direct product, non-abelian, supersoluble, monomial
Aliases: C2×He3⋊2Q8, C62.6D6, (C3×C6)⋊Dic6, He3⋊4(C2×Q8), (C2×He3)⋊2Q8, C3⋊Dic3.8D6, C32⋊2(C2×Dic6), C6.10(C32⋊2Q8), C32⋊C12.7C22, (C2×He3).10C23, C22.8(C32⋊D6), He3⋊3C4.13C22, (C22×He3).6C22, C6.84(C2×S32), (C2×C6).52S32, (C2×C3⋊Dic3).4S3, C2.12(C2×C32⋊D6), C3.2(C2×C32⋊2Q8), (C2×C32⋊C12).4C2, (C3×C6).10(C22×S3), (C2×He3⋊3C4).5C2, SmallGroup(432,316)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×He3⋊2Q8
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, be=eb, fbf-1=b-1, cd=dc, ece-1=fcf-1=c-1, ede-1=d-1, df=fd, fef-1=e-1 >
Subgroups: 739 in 149 conjugacy classes, 45 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, Q8, C32, C32, Dic3, C12, C2×C6, C2×C6, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, He3, C3×Dic3, C3⋊Dic3, C62, C62, C2×Dic6, C2×He3, C2×He3, C32⋊2Q8, C6×Dic3, C2×C3⋊Dic3, C32⋊C12, He3⋊3C4, C22×He3, C2×C32⋊2Q8, He3⋊2Q8, C2×C32⋊C12, C2×He3⋊3C4, C2×He3⋊2Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, C22×S3, S32, C2×Dic6, C32⋊2Q8, C2×S32, C32⋊D6, C2×C32⋊2Q8, He3⋊2Q8, C2×C32⋊D6, C2×He3⋊2Q8
►On 144 points
Generators in S144
(1 27)(2 28)(3 25)(4 26)(5 38)(6 39)(7 40)(8 37)(9 73)(10 74)(11 75)(12 76)(13 69)(14 70)(15 71)(16 72)(17 46)(18 47)(19 48)(20 45)(21 58)(22 59)(23 60)(24 57)(29 68)(30 65)(31 66)(32 67)(33 102)(34 103)(35 104)(36 101)(41 112)(42 109)(43 110)(44 111)(49 95)(50 96)(51 93)(52 94)(53 130)(54 131)(55 132)(56 129)(61 88)(62 85)(63 86)(64 87)(77 117)(78 118)(79 119)(80 120)(81 121)(82 122)(83 123)(84 124)(89 97)(90 98)(91 99)(92 100)(105 133)(106 134)(107 135)(108 136)(113 137)(114 138)(115 139)(116 140)(125 141)(126 142)(127 143)(128 144)
(5 49 66)(6 50 67)(7 51 68)(8 52 65)(21 112 103)(22 109 104)(23 110 101)(24 111 102)(29 40 93)(30 37 94)(31 38 95)(32 39 96)(33 57 44)(34 58 41)(35 59 42)(36 60 43)(53 144 117)(54 141 118)(55 142 119)(56 143 120)(61 115 99)(62 116 100)(63 113 97)(64 114 98)(77 130 128)(78 131 125)(79 132 126)(80 129 127)(85 140 92)(86 137 89)(87 138 90)(88 139 91)
(1 69 19)(2 20 70)(3 71 17)(4 18 72)(5 49 66)(6 67 50)(7 51 68)(8 65 52)(9 82 106)(10 107 83)(11 84 108)(12 105 81)(13 48 27)(14 28 45)(15 46 25)(16 26 47)(21 112 103)(22 104 109)(23 110 101)(24 102 111)(29 40 93)(30 94 37)(31 38 95)(32 96 39)(33 44 57)(34 58 41)(35 42 59)(36 60 43)(53 117 144)(54 141 118)(55 119 142)(56 143 120)(61 99 115)(62 116 100)(63 97 113)(64 114 98)(73 122 134)(74 135 123)(75 124 136)(76 133 121)(77 128 130)(78 131 125)(79 126 132)(80 129 127)(85 140 92)(86 89 137)(87 138 90)(88 91 139)
(1 32 23)(2 24 29)(3 30 21)(4 22 31)(5 47 35)(6 36 48)(7 45 33)(8 34 46)(9 97 118)(10 119 98)(11 99 120)(12 117 100)(13 50 43)(14 44 51)(15 52 41)(16 42 49)(17 37 103)(18 104 38)(19 39 101)(20 102 40)(25 65 58)(26 59 66)(27 67 60)(28 57 68)(53 116 81)(54 82 113)(55 114 83)(56 84 115)(61 143 108)(62 105 144)(63 141 106)(64 107 142)(69 96 110)(70 111 93)(71 94 112)(72 109 95)(73 89 78)(74 79 90)(75 91 80)(76 77 92)(85 133 128)(86 125 134)(87 135 126)(88 127 136)(121 130 140)(122 137 131)(123 132 138)(124 139 129)
(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 9 3 11)(2 12 4 10)(5 138 7 140)(6 137 8 139)(13 134 15 136)(14 133 16 135)(17 84 19 82)(18 83 20 81)(21 120 23 118)(22 119 24 117)(25 75 27 73)(26 74 28 76)(29 100 31 98)(30 99 32 97)(33 130 35 132)(34 129 36 131)(37 115 39 113)(38 114 40 116)(41 127 43 125)(42 126 44 128)(45 121 47 123)(46 124 48 122)(49 87 51 85)(50 86 52 88)(53 104 55 102)(54 103 56 101)(57 77 59 79)(58 80 60 78)(61 96 63 94)(62 95 64 93)(65 91 67 89)(66 90 68 92)(69 106 71 108)(70 105 72 107)(109 142 111 144)(110 141 112 143)
G:=sub<Sym(144)| (1,27)(2,28)(3,25)(4,26)(5,38)(6,39)(7,40)(8,37)(9,73)(10,74)(11,75)(12,76)(13,69)(14,70)(15,71)(16,72)(17,46)(18,47)(19,48)(20,45)(21,58)(22,59)(23,60)(24,57)(29,68)(30,65)(31,66)(32,67)(33,102)(34,103)(35,104)(36,101)(41,112)(42,109)(43,110)(44,111)(49,95)(50,96)(51,93)(52,94)(53,130)(54,131)(55,132)(56,129)(61,88)(62,85)(63,86)(64,87)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(89,97)(90,98)(91,99)(92,100)(105,133)(106,134)(107,135)(108,136)(113,137)(114,138)(115,139)(116,140)(125,141)(126,142)(127,143)(128,144), (5,49,66)(6,50,67)(7,51,68)(8,52,65)(21,112,103)(22,109,104)(23,110,101)(24,111,102)(29,40,93)(30,37,94)(31,38,95)(32,39,96)(33,57,44)(34,58,41)(35,59,42)(36,60,43)(53,144,117)(54,141,118)(55,142,119)(56,143,120)(61,115,99)(62,116,100)(63,113,97)(64,114,98)(77,130,128)(78,131,125)(79,132,126)(80,129,127)(85,140,92)(86,137,89)(87,138,90)(88,139,91), (1,69,19)(2,20,70)(3,71,17)(4,18,72)(5,49,66)(6,67,50)(7,51,68)(8,65,52)(9,82,106)(10,107,83)(11,84,108)(12,105,81)(13,48,27)(14,28,45)(15,46,25)(16,26,47)(21,112,103)(22,104,109)(23,110,101)(24,102,111)(29,40,93)(30,94,37)(31,38,95)(32,96,39)(33,44,57)(34,58,41)(35,42,59)(36,60,43)(53,117,144)(54,141,118)(55,119,142)(56,143,120)(61,99,115)(62,116,100)(63,97,113)(64,114,98)(73,122,134)(74,135,123)(75,124,136)(76,133,121)(77,128,130)(78,131,125)(79,126,132)(80,129,127)(85,140,92)(86,89,137)(87,138,90)(88,91,139), (1,32,23)(2,24,29)(3,30,21)(4,22,31)(5,47,35)(6,36,48)(7,45,33)(8,34,46)(9,97,118)(10,119,98)(11,99,120)(12,117,100)(13,50,43)(14,44,51)(15,52,41)(16,42,49)(17,37,103)(18,104,38)(19,39,101)(20,102,40)(25,65,58)(26,59,66)(27,67,60)(28,57,68)(53,116,81)(54,82,113)(55,114,83)(56,84,115)(61,143,108)(62,105,144)(63,141,106)(64,107,142)(69,96,110)(70,111,93)(71,94,112)(72,109,95)(73,89,78)(74,79,90)(75,91,80)(76,77,92)(85,133,128)(86,125,134)(87,135,126)(88,127,136)(121,130,140)(122,137,131)(123,132,138)(124,139,129), (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), (1,9,3,11)(2,12,4,10)(5,138,7,140)(6,137,8,139)(13,134,15,136)(14,133,16,135)(17,84,19,82)(18,83,20,81)(21,120,23,118)(22,119,24,117)(25,75,27,73)(26,74,28,76)(29,100,31,98)(30,99,32,97)(33,130,35,132)(34,129,36,131)(37,115,39,113)(38,114,40,116)(41,127,43,125)(42,126,44,128)(45,121,47,123)(46,124,48,122)(49,87,51,85)(50,86,52,88)(53,104,55,102)(54,103,56,101)(57,77,59,79)(58,80,60,78)(61,96,63,94)(62,95,64,93)(65,91,67,89)(66,90,68,92)(69,106,71,108)(70,105,72,107)(109,142,111,144)(110,141,112,143)>;
G:=Group( (1,27)(2,28)(3,25)(4,26)(5,38)(6,39)(7,40)(8,37)(9,73)(10,74)(11,75)(12,76)(13,69)(14,70)(15,71)(16,72)(17,46)(18,47)(19,48)(20,45)(21,58)(22,59)(23,60)(24,57)(29,68)(30,65)(31,66)(32,67)(33,102)(34,103)(35,104)(36,101)(41,112)(42,109)(43,110)(44,111)(49,95)(50,96)(51,93)(52,94)(53,130)(54,131)(55,132)(56,129)(61,88)(62,85)(63,86)(64,87)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(89,97)(90,98)(91,99)(92,100)(105,133)(106,134)(107,135)(108,136)(113,137)(114,138)(115,139)(116,140)(125,141)(126,142)(127,143)(128,144), (5,49,66)(6,50,67)(7,51,68)(8,52,65)(21,112,103)(22,109,104)(23,110,101)(24,111,102)(29,40,93)(30,37,94)(31,38,95)(32,39,96)(33,57,44)(34,58,41)(35,59,42)(36,60,43)(53,144,117)(54,141,118)(55,142,119)(56,143,120)(61,115,99)(62,116,100)(63,113,97)(64,114,98)(77,130,128)(78,131,125)(79,132,126)(80,129,127)(85,140,92)(86,137,89)(87,138,90)(88,139,91), (1,69,19)(2,20,70)(3,71,17)(4,18,72)(5,49,66)(6,67,50)(7,51,68)(8,65,52)(9,82,106)(10,107,83)(11,84,108)(12,105,81)(13,48,27)(14,28,45)(15,46,25)(16,26,47)(21,112,103)(22,104,109)(23,110,101)(24,102,111)(29,40,93)(30,94,37)(31,38,95)(32,96,39)(33,44,57)(34,58,41)(35,42,59)(36,60,43)(53,117,144)(54,141,118)(55,119,142)(56,143,120)(61,99,115)(62,116,100)(63,97,113)(64,114,98)(73,122,134)(74,135,123)(75,124,136)(76,133,121)(77,128,130)(78,131,125)(79,126,132)(80,129,127)(85,140,92)(86,89,137)(87,138,90)(88,91,139), (1,32,23)(2,24,29)(3,30,21)(4,22,31)(5,47,35)(6,36,48)(7,45,33)(8,34,46)(9,97,118)(10,119,98)(11,99,120)(12,117,100)(13,50,43)(14,44,51)(15,52,41)(16,42,49)(17,37,103)(18,104,38)(19,39,101)(20,102,40)(25,65,58)(26,59,66)(27,67,60)(28,57,68)(53,116,81)(54,82,113)(55,114,83)(56,84,115)(61,143,108)(62,105,144)(63,141,106)(64,107,142)(69,96,110)(70,111,93)(71,94,112)(72,109,95)(73,89,78)(74,79,90)(75,91,80)(76,77,92)(85,133,128)(86,125,134)(87,135,126)(88,127,136)(121,130,140)(122,137,131)(123,132,138)(124,139,129), (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), (1,9,3,11)(2,12,4,10)(5,138,7,140)(6,137,8,139)(13,134,15,136)(14,133,16,135)(17,84,19,82)(18,83,20,81)(21,120,23,118)(22,119,24,117)(25,75,27,73)(26,74,28,76)(29,100,31,98)(30,99,32,97)(33,130,35,132)(34,129,36,131)(37,115,39,113)(38,114,40,116)(41,127,43,125)(42,126,44,128)(45,121,47,123)(46,124,48,122)(49,87,51,85)(50,86,52,88)(53,104,55,102)(54,103,56,101)(57,77,59,79)(58,80,60,78)(61,96,63,94)(62,95,64,93)(65,91,67,89)(66,90,68,92)(69,106,71,108)(70,105,72,107)(109,142,111,144)(110,141,112,143) );
G=PermutationGroup([[(1,27),(2,28),(3,25),(4,26),(5,38),(6,39),(7,40),(8,37),(9,73),(10,74),(11,75),(12,76),(13,69),(14,70),(15,71),(16,72),(17,46),(18,47),(19,48),(20,45),(21,58),(22,59),(23,60),(24,57),(29,68),(30,65),(31,66),(32,67),(33,102),(34,103),(35,104),(36,101),(41,112),(42,109),(43,110),(44,111),(49,95),(50,96),(51,93),(52,94),(53,130),(54,131),(55,132),(56,129),(61,88),(62,85),(63,86),(64,87),(77,117),(78,118),(79,119),(80,120),(81,121),(82,122),(83,123),(84,124),(89,97),(90,98),(91,99),(92,100),(105,133),(106,134),(107,135),(108,136),(113,137),(114,138),(115,139),(116,140),(125,141),(126,142),(127,143),(128,144)], [(5,49,66),(6,50,67),(7,51,68),(8,52,65),(21,112,103),(22,109,104),(23,110,101),(24,111,102),(29,40,93),(30,37,94),(31,38,95),(32,39,96),(33,57,44),(34,58,41),(35,59,42),(36,60,43),(53,144,117),(54,141,118),(55,142,119),(56,143,120),(61,115,99),(62,116,100),(63,113,97),(64,114,98),(77,130,128),(78,131,125),(79,132,126),(80,129,127),(85,140,92),(86,137,89),(87,138,90),(88,139,91)], [(1,69,19),(2,20,70),(3,71,17),(4,18,72),(5,49,66),(6,67,50),(7,51,68),(8,65,52),(9,82,106),(10,107,83),(11,84,108),(12,105,81),(13,48,27),(14,28,45),(15,46,25),(16,26,47),(21,112,103),(22,104,109),(23,110,101),(24,102,111),(29,40,93),(30,94,37),(31,38,95),(32,96,39),(33,44,57),(34,58,41),(35,42,59),(36,60,43),(53,117,144),(54,141,118),(55,119,142),(56,143,120),(61,99,115),(62,116,100),(63,97,113),(64,114,98),(73,122,134),(74,135,123),(75,124,136),(76,133,121),(77,128,130),(78,131,125),(79,126,132),(80,129,127),(85,140,92),(86,89,137),(87,138,90),(88,91,139)], [(1,32,23),(2,24,29),(3,30,21),(4,22,31),(5,47,35),(6,36,48),(7,45,33),(8,34,46),(9,97,118),(10,119,98),(11,99,120),(12,117,100),(13,50,43),(14,44,51),(15,52,41),(16,42,49),(17,37,103),(18,104,38),(19,39,101),(20,102,40),(25,65,58),(26,59,66),(27,67,60),(28,57,68),(53,116,81),(54,82,113),(55,114,83),(56,84,115),(61,143,108),(62,105,144),(63,141,106),(64,107,142),(69,96,110),(70,111,93),(71,94,112),(72,109,95),(73,89,78),(74,79,90),(75,91,80),(76,77,92),(85,133,128),(86,125,134),(87,135,126),(88,127,136),(121,130,140),(122,137,131),(123,132,138),(124,139,129)], [(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)], [(1,9,3,11),(2,12,4,10),(5,138,7,140),(6,137,8,139),(13,134,15,136),(14,133,16,135),(17,84,19,82),(18,83,20,81),(21,120,23,118),(22,119,24,117),(25,75,27,73),(26,74,28,76),(29,100,31,98),(30,99,32,97),(33,130,35,132),(34,129,36,131),(37,115,39,113),(38,114,40,116),(41,127,43,125),(42,126,44,128),(45,121,47,123),(46,124,48,122),(49,87,51,85),(50,86,52,88),(53,104,55,102),(54,103,56,101),(57,77,59,79),(58,80,60,78),(61,96,63,94),(62,95,64,93),(65,91,67,89),(66,90,68,92),(69,106,71,108),(70,105,72,107),(109,142,111,144),(110,141,112,143)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | ··· | 4F | 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 | 18 | ··· | 18 | 2 | 2 | 2 | 6 | ··· | 6 | 12 | 12 | 12 | 18 | ··· | 18 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 |
| type | + | + | + | + | + | - | + | + | - | + | - | + | + | - | + |
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | Dic6 | S32 | C32⋊2Q8 | C2×S32 | C32⋊D6 | He3⋊2Q8 | C2×C32⋊D6 |
| kernel | C2×He3⋊2Q8 | He3⋊2Q8 | C2×C32⋊C12 | C2×He3⋊3C4 | C2×C3⋊Dic3 | C2×He3 | C3⋊Dic3 | C62 | C3×C6 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 2 | 4 | 2 | 8 | 1 | 2 | 1 | 2 | 4 | 2 |
Matrix representation of C2×He3⋊2Q8 ►in GL10(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 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 | 12 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 10 | 0 | 3 |
| 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 | 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 | 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 | 8 | 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 | 0 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 |
| 7 | 1 | 7 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 6 | 12 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 2 | 6 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 11 | 12 | 1 | 7 | 0 | 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 | 0 | 0 | 0 | 1 |
| 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 | 12 | 0 | 0 | 0 |
| 0 | 2 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 |
G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,1,10,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1,1,10,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[12,0,12,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,8,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,12,12],[7,12,1,11,0,0,0,0,0,0,1,6,2,12,0,0,0,0,0,0,7,12,6,1,0,0,0,0,0,0,1,6,12,7,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0],[0,2,0,4,0,0,0,0,0,0,2,0,4,0,0,0,0,0,0,0,0,9,0,11,0,0,0,0,0,0,9,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0] >;
C2×He3⋊2Q8 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3\rtimes_2Q_8 % in TeX
G:=Group("C2xHe3:2Q8"); // GroupNames label
G:=SmallGroup(432,316);
// by ID
G=gap.SmallGroup(432,316);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,64,571,4037,537,14118,7069]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,d*b*d^-1=b*c^-1,b*e=e*b,f*b*f^-1=b^-1,c*d=d*c,e*c*e^-1=f*c*f^-1=c^-1,e*d*e^-1=d^-1,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations