direct product, non-abelian, supersoluble, monomial
Aliases: C2×He3⋊4Q8, C62.44D6, He3⋊6(C2×Q8), (C2×He3)⋊4Q8, (C3×C6)⋊4Dic6, (C6×C12).15S3, (C3×C12).54D6, C32⋊5(C2×Dic6), (C2×He3).29C23, (C4×He3).39C22, C6.11(C32⋊4Q8), He3⋊3C4.16C22, (C22×He3).31C22, (C2×C4×He3).9C2, C12.82(C2×C3⋊S3), C6.62(C22×C3⋊S3), (C2×C12).22(C3⋊S3), C3.2(C2×C32⋊4Q8), (C2×He3⋊3C4).9C2, (C3×C6).39(C22×S3), C4.11(C2×He3⋊C2), (C2×C4).4(He3⋊C2), C22.8(C2×He3⋊C2), C2.3(C22×He3⋊C2), (C2×C6).56(C2×C3⋊S3), SmallGroup(432,384)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×He3⋊4Q8
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, ce=ec, cf=fc, de=ed, fdf-1=d-1, fef-1=e-1 >
Subgroups: 625 in 209 conjugacy classes, 67 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, He3, C3×Dic3, C3×C12, C62, C2×Dic6, C6×Q8, C2×He3, C2×He3, C3×Dic6, C6×Dic3, C6×C12, He3⋊3C4, C4×He3, C22×He3, C6×Dic6, He3⋊4Q8, C2×He3⋊3C4, C2×C4×He3, C2×He3⋊4Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C3⋊S3, Dic6, C22×S3, C2×C3⋊S3, C2×Dic6, He3⋊C2, C32⋊4Q8, C22×C3⋊S3, C2×He3⋊C2, C2×C32⋊4Q8, He3⋊4Q8, C22×He3⋊C2, C2×He3⋊4Q8
►On 144 points
Generators in S144
(1 42)(2 43)(3 44)(4 41)(5 31)(6 32)(7 29)(8 30)(9 85)(10 86)(11 87)(12 88)(13 38)(14 39)(15 40)(16 37)(17 47)(18 48)(19 45)(20 46)(21 58)(22 59)(23 60)(24 57)(25 111)(26 112)(27 109)(28 110)(33 71)(34 72)(35 69)(36 70)(49 103)(50 104)(51 101)(52 102)(53 75)(54 76)(55 73)(56 74)(61 132)(62 129)(63 130)(64 131)(65 95)(66 96)(67 93)(68 94)(77 120)(78 117)(79 118)(80 119)(81 124)(82 121)(83 122)(84 123)(89 114)(90 115)(91 116)(92 113)(97 140)(98 137)(99 138)(100 139)(105 134)(106 135)(107 136)(108 133)(125 142)(126 143)(127 144)(128 141)
(5 35 28)(6 36 25)(7 33 26)(8 34 27)(9 141 118)(10 142 119)(11 143 120)(12 144 117)(21 104 95)(22 101 96)(23 102 93)(24 103 94)(29 71 112)(30 72 109)(31 69 110)(32 70 111)(49 68 57)(50 65 58)(51 66 59)(52 67 60)(61 97 108)(62 98 105)(63 99 106)(64 100 107)(77 87 126)(78 88 127)(79 85 128)(80 86 125)(129 137 134)(130 138 135)(131 139 136)(132 140 133)
(1 37 19)(2 38 20)(3 39 17)(4 40 18)(5 35 28)(6 36 25)(7 33 26)(8 34 27)(9 118 141)(10 119 142)(11 120 143)(12 117 144)(13 46 43)(14 47 44)(15 48 41)(16 45 42)(21 95 104)(22 96 101)(23 93 102)(24 94 103)(29 71 112)(30 72 109)(31 69 110)(32 70 111)(49 57 68)(50 58 65)(51 59 66)(52 60 67)(53 114 81)(54 115 82)(55 116 83)(56 113 84)(61 97 108)(62 98 105)(63 99 106)(64 100 107)(73 91 122)(74 92 123)(75 89 124)(76 90 121)(77 126 87)(78 127 88)(79 128 85)(80 125 86)(129 137 134)(130 138 135)(131 139 136)(132 140 133)
(1 96 31)(2 93 32)(3 94 29)(4 95 30)(5 42 66)(6 43 67)(7 44 68)(8 41 65)(9 99 82)(10 100 83)(11 97 84)(12 98 81)(13 52 36)(14 49 33)(15 50 34)(16 51 35)(17 24 112)(18 21 109)(19 22 110)(20 23 111)(25 46 60)(26 47 57)(27 48 58)(28 45 59)(37 101 69)(38 102 70)(39 103 71)(40 104 72)(53 117 105)(54 118 106)(55 119 107)(56 120 108)(61 113 143)(62 114 144)(63 115 141)(64 116 142)(73 80 136)(74 77 133)(75 78 134)(76 79 135)(85 138 121)(86 139 122)(87 140 123)(88 137 124)(89 127 129)(90 128 130)(91 125 131)(92 126 132)
(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 54 3 56)(2 53 4 55)(5 79 7 77)(6 78 8 80)(9 112 11 110)(10 111 12 109)(13 89 15 91)(14 92 16 90)(17 84 19 82)(18 83 20 81)(21 100 23 98)(22 99 24 97)(25 88 27 86)(26 87 28 85)(29 120 31 118)(30 119 32 117)(33 126 35 128)(34 125 36 127)(37 115 39 113)(38 114 40 116)(41 73 43 75)(42 76 44 74)(45 121 47 123)(46 124 48 122)(49 132 51 130)(50 131 52 129)(57 140 59 138)(58 139 60 137)(61 101 63 103)(62 104 64 102)(65 136 67 134)(66 135 68 133)(69 141 71 143)(70 144 72 142)(93 105 95 107)(94 108 96 106)
G:=sub<Sym(144)| (1,42)(2,43)(3,44)(4,41)(5,31)(6,32)(7,29)(8,30)(9,85)(10,86)(11,87)(12,88)(13,38)(14,39)(15,40)(16,37)(17,47)(18,48)(19,45)(20,46)(21,58)(22,59)(23,60)(24,57)(25,111)(26,112)(27,109)(28,110)(33,71)(34,72)(35,69)(36,70)(49,103)(50,104)(51,101)(52,102)(53,75)(54,76)(55,73)(56,74)(61,132)(62,129)(63,130)(64,131)(65,95)(66,96)(67,93)(68,94)(77,120)(78,117)(79,118)(80,119)(81,124)(82,121)(83,122)(84,123)(89,114)(90,115)(91,116)(92,113)(97,140)(98,137)(99,138)(100,139)(105,134)(106,135)(107,136)(108,133)(125,142)(126,143)(127,144)(128,141), (5,35,28)(6,36,25)(7,33,26)(8,34,27)(9,141,118)(10,142,119)(11,143,120)(12,144,117)(21,104,95)(22,101,96)(23,102,93)(24,103,94)(29,71,112)(30,72,109)(31,69,110)(32,70,111)(49,68,57)(50,65,58)(51,66,59)(52,67,60)(61,97,108)(62,98,105)(63,99,106)(64,100,107)(77,87,126)(78,88,127)(79,85,128)(80,86,125)(129,137,134)(130,138,135)(131,139,136)(132,140,133), (1,37,19)(2,38,20)(3,39,17)(4,40,18)(5,35,28)(6,36,25)(7,33,26)(8,34,27)(9,118,141)(10,119,142)(11,120,143)(12,117,144)(13,46,43)(14,47,44)(15,48,41)(16,45,42)(21,95,104)(22,96,101)(23,93,102)(24,94,103)(29,71,112)(30,72,109)(31,69,110)(32,70,111)(49,57,68)(50,58,65)(51,59,66)(52,60,67)(53,114,81)(54,115,82)(55,116,83)(56,113,84)(61,97,108)(62,98,105)(63,99,106)(64,100,107)(73,91,122)(74,92,123)(75,89,124)(76,90,121)(77,126,87)(78,127,88)(79,128,85)(80,125,86)(129,137,134)(130,138,135)(131,139,136)(132,140,133), (1,96,31)(2,93,32)(3,94,29)(4,95,30)(5,42,66)(6,43,67)(7,44,68)(8,41,65)(9,99,82)(10,100,83)(11,97,84)(12,98,81)(13,52,36)(14,49,33)(15,50,34)(16,51,35)(17,24,112)(18,21,109)(19,22,110)(20,23,111)(25,46,60)(26,47,57)(27,48,58)(28,45,59)(37,101,69)(38,102,70)(39,103,71)(40,104,72)(53,117,105)(54,118,106)(55,119,107)(56,120,108)(61,113,143)(62,114,144)(63,115,141)(64,116,142)(73,80,136)(74,77,133)(75,78,134)(76,79,135)(85,138,121)(86,139,122)(87,140,123)(88,137,124)(89,127,129)(90,128,130)(91,125,131)(92,126,132), (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), (1,54,3,56)(2,53,4,55)(5,79,7,77)(6,78,8,80)(9,112,11,110)(10,111,12,109)(13,89,15,91)(14,92,16,90)(17,84,19,82)(18,83,20,81)(21,100,23,98)(22,99,24,97)(25,88,27,86)(26,87,28,85)(29,120,31,118)(30,119,32,117)(33,126,35,128)(34,125,36,127)(37,115,39,113)(38,114,40,116)(41,73,43,75)(42,76,44,74)(45,121,47,123)(46,124,48,122)(49,132,51,130)(50,131,52,129)(57,140,59,138)(58,139,60,137)(61,101,63,103)(62,104,64,102)(65,136,67,134)(66,135,68,133)(69,141,71,143)(70,144,72,142)(93,105,95,107)(94,108,96,106)>;
G:=Group( (1,42)(2,43)(3,44)(4,41)(5,31)(6,32)(7,29)(8,30)(9,85)(10,86)(11,87)(12,88)(13,38)(14,39)(15,40)(16,37)(17,47)(18,48)(19,45)(20,46)(21,58)(22,59)(23,60)(24,57)(25,111)(26,112)(27,109)(28,110)(33,71)(34,72)(35,69)(36,70)(49,103)(50,104)(51,101)(52,102)(53,75)(54,76)(55,73)(56,74)(61,132)(62,129)(63,130)(64,131)(65,95)(66,96)(67,93)(68,94)(77,120)(78,117)(79,118)(80,119)(81,124)(82,121)(83,122)(84,123)(89,114)(90,115)(91,116)(92,113)(97,140)(98,137)(99,138)(100,139)(105,134)(106,135)(107,136)(108,133)(125,142)(126,143)(127,144)(128,141), (5,35,28)(6,36,25)(7,33,26)(8,34,27)(9,141,118)(10,142,119)(11,143,120)(12,144,117)(21,104,95)(22,101,96)(23,102,93)(24,103,94)(29,71,112)(30,72,109)(31,69,110)(32,70,111)(49,68,57)(50,65,58)(51,66,59)(52,67,60)(61,97,108)(62,98,105)(63,99,106)(64,100,107)(77,87,126)(78,88,127)(79,85,128)(80,86,125)(129,137,134)(130,138,135)(131,139,136)(132,140,133), (1,37,19)(2,38,20)(3,39,17)(4,40,18)(5,35,28)(6,36,25)(7,33,26)(8,34,27)(9,118,141)(10,119,142)(11,120,143)(12,117,144)(13,46,43)(14,47,44)(15,48,41)(16,45,42)(21,95,104)(22,96,101)(23,93,102)(24,94,103)(29,71,112)(30,72,109)(31,69,110)(32,70,111)(49,57,68)(50,58,65)(51,59,66)(52,60,67)(53,114,81)(54,115,82)(55,116,83)(56,113,84)(61,97,108)(62,98,105)(63,99,106)(64,100,107)(73,91,122)(74,92,123)(75,89,124)(76,90,121)(77,126,87)(78,127,88)(79,128,85)(80,125,86)(129,137,134)(130,138,135)(131,139,136)(132,140,133), (1,96,31)(2,93,32)(3,94,29)(4,95,30)(5,42,66)(6,43,67)(7,44,68)(8,41,65)(9,99,82)(10,100,83)(11,97,84)(12,98,81)(13,52,36)(14,49,33)(15,50,34)(16,51,35)(17,24,112)(18,21,109)(19,22,110)(20,23,111)(25,46,60)(26,47,57)(27,48,58)(28,45,59)(37,101,69)(38,102,70)(39,103,71)(40,104,72)(53,117,105)(54,118,106)(55,119,107)(56,120,108)(61,113,143)(62,114,144)(63,115,141)(64,116,142)(73,80,136)(74,77,133)(75,78,134)(76,79,135)(85,138,121)(86,139,122)(87,140,123)(88,137,124)(89,127,129)(90,128,130)(91,125,131)(92,126,132), (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), (1,54,3,56)(2,53,4,55)(5,79,7,77)(6,78,8,80)(9,112,11,110)(10,111,12,109)(13,89,15,91)(14,92,16,90)(17,84,19,82)(18,83,20,81)(21,100,23,98)(22,99,24,97)(25,88,27,86)(26,87,28,85)(29,120,31,118)(30,119,32,117)(33,126,35,128)(34,125,36,127)(37,115,39,113)(38,114,40,116)(41,73,43,75)(42,76,44,74)(45,121,47,123)(46,124,48,122)(49,132,51,130)(50,131,52,129)(57,140,59,138)(58,139,60,137)(61,101,63,103)(62,104,64,102)(65,136,67,134)(66,135,68,133)(69,141,71,143)(70,144,72,142)(93,105,95,107)(94,108,96,106) );
G=PermutationGroup([[(1,42),(2,43),(3,44),(4,41),(5,31),(6,32),(7,29),(8,30),(9,85),(10,86),(11,87),(12,88),(13,38),(14,39),(15,40),(16,37),(17,47),(18,48),(19,45),(20,46),(21,58),(22,59),(23,60),(24,57),(25,111),(26,112),(27,109),(28,110),(33,71),(34,72),(35,69),(36,70),(49,103),(50,104),(51,101),(52,102),(53,75),(54,76),(55,73),(56,74),(61,132),(62,129),(63,130),(64,131),(65,95),(66,96),(67,93),(68,94),(77,120),(78,117),(79,118),(80,119),(81,124),(82,121),(83,122),(84,123),(89,114),(90,115),(91,116),(92,113),(97,140),(98,137),(99,138),(100,139),(105,134),(106,135),(107,136),(108,133),(125,142),(126,143),(127,144),(128,141)], [(5,35,28),(6,36,25),(7,33,26),(8,34,27),(9,141,118),(10,142,119),(11,143,120),(12,144,117),(21,104,95),(22,101,96),(23,102,93),(24,103,94),(29,71,112),(30,72,109),(31,69,110),(32,70,111),(49,68,57),(50,65,58),(51,66,59),(52,67,60),(61,97,108),(62,98,105),(63,99,106),(64,100,107),(77,87,126),(78,88,127),(79,85,128),(80,86,125),(129,137,134),(130,138,135),(131,139,136),(132,140,133)], [(1,37,19),(2,38,20),(3,39,17),(4,40,18),(5,35,28),(6,36,25),(7,33,26),(8,34,27),(9,118,141),(10,119,142),(11,120,143),(12,117,144),(13,46,43),(14,47,44),(15,48,41),(16,45,42),(21,95,104),(22,96,101),(23,93,102),(24,94,103),(29,71,112),(30,72,109),(31,69,110),(32,70,111),(49,57,68),(50,58,65),(51,59,66),(52,60,67),(53,114,81),(54,115,82),(55,116,83),(56,113,84),(61,97,108),(62,98,105),(63,99,106),(64,100,107),(73,91,122),(74,92,123),(75,89,124),(76,90,121),(77,126,87),(78,127,88),(79,128,85),(80,125,86),(129,137,134),(130,138,135),(131,139,136),(132,140,133)], [(1,96,31),(2,93,32),(3,94,29),(4,95,30),(5,42,66),(6,43,67),(7,44,68),(8,41,65),(9,99,82),(10,100,83),(11,97,84),(12,98,81),(13,52,36),(14,49,33),(15,50,34),(16,51,35),(17,24,112),(18,21,109),(19,22,110),(20,23,111),(25,46,60),(26,47,57),(27,48,58),(28,45,59),(37,101,69),(38,102,70),(39,103,71),(40,104,72),(53,117,105),(54,118,106),(55,119,107),(56,120,108),(61,113,143),(62,114,144),(63,115,141),(64,116,142),(73,80,136),(74,77,133),(75,78,134),(76,79,135),(85,138,121),(86,139,122),(87,140,123),(88,137,124),(89,127,129),(90,128,130),(91,125,131),(92,126,132)], [(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)], [(1,54,3,56),(2,53,4,55),(5,79,7,77),(6,78,8,80),(9,112,11,110),(10,111,12,109),(13,89,15,91),(14,92,16,90),(17,84,19,82),(18,83,20,81),(21,100,23,98),(22,99,24,97),(25,88,27,86),(26,87,28,85),(29,120,31,118),(30,119,32,117),(33,126,35,128),(34,125,36,127),(37,115,39,113),(38,114,40,116),(41,73,43,75),(42,76,44,74),(45,121,47,123),(46,124,48,122),(49,132,51,130),(50,131,52,129),(57,140,59,138),(58,139,60,137),(61,101,63,103),(62,104,64,102),(65,136,67,134),(66,135,68,133),(69,141,71,143),(70,144,72,142),(93,105,95,107),(94,108,96,106)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6R | 12A | 12B | 12C | 12D | 12E | ··· | 12T | 12U | ··· | 12AB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 18 | 18 | 18 | 18 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 18 | ··· | 18 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 |
| type | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | Dic6 | He3⋊C2 | C2×He3⋊C2 | C2×He3⋊C2 | He3⋊4Q8 |
| kernel | C2×He3⋊4Q8 | He3⋊4Q8 | C2×He3⋊3C4 | C2×C4×He3 | C6×C12 | C2×He3 | C3×C12 | C62 | C3×C6 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 8 | 4 | 16 | 4 | 8 | 4 | 4 |
Matrix representation of C2×He3⋊4Q8 ►in GL7(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 10 | 6 | 0 | 0 | 0 | 0 | 0 |
| 7 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 | 0 |
| 0 | 0 | 7 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 5 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 | 0 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3],[0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[10,7,0,0,0,0,0,6,3,0,0,0,0,0,0,0,10,7,0,0,0,0,0,6,3,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[5,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,9,11,0,0,0,0,0,2,4,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0] >;
C2×He3⋊4Q8 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3\rtimes_4Q_8 % in TeX
G:=Group("C2xHe3:4Q8"); // GroupNames label
G:=SmallGroup(432,384);
// by ID
G=gap.SmallGroup(432,384);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,254,58,1124,4037,537]);
// 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,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f^-1=d^-1,f*e*f^-1=e^-1>;
// generators/relations