direct product, non-abelian, soluble
Aliases: C2×Q8⋊He3, C62.5A4, (C2×Q8)⋊He3, Q8⋊(C2×He3), C6.24(C6×A4), (Q8×C32)⋊8C6, (C6×SL2(𝔽3))⋊C3, (C3×C6)⋊2SL2(𝔽3), (C6×Q8).5C32, (C3×SL2(𝔽3))⋊2C6, C3.5(C6×SL2(𝔽3)), C6.5(C3×SL2(𝔽3)), C22.2(C32⋊A4), C32⋊3(C2×SL2(𝔽3)), (Q8×C3×C6)⋊1C3, (C3×C6).7(C2×A4), (C2×C6).21(C3×A4), C2.2(C2×C32⋊A4), (C3×Q8).8(C3×C6), SmallGroup(432,336)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8⋊He3
G = < a,b,c,d,e,f | a2=b4=d3=e3=f3=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, fbf-1=c, cd=dc, ce=ec, fcf-1=bc, de=ed, fdf-1=de-1, ef=fe >
Subgroups: 400 in 101 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, Q8, Q8, C32, C32, C12, C2×C6, C2×C6, C2×Q8, C3×C6, C3×C6, C3×C6, SL2(𝔽3), C2×C12, C3×Q8, C3×Q8, He3, C3×C12, C62, C62, C2×SL2(𝔽3), C6×Q8, C6×Q8, C2×He3, C3×SL2(𝔽3), C6×C12, Q8×C32, Q8×C32, C22×He3, C6×SL2(𝔽3), Q8×C3×C6, Q8⋊He3, C2×Q8⋊He3
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, SL2(𝔽3), C2×A4, He3, C3×A4, C2×SL2(𝔽3), C2×He3, C3×SL2(𝔽3), C6×A4, C32⋊A4, C6×SL2(𝔽3), Q8⋊He3, C2×C32⋊A4, C2×Q8⋊He3
►On 144 points
Generators in S144
(1 82)(2 83)(3 84)(4 81)(5 75)(6 76)(7 73)(8 74)(9 79)(10 80)(11 77)(12 78)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(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 12 3 10)(2 11 4 9)(5 144 7 142)(6 143 8 141)(13 17 15 19)(14 20 16 18)(21 27 23 25)(22 26 24 28)(29 36 31 34)(30 35 32 33)(37 44 39 42)(38 43 40 41)(45 49 47 51)(46 52 48 50)(53 59 55 57)(54 58 56 60)(61 67 63 65)(62 66 64 68)(69 76 71 74)(70 75 72 73)(77 81 79 83)(78 84 80 82)(85 89 87 91)(86 92 88 90)(93 99 95 97)(94 98 96 100)(101 108 103 106)(102 107 104 105)(109 116 111 114)(110 115 112 113)(117 121 119 123)(118 124 120 122)(125 131 127 129)(126 130 128 132)(133 139 135 137)(134 138 136 140)
(5 130 138)(6 131 139)(7 132 140)(8 129 137)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 50 41)(34 51 42)(35 52 43)(36 49 44)(53 61 69)(54 62 70)(55 63 71)(56 64 72)(57 65 74)(58 66 75)(59 67 76)(60 68 73)(101 117 109)(102 118 110)(103 119 111)(104 120 112)(105 122 113)(106 123 114)(107 124 115)(108 121 116)(125 133 141)(126 134 142)(127 135 143)(128 136 144)
(1 28 18)(2 25 19)(3 26 20)(4 27 17)(5 138 130)(6 139 131)(7 140 132)(8 137 129)(9 23 15)(10 24 16)(11 21 13)(12 22 14)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 50 41)(34 51 42)(35 52 43)(36 49 44)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(57 74 65)(58 75 66)(59 76 67)(60 73 68)(77 93 85)(78 94 86)(79 95 87)(80 96 88)(81 99 89)(82 100 90)(83 97 91)(84 98 92)(101 117 109)(102 118 110)(103 119 111)(104 120 112)(105 122 113)(106 123 114)(107 124 115)(108 121 116)(125 141 133)(126 142 134)(127 143 135)(128 144 136)
(1 53 32)(2 58 33)(3 55 30)(4 60 35)(5 122 97)(6 117 93)(7 124 99)(8 119 95)(9 57 31)(10 56 34)(11 59 29)(12 54 36)(13 67 37)(14 62 44)(15 65 39)(16 64 42)(17 68 43)(18 61 40)(19 66 41)(20 63 38)(21 76 45)(22 70 49)(23 74 47)(24 72 51)(25 75 50)(26 71 46)(27 73 52)(28 69 48)(77 131 101)(78 126 108)(79 129 103)(80 128 106)(81 132 107)(82 125 104)(83 130 105)(84 127 102)(85 139 109)(86 134 116)(87 137 111)(88 136 114)(89 140 115)(90 133 112)(91 138 113)(92 135 110)(94 142 121)(96 144 123)(98 143 118)(100 141 120)
G:=sub<Sym(144)| (1,82)(2,83)(3,84)(4,81)(5,75)(6,76)(7,73)(8,74)(9,79)(10,80)(11,77)(12,78)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (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), (1,12,3,10)(2,11,4,9)(5,144,7,142)(6,143,8,141)(13,17,15,19)(14,20,16,18)(21,27,23,25)(22,26,24,28)(29,36,31,34)(30,35,32,33)(37,44,39,42)(38,43,40,41)(45,49,47,51)(46,52,48,50)(53,59,55,57)(54,58,56,60)(61,67,63,65)(62,66,64,68)(69,76,71,74)(70,75,72,73)(77,81,79,83)(78,84,80,82)(85,89,87,91)(86,92,88,90)(93,99,95,97)(94,98,96,100)(101,108,103,106)(102,107,104,105)(109,116,111,114)(110,115,112,113)(117,121,119,123)(118,124,120,122)(125,131,127,129)(126,130,128,132)(133,139,135,137)(134,138,136,140), (5,130,138)(6,131,139)(7,132,140)(8,129,137)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,50,41)(34,51,42)(35,52,43)(36,49,44)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(57,65,74)(58,66,75)(59,67,76)(60,68,73)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(105,122,113)(106,123,114)(107,124,115)(108,121,116)(125,133,141)(126,134,142)(127,135,143)(128,136,144), (1,28,18)(2,25,19)(3,26,20)(4,27,17)(5,138,130)(6,139,131)(7,140,132)(8,137,129)(9,23,15)(10,24,16)(11,21,13)(12,22,14)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,50,41)(34,51,42)(35,52,43)(36,49,44)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,74,65)(58,75,66)(59,76,67)(60,73,68)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(81,99,89)(82,100,90)(83,97,91)(84,98,92)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(105,122,113)(106,123,114)(107,124,115)(108,121,116)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (1,53,32)(2,58,33)(3,55,30)(4,60,35)(5,122,97)(6,117,93)(7,124,99)(8,119,95)(9,57,31)(10,56,34)(11,59,29)(12,54,36)(13,67,37)(14,62,44)(15,65,39)(16,64,42)(17,68,43)(18,61,40)(19,66,41)(20,63,38)(21,76,45)(22,70,49)(23,74,47)(24,72,51)(25,75,50)(26,71,46)(27,73,52)(28,69,48)(77,131,101)(78,126,108)(79,129,103)(80,128,106)(81,132,107)(82,125,104)(83,130,105)(84,127,102)(85,139,109)(86,134,116)(87,137,111)(88,136,114)(89,140,115)(90,133,112)(91,138,113)(92,135,110)(94,142,121)(96,144,123)(98,143,118)(100,141,120)>;
G:=Group( (1,82)(2,83)(3,84)(4,81)(5,75)(6,76)(7,73)(8,74)(9,79)(10,80)(11,77)(12,78)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (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), (1,12,3,10)(2,11,4,9)(5,144,7,142)(6,143,8,141)(13,17,15,19)(14,20,16,18)(21,27,23,25)(22,26,24,28)(29,36,31,34)(30,35,32,33)(37,44,39,42)(38,43,40,41)(45,49,47,51)(46,52,48,50)(53,59,55,57)(54,58,56,60)(61,67,63,65)(62,66,64,68)(69,76,71,74)(70,75,72,73)(77,81,79,83)(78,84,80,82)(85,89,87,91)(86,92,88,90)(93,99,95,97)(94,98,96,100)(101,108,103,106)(102,107,104,105)(109,116,111,114)(110,115,112,113)(117,121,119,123)(118,124,120,122)(125,131,127,129)(126,130,128,132)(133,139,135,137)(134,138,136,140), (5,130,138)(6,131,139)(7,132,140)(8,129,137)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,50,41)(34,51,42)(35,52,43)(36,49,44)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(57,65,74)(58,66,75)(59,67,76)(60,68,73)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(105,122,113)(106,123,114)(107,124,115)(108,121,116)(125,133,141)(126,134,142)(127,135,143)(128,136,144), (1,28,18)(2,25,19)(3,26,20)(4,27,17)(5,138,130)(6,139,131)(7,140,132)(8,137,129)(9,23,15)(10,24,16)(11,21,13)(12,22,14)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,50,41)(34,51,42)(35,52,43)(36,49,44)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,74,65)(58,75,66)(59,76,67)(60,73,68)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(81,99,89)(82,100,90)(83,97,91)(84,98,92)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(105,122,113)(106,123,114)(107,124,115)(108,121,116)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (1,53,32)(2,58,33)(3,55,30)(4,60,35)(5,122,97)(6,117,93)(7,124,99)(8,119,95)(9,57,31)(10,56,34)(11,59,29)(12,54,36)(13,67,37)(14,62,44)(15,65,39)(16,64,42)(17,68,43)(18,61,40)(19,66,41)(20,63,38)(21,76,45)(22,70,49)(23,74,47)(24,72,51)(25,75,50)(26,71,46)(27,73,52)(28,69,48)(77,131,101)(78,126,108)(79,129,103)(80,128,106)(81,132,107)(82,125,104)(83,130,105)(84,127,102)(85,139,109)(86,134,116)(87,137,111)(88,136,114)(89,140,115)(90,133,112)(91,138,113)(92,135,110)(94,142,121)(96,144,123)(98,143,118)(100,141,120) );
G=PermutationGroup([[(1,82),(2,83),(3,84),(4,81),(5,75),(6,76),(7,73),(8,74),(9,79),(10,80),(11,77),(12,78),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(49,121),(50,122),(51,123),(52,124),(53,125),(54,126),(55,127),(56,128),(57,129),(58,130),(59,131),(60,132),(61,133),(62,134),(63,135),(64,136),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144)], [(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)], [(1,12,3,10),(2,11,4,9),(5,144,7,142),(6,143,8,141),(13,17,15,19),(14,20,16,18),(21,27,23,25),(22,26,24,28),(29,36,31,34),(30,35,32,33),(37,44,39,42),(38,43,40,41),(45,49,47,51),(46,52,48,50),(53,59,55,57),(54,58,56,60),(61,67,63,65),(62,66,64,68),(69,76,71,74),(70,75,72,73),(77,81,79,83),(78,84,80,82),(85,89,87,91),(86,92,88,90),(93,99,95,97),(94,98,96,100),(101,108,103,106),(102,107,104,105),(109,116,111,114),(110,115,112,113),(117,121,119,123),(118,124,120,122),(125,131,127,129),(126,130,128,132),(133,139,135,137),(134,138,136,140)], [(5,130,138),(6,131,139),(7,132,140),(8,129,137),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(33,50,41),(34,51,42),(35,52,43),(36,49,44),(53,61,69),(54,62,70),(55,63,71),(56,64,72),(57,65,74),(58,66,75),(59,67,76),(60,68,73),(101,117,109),(102,118,110),(103,119,111),(104,120,112),(105,122,113),(106,123,114),(107,124,115),(108,121,116),(125,133,141),(126,134,142),(127,135,143),(128,136,144)], [(1,28,18),(2,25,19),(3,26,20),(4,27,17),(5,138,130),(6,139,131),(7,140,132),(8,137,129),(9,23,15),(10,24,16),(11,21,13),(12,22,14),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(33,50,41),(34,51,42),(35,52,43),(36,49,44),(53,69,61),(54,70,62),(55,71,63),(56,72,64),(57,74,65),(58,75,66),(59,76,67),(60,73,68),(77,93,85),(78,94,86),(79,95,87),(80,96,88),(81,99,89),(82,100,90),(83,97,91),(84,98,92),(101,117,109),(102,118,110),(103,119,111),(104,120,112),(105,122,113),(106,123,114),(107,124,115),(108,121,116),(125,141,133),(126,142,134),(127,143,135),(128,144,136)], [(1,53,32),(2,58,33),(3,55,30),(4,60,35),(5,122,97),(6,117,93),(7,124,99),(8,119,95),(9,57,31),(10,56,34),(11,59,29),(12,54,36),(13,67,37),(14,62,44),(15,65,39),(16,64,42),(17,68,43),(18,61,40),(19,66,41),(20,63,38),(21,76,45),(22,70,49),(23,74,47),(24,72,51),(25,75,50),(26,71,46),(27,73,52),(28,69,48),(77,131,101),(78,126,108),(79,129,103),(80,128,106),(81,132,107),(82,125,104),(83,130,105),(84,127,102),(85,139,109),(86,134,116),(87,137,111),(88,136,114),(89,140,115),(90,133,112),(91,138,113),(92,135,110),(94,142,121),(96,144,123),(98,143,118),(100,141,120)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6L | 6M | ··· | 6AD | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 12 | ··· | 12 | 6 | 6 | 1 | ··· | 1 | 3 | ··· | 3 | 12 | ··· | 12 | 6 | ··· | 6 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 |
| type | + | + | - | + | + | |||||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | SL2(𝔽3) | SL2(𝔽3) | C3×SL2(𝔽3) | A4 | C2×A4 | He3 | C3×A4 | C2×He3 | C6×A4 | C32⋊A4 | C2×C32⋊A4 | Q8⋊He3 |
| kernel | C2×Q8⋊He3 | Q8⋊He3 | C6×SL2(𝔽3) | Q8×C3×C6 | C3×SL2(𝔽3) | Q8×C32 | C3×C6 | C3×C6 | C6 | C62 | C3×C6 | C2×Q8 | C2×C6 | Q8 | C6 | C22 | C2 | C2 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 2 | 4 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 4 |
Matrix representation of C2×Q8⋊He3 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 3 | 4 | 0 | 0 | 0 |
| 4 | 10 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 1 | 10 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[3,4,0,0,0,4,10,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[0,12,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[3,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,10,9,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;
C2×Q8⋊He3 in GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes {\rm He}_3 % in TeX
G:=Group("C2xQ8:He3"); // GroupNames label
G:=SmallGroup(432,336);
// by ID
G=gap.SmallGroup(432,336);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,261,1901,172,3414,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^4=d^3=e^3=f^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,f*b*f^-1=c,c*d=d*c,c*e=e*c,f*c*f^-1=b*c,d*e=e*d,f*d*f^-1=d*e^-1,e*f=f*e>;
// generators/relations