3).12(C2×C4), (C2×He3⋊3C4).6C2, SmallGroup(432,94)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3⋊C42 |
Generators and relations for He3⋊C42
G = < a,b,c,d,e | a3=b3=c3=d4=e4=1, ab=ba, cac-1=ab-1, dad-1=a-1, ae=ea, bc=cb, dbd-1=ebe-1=b-1, cd=dc, ece-1=c-1, de=ed >
Subgroups: 495 in 121 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C32, C32, Dic3, C12, C2×C6, C2×C6, C42, C3×C6, C3×C6, C2×Dic3, C2×C12, He3, C3×Dic3, C3⋊Dic3, C62, C62, C4×Dic3, C2×He3, C2×He3, C6×Dic3, C2×C3⋊Dic3, C32⋊C12, He3⋊3C4, C22×He3, Dic32, C2×C32⋊C12, C2×He3⋊3C4, He3⋊C42
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, C4×S3, C2×Dic3, S32, C4×Dic3, S3×Dic3, C6.D6, C32⋊D6, Dic32, C6.S32, He3⋊(C2×C4), He3⋊C42
►On 144 points
Generators in S144
(1 51 117)(2 118 52)(3 49 119)(4 120 50)(5 33 64)(6 61 34)(7 35 62)(8 63 36)(9 69 40)(10 37 70)(11 71 38)(12 39 72)(13 42 20)(14 17 43)(15 44 18)(16 19 41)(21 55 129)(22 130 56)(23 53 131)(24 132 54)(25 68 133)(26 134 65)(27 66 135)(28 136 67)(29 60 123)(30 124 57)(31 58 121)(32 122 59)(45 74 103)(46 104 75)(47 76 101)(48 102 73)(77 113 86)(78 87 114)(79 115 88)(80 85 116)(81 141 110)(82 111 142)(83 143 112)(84 109 144)(89 94 126)(90 127 95)(91 96 128)(92 125 93)(97 106 140)(98 137 107)(99 108 138)(100 139 105)
(1 114 15)(2 16 115)(3 116 13)(4 14 113)(5 136 139)(6 140 133)(7 134 137)(8 138 135)(9 141 47)(10 48 142)(11 143 45)(12 46 144)(17 86 120)(18 117 87)(19 88 118)(20 119 85)(21 125 124)(22 121 126)(23 127 122)(24 123 128)(25 61 97)(26 98 62)(27 63 99)(28 100 64)(29 91 132)(30 129 92)(31 89 130)(32 131 90)(33 67 105)(34 106 68)(35 65 107)(36 108 66)(37 102 82)(38 83 103)(39 104 84)(40 81 101)(41 79 52)(42 49 80)(43 77 50)(44 51 78)(53 95 59)(54 60 96)(55 93 57)(56 58 94)(69 110 76)(70 73 111)(71 112 74)(72 75 109)
(5 136 139)(6 133 140)(7 134 137)(8 135 138)(9 47 141)(10 48 142)(11 45 143)(12 46 144)(17 86 120)(18 87 117)(19 88 118)(20 85 119)(29 91 132)(30 92 129)(31 89 130)(32 90 131)(33 105 67)(34 106 68)(35 107 65)(36 108 66)(37 82 102)(38 83 103)(39 84 104)(40 81 101)(41 52 79)(42 49 80)(43 50 77)(44 51 78)(53 95 59)(54 96 60)(55 93 57)(56 94 58)
(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 76 126 26)(2 73 127 27)(3 74 128 28)(4 75 125 25)(5 80 83 132)(6 77 84 129)(7 78 81 130)(8 79 82 131)(9 58 107 18)(10 59 108 19)(11 60 105 20)(12 57 106 17)(13 71 123 100)(14 72 124 97)(15 69 121 98)(16 70 122 99)(21 61 113 109)(22 62 114 110)(23 63 115 111)(24 64 116 112)(29 139 42 38)(30 140 43 39)(31 137 44 40)(32 138 41 37)(33 85 143 54)(34 86 144 55)(35 87 141 56)(36 88 142 53)(45 96 67 119)(46 93 68 120)(47 94 65 117)(48 95 66 118)(49 103 91 136)(50 104 92 133)(51 101 89 134)(52 102 90 135)
G:=sub<Sym(144)| (1,51,117)(2,118,52)(3,49,119)(4,120,50)(5,33,64)(6,61,34)(7,35,62)(8,63,36)(9,69,40)(10,37,70)(11,71,38)(12,39,72)(13,42,20)(14,17,43)(15,44,18)(16,19,41)(21,55,129)(22,130,56)(23,53,131)(24,132,54)(25,68,133)(26,134,65)(27,66,135)(28,136,67)(29,60,123)(30,124,57)(31,58,121)(32,122,59)(45,74,103)(46,104,75)(47,76,101)(48,102,73)(77,113,86)(78,87,114)(79,115,88)(80,85,116)(81,141,110)(82,111,142)(83,143,112)(84,109,144)(89,94,126)(90,127,95)(91,96,128)(92,125,93)(97,106,140)(98,137,107)(99,108,138)(100,139,105), (1,114,15)(2,16,115)(3,116,13)(4,14,113)(5,136,139)(6,140,133)(7,134,137)(8,138,135)(9,141,47)(10,48,142)(11,143,45)(12,46,144)(17,86,120)(18,117,87)(19,88,118)(20,119,85)(21,125,124)(22,121,126)(23,127,122)(24,123,128)(25,61,97)(26,98,62)(27,63,99)(28,100,64)(29,91,132)(30,129,92)(31,89,130)(32,131,90)(33,67,105)(34,106,68)(35,65,107)(36,108,66)(37,102,82)(38,83,103)(39,104,84)(40,81,101)(41,79,52)(42,49,80)(43,77,50)(44,51,78)(53,95,59)(54,60,96)(55,93,57)(56,58,94)(69,110,76)(70,73,111)(71,112,74)(72,75,109), (5,136,139)(6,133,140)(7,134,137)(8,135,138)(9,47,141)(10,48,142)(11,45,143)(12,46,144)(17,86,120)(18,87,117)(19,88,118)(20,85,119)(29,91,132)(30,92,129)(31,89,130)(32,90,131)(33,105,67)(34,106,68)(35,107,65)(36,108,66)(37,82,102)(38,83,103)(39,84,104)(40,81,101)(41,52,79)(42,49,80)(43,50,77)(44,51,78)(53,95,59)(54,96,60)(55,93,57)(56,94,58), (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), (1,76,126,26)(2,73,127,27)(3,74,128,28)(4,75,125,25)(5,80,83,132)(6,77,84,129)(7,78,81,130)(8,79,82,131)(9,58,107,18)(10,59,108,19)(11,60,105,20)(12,57,106,17)(13,71,123,100)(14,72,124,97)(15,69,121,98)(16,70,122,99)(21,61,113,109)(22,62,114,110)(23,63,115,111)(24,64,116,112)(29,139,42,38)(30,140,43,39)(31,137,44,40)(32,138,41,37)(33,85,143,54)(34,86,144,55)(35,87,141,56)(36,88,142,53)(45,96,67,119)(46,93,68,120)(47,94,65,117)(48,95,66,118)(49,103,91,136)(50,104,92,133)(51,101,89,134)(52,102,90,135)>;
G:=Group( (1,51,117)(2,118,52)(3,49,119)(4,120,50)(5,33,64)(6,61,34)(7,35,62)(8,63,36)(9,69,40)(10,37,70)(11,71,38)(12,39,72)(13,42,20)(14,17,43)(15,44,18)(16,19,41)(21,55,129)(22,130,56)(23,53,131)(24,132,54)(25,68,133)(26,134,65)(27,66,135)(28,136,67)(29,60,123)(30,124,57)(31,58,121)(32,122,59)(45,74,103)(46,104,75)(47,76,101)(48,102,73)(77,113,86)(78,87,114)(79,115,88)(80,85,116)(81,141,110)(82,111,142)(83,143,112)(84,109,144)(89,94,126)(90,127,95)(91,96,128)(92,125,93)(97,106,140)(98,137,107)(99,108,138)(100,139,105), (1,114,15)(2,16,115)(3,116,13)(4,14,113)(5,136,139)(6,140,133)(7,134,137)(8,138,135)(9,141,47)(10,48,142)(11,143,45)(12,46,144)(17,86,120)(18,117,87)(19,88,118)(20,119,85)(21,125,124)(22,121,126)(23,127,122)(24,123,128)(25,61,97)(26,98,62)(27,63,99)(28,100,64)(29,91,132)(30,129,92)(31,89,130)(32,131,90)(33,67,105)(34,106,68)(35,65,107)(36,108,66)(37,102,82)(38,83,103)(39,104,84)(40,81,101)(41,79,52)(42,49,80)(43,77,50)(44,51,78)(53,95,59)(54,60,96)(55,93,57)(56,58,94)(69,110,76)(70,73,111)(71,112,74)(72,75,109), (5,136,139)(6,133,140)(7,134,137)(8,135,138)(9,47,141)(10,48,142)(11,45,143)(12,46,144)(17,86,120)(18,87,117)(19,88,118)(20,85,119)(29,91,132)(30,92,129)(31,89,130)(32,90,131)(33,105,67)(34,106,68)(35,107,65)(36,108,66)(37,82,102)(38,83,103)(39,84,104)(40,81,101)(41,52,79)(42,49,80)(43,50,77)(44,51,78)(53,95,59)(54,96,60)(55,93,57)(56,94,58), (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), (1,76,126,26)(2,73,127,27)(3,74,128,28)(4,75,125,25)(5,80,83,132)(6,77,84,129)(7,78,81,130)(8,79,82,131)(9,58,107,18)(10,59,108,19)(11,60,105,20)(12,57,106,17)(13,71,123,100)(14,72,124,97)(15,69,121,98)(16,70,122,99)(21,61,113,109)(22,62,114,110)(23,63,115,111)(24,64,116,112)(29,139,42,38)(30,140,43,39)(31,137,44,40)(32,138,41,37)(33,85,143,54)(34,86,144,55)(35,87,141,56)(36,88,142,53)(45,96,67,119)(46,93,68,120)(47,94,65,117)(48,95,66,118)(49,103,91,136)(50,104,92,133)(51,101,89,134)(52,102,90,135) );
G=PermutationGroup([[(1,51,117),(2,118,52),(3,49,119),(4,120,50),(5,33,64),(6,61,34),(7,35,62),(8,63,36),(9,69,40),(10,37,70),(11,71,38),(12,39,72),(13,42,20),(14,17,43),(15,44,18),(16,19,41),(21,55,129),(22,130,56),(23,53,131),(24,132,54),(25,68,133),(26,134,65),(27,66,135),(28,136,67),(29,60,123),(30,124,57),(31,58,121),(32,122,59),(45,74,103),(46,104,75),(47,76,101),(48,102,73),(77,113,86),(78,87,114),(79,115,88),(80,85,116),(81,141,110),(82,111,142),(83,143,112),(84,109,144),(89,94,126),(90,127,95),(91,96,128),(92,125,93),(97,106,140),(98,137,107),(99,108,138),(100,139,105)], [(1,114,15),(2,16,115),(3,116,13),(4,14,113),(5,136,139),(6,140,133),(7,134,137),(8,138,135),(9,141,47),(10,48,142),(11,143,45),(12,46,144),(17,86,120),(18,117,87),(19,88,118),(20,119,85),(21,125,124),(22,121,126),(23,127,122),(24,123,128),(25,61,97),(26,98,62),(27,63,99),(28,100,64),(29,91,132),(30,129,92),(31,89,130),(32,131,90),(33,67,105),(34,106,68),(35,65,107),(36,108,66),(37,102,82),(38,83,103),(39,104,84),(40,81,101),(41,79,52),(42,49,80),(43,77,50),(44,51,78),(53,95,59),(54,60,96),(55,93,57),(56,58,94),(69,110,76),(70,73,111),(71,112,74),(72,75,109)], [(5,136,139),(6,133,140),(7,134,137),(8,135,138),(9,47,141),(10,48,142),(11,45,143),(12,46,144),(17,86,120),(18,87,117),(19,88,118),(20,85,119),(29,91,132),(30,92,129),(31,89,130),(32,90,131),(33,105,67),(34,106,68),(35,107,65),(36,108,66),(37,82,102),(38,83,103),(39,84,104),(40,81,101),(41,52,79),(42,49,80),(43,50,77),(44,51,78),(53,95,59),(54,96,60),(55,93,57),(56,94,58)], [(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)], [(1,76,126,26),(2,73,127,27),(3,74,128,28),(4,75,125,25),(5,80,83,132),(6,77,84,129),(7,78,81,130),(8,79,82,131),(9,58,107,18),(10,59,108,19),(11,60,105,20),(12,57,106,17),(13,71,123,100),(14,72,124,97),(15,69,121,98),(16,70,122,99),(21,61,113,109),(22,62,114,110),(23,63,115,111),(24,64,116,112),(29,139,42,38),(30,140,43,39),(31,137,44,40),(32,138,41,37),(33,85,143,54),(34,86,144,55),(35,87,141,56),(36,88,142,53),(45,96,67,119),(46,93,68,120),(47,94,65,117),(48,95,66,118),(49,103,91,136),(50,104,92,133),(51,101,89,134),(52,102,90,135)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | ··· | 4L | 6A | 6B | 6C | 6D | ··· | 6I | 6J | 6K | 6L | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 12 | 9 | ··· | 9 | 2 | 2 | 2 | 6 | ··· | 6 | 12 | 12 | 12 | 18 | ··· | 18 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | C4×S3 | S32 | S3×Dic3 | C6.D6 | C32⋊D6 | C6.S32 | He3⋊(C2×C4) |
| kernel | He3⋊C42 | C2×C32⋊C12 | C2×He3⋊3C4 | C32⋊C12 | He3⋊3C4 | C2×C3⋊Dic3 | C3⋊Dic3 | C62 | C3×C6 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 2 | 4 | 2 | 8 | 1 | 2 | 1 | 2 | 4 | 2 |
Matrix representation of He3⋊C42 ►in GL10(𝔽13)
| 12 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 2 | 0 | 0 | 3 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 2 | 0 | 0 | 3 |
| 5 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 1 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 1 | 12 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 4 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 2 | 4 | 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 12 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 12 | 0 | 1 |
| 0 | 0 | 0 | 0 | 11 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 | 9 | 0 | 0 | 9 |
G:=sub<GL(10,GF(13))| [12,0,12,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,9,0,0,1,1,1,0,0,0,0,0,9,0,0,2,0,0,0,0,0,0,0,9,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,1,4,0,12,1,0,0,0,0,0,3,0,0,11,0,0,0,0,0,0,0,9,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,3],[5,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,8,0,8,0,0,0,0,0,0,0,0,8,0,8,0,0,0,0,0,0,0,0,0,0,9,0,0,2,2,2,0,0,0,0,0,0,9,0,0,4,0,0,0,0,0,9,0,0,4,0,0,0,0,0,11,1,1,4,0,0,0,0,0,0,0,0,12,0,0,4,0,0,0,0,0,12,0,0,4,0],[1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,4,0,0,11,11,11,0,0,0,0,0,4,0,0,9,0,0,0,0,0,0,0,4,0,0,9,0,0,0,0,2,12,12,9,0,0,0,0,0,0,0,1,0,0,9,0,0,0,0,0,0,0,1,0,0,9] >;
He3⋊C42 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes C_4^2 % in TeX
G:=Group("He3:C4^2"); // GroupNames label
G:=SmallGroup(432,94);
// by ID
G=gap.SmallGroup(432,94);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,36,571,4037,537,14118,7069]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^4=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e^-1=b^-1,c*d=d*c,e*c*e^-1=c^-1,d*e=e*d>;
// generators/relations