direct product, metabelian, nilpotent (class 2), monomial
Aliases: C2×Q8×He3, C12.30C62, C32⋊7(C6×Q8), (C6×C12).13C6, (Q8×C32)⋊9C6, C6.7(Q8×C32), (C2×C6).35C62, C6.23(C2×C62), C62.40(C2×C6), C4.4(C22×He3), C2.3(C23×He3), (C6×Q8).7C32, (C2×He3).41C23, (C4×He3).53C22, C22.6(C22×He3), (C22×He3).40C22, (Q8×C3×C6)⋊2C3, C3.2(Q8×C3×C6), (C3×C6)⋊4(C3×Q8), (C2×C4×He3).15C2, (C2×C4).3(C2×He3), (C2×C12).20(C3×C6), (C3×C12).24(C2×C6), (C3×Q8).22(C3×C6), (C3×C6).33(C22×C6), SmallGroup(432,407)
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, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf-1=de-1, ef=fe >
Subgroups: 361 in 209 conjugacy classes, 133 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, Q8, C32, C12, C12, C2×C6, C2×C6, C2×Q8, C3×C6, C2×C12, C2×C12, C3×Q8, C3×Q8, He3, C3×C12, C62, C6×Q8, C6×Q8, C2×He3, C2×He3, C6×C12, Q8×C32, C4×He3, C22×He3, Q8×C3×C6, C2×C4×He3, Q8×He3, C2×Q8×He3
Quotients: C1, C2, C3, C22, C6, Q8, C23, C32, C2×C6, C2×Q8, C3×C6, C3×Q8, C22×C6, He3, C62, C6×Q8, C2×He3, Q8×C32, C2×C62, C22×He3, Q8×C3×C6, Q8×He3, C23×He3, C2×Q8×He3
►On 144 points
Generators in S144
(1 8)(2 5)(3 6)(4 7)(9 22)(10 23)(11 24)(12 21)(13 26)(14 27)(15 28)(16 25)(17 48)(18 45)(19 46)(20 47)(29 69)(30 70)(31 71)(32 72)(33 96)(34 93)(35 94)(36 95)(37 64)(38 61)(39 62)(40 63)(41 68)(42 65)(43 66)(44 67)(49 112)(50 109)(51 110)(52 111)(53 76)(54 73)(55 74)(56 75)(57 130)(58 131)(59 132)(60 129)(77 100)(78 97)(79 98)(80 99)(81 114)(82 115)(83 116)(84 113)(85 121)(86 122)(87 123)(88 124)(89 101)(90 102)(91 103)(92 104)(105 140)(106 137)(107 138)(108 139)(117 134)(118 135)(119 136)(120 133)(125 142)(126 143)(127 144)(128 141)
(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 81 3 83)(2 84 4 82)(5 113 7 115)(6 116 8 114)(9 99 11 97)(10 98 12 100)(13 104 15 102)(14 103 16 101)(17 124 19 122)(18 123 20 121)(21 77 23 79)(22 80 24 78)(25 89 27 91)(26 92 28 90)(29 143 31 141)(30 142 32 144)(33 53 35 55)(34 56 36 54)(37 136 39 134)(38 135 40 133)(41 140 43 138)(42 139 44 137)(45 87 47 85)(46 86 48 88)(49 58 51 60)(50 57 52 59)(61 118 63 120)(62 117 64 119)(65 108 67 106)(66 107 68 105)(69 126 71 128)(70 125 72 127)(73 93 75 95)(74 96 76 94)(109 130 111 132)(110 129 112 131)
(9 31 61)(10 32 62)(11 29 63)(12 30 64)(21 70 37)(22 71 38)(23 72 39)(24 69 40)(33 49 66)(34 50 67)(35 51 68)(36 52 65)(41 94 110)(42 95 111)(43 96 112)(44 93 109)(53 58 107)(54 59 108)(55 60 105)(56 57 106)(73 132 139)(74 129 140)(75 130 137)(76 131 138)(77 125 136)(78 126 133)(79 127 134)(80 128 135)(97 143 120)(98 144 117)(99 141 118)(100 142 119)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 45 26)(6 46 27)(7 47 28)(8 48 25)(9 31 61)(10 32 62)(11 29 63)(12 30 64)(21 70 37)(22 71 38)(23 72 39)(24 69 40)(33 66 49)(34 67 50)(35 68 51)(36 65 52)(41 110 94)(42 111 95)(43 112 96)(44 109 93)(53 107 58)(54 108 59)(55 105 60)(56 106 57)(73 139 132)(74 140 129)(75 137 130)(76 138 131)(77 125 136)(78 126 133)(79 127 134)(80 128 135)(81 124 101)(82 121 102)(83 122 103)(84 123 104)(85 90 115)(86 91 116)(87 92 113)(88 89 114)(97 143 120)(98 144 117)(99 141 118)(100 142 119)
(1 42 71)(2 43 72)(3 44 69)(4 41 70)(5 66 32)(6 67 29)(7 68 30)(8 65 31)(9 25 36)(10 26 33)(11 27 34)(12 28 35)(13 96 23)(14 93 24)(15 94 21)(16 95 22)(17 111 38)(18 112 39)(19 109 40)(20 110 37)(45 49 62)(46 50 63)(47 51 64)(48 52 61)(53 98 92)(54 99 89)(55 100 90)(56 97 91)(57 120 86)(58 117 87)(59 118 88)(60 119 85)(73 80 101)(74 77 102)(75 78 103)(76 79 104)(81 139 128)(82 140 125)(83 137 126)(84 138 127)(105 142 115)(106 143 116)(107 144 113)(108 141 114)(121 129 136)(122 130 133)(123 131 134)(124 132 135)
G:=sub<Sym(144)| (1,8)(2,5)(3,6)(4,7)(9,22)(10,23)(11,24)(12,21)(13,26)(14,27)(15,28)(16,25)(17,48)(18,45)(19,46)(20,47)(29,69)(30,70)(31,71)(32,72)(33,96)(34,93)(35,94)(36,95)(37,64)(38,61)(39,62)(40,63)(41,68)(42,65)(43,66)(44,67)(49,112)(50,109)(51,110)(52,111)(53,76)(54,73)(55,74)(56,75)(57,130)(58,131)(59,132)(60,129)(77,100)(78,97)(79,98)(80,99)(81,114)(82,115)(83,116)(84,113)(85,121)(86,122)(87,123)(88,124)(89,101)(90,102)(91,103)(92,104)(105,140)(106,137)(107,138)(108,139)(117,134)(118,135)(119,136)(120,133)(125,142)(126,143)(127,144)(128,141), (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), (1,81,3,83)(2,84,4,82)(5,113,7,115)(6,116,8,114)(9,99,11,97)(10,98,12,100)(13,104,15,102)(14,103,16,101)(17,124,19,122)(18,123,20,121)(21,77,23,79)(22,80,24,78)(25,89,27,91)(26,92,28,90)(29,143,31,141)(30,142,32,144)(33,53,35,55)(34,56,36,54)(37,136,39,134)(38,135,40,133)(41,140,43,138)(42,139,44,137)(45,87,47,85)(46,86,48,88)(49,58,51,60)(50,57,52,59)(61,118,63,120)(62,117,64,119)(65,108,67,106)(66,107,68,105)(69,126,71,128)(70,125,72,127)(73,93,75,95)(74,96,76,94)(109,130,111,132)(110,129,112,131), (9,31,61)(10,32,62)(11,29,63)(12,30,64)(21,70,37)(22,71,38)(23,72,39)(24,69,40)(33,49,66)(34,50,67)(35,51,68)(36,52,65)(41,94,110)(42,95,111)(43,96,112)(44,93,109)(53,58,107)(54,59,108)(55,60,105)(56,57,106)(73,132,139)(74,129,140)(75,130,137)(76,131,138)(77,125,136)(78,126,133)(79,127,134)(80,128,135)(97,143,120)(98,144,117)(99,141,118)(100,142,119), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,45,26)(6,46,27)(7,47,28)(8,48,25)(9,31,61)(10,32,62)(11,29,63)(12,30,64)(21,70,37)(22,71,38)(23,72,39)(24,69,40)(33,66,49)(34,67,50)(35,68,51)(36,65,52)(41,110,94)(42,111,95)(43,112,96)(44,109,93)(53,107,58)(54,108,59)(55,105,60)(56,106,57)(73,139,132)(74,140,129)(75,137,130)(76,138,131)(77,125,136)(78,126,133)(79,127,134)(80,128,135)(81,124,101)(82,121,102)(83,122,103)(84,123,104)(85,90,115)(86,91,116)(87,92,113)(88,89,114)(97,143,120)(98,144,117)(99,141,118)(100,142,119), (1,42,71)(2,43,72)(3,44,69)(4,41,70)(5,66,32)(6,67,29)(7,68,30)(8,65,31)(9,25,36)(10,26,33)(11,27,34)(12,28,35)(13,96,23)(14,93,24)(15,94,21)(16,95,22)(17,111,38)(18,112,39)(19,109,40)(20,110,37)(45,49,62)(46,50,63)(47,51,64)(48,52,61)(53,98,92)(54,99,89)(55,100,90)(56,97,91)(57,120,86)(58,117,87)(59,118,88)(60,119,85)(73,80,101)(74,77,102)(75,78,103)(76,79,104)(81,139,128)(82,140,125)(83,137,126)(84,138,127)(105,142,115)(106,143,116)(107,144,113)(108,141,114)(121,129,136)(122,130,133)(123,131,134)(124,132,135)>;
G:=Group( (1,8)(2,5)(3,6)(4,7)(9,22)(10,23)(11,24)(12,21)(13,26)(14,27)(15,28)(16,25)(17,48)(18,45)(19,46)(20,47)(29,69)(30,70)(31,71)(32,72)(33,96)(34,93)(35,94)(36,95)(37,64)(38,61)(39,62)(40,63)(41,68)(42,65)(43,66)(44,67)(49,112)(50,109)(51,110)(52,111)(53,76)(54,73)(55,74)(56,75)(57,130)(58,131)(59,132)(60,129)(77,100)(78,97)(79,98)(80,99)(81,114)(82,115)(83,116)(84,113)(85,121)(86,122)(87,123)(88,124)(89,101)(90,102)(91,103)(92,104)(105,140)(106,137)(107,138)(108,139)(117,134)(118,135)(119,136)(120,133)(125,142)(126,143)(127,144)(128,141), (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), (1,81,3,83)(2,84,4,82)(5,113,7,115)(6,116,8,114)(9,99,11,97)(10,98,12,100)(13,104,15,102)(14,103,16,101)(17,124,19,122)(18,123,20,121)(21,77,23,79)(22,80,24,78)(25,89,27,91)(26,92,28,90)(29,143,31,141)(30,142,32,144)(33,53,35,55)(34,56,36,54)(37,136,39,134)(38,135,40,133)(41,140,43,138)(42,139,44,137)(45,87,47,85)(46,86,48,88)(49,58,51,60)(50,57,52,59)(61,118,63,120)(62,117,64,119)(65,108,67,106)(66,107,68,105)(69,126,71,128)(70,125,72,127)(73,93,75,95)(74,96,76,94)(109,130,111,132)(110,129,112,131), (9,31,61)(10,32,62)(11,29,63)(12,30,64)(21,70,37)(22,71,38)(23,72,39)(24,69,40)(33,49,66)(34,50,67)(35,51,68)(36,52,65)(41,94,110)(42,95,111)(43,96,112)(44,93,109)(53,58,107)(54,59,108)(55,60,105)(56,57,106)(73,132,139)(74,129,140)(75,130,137)(76,131,138)(77,125,136)(78,126,133)(79,127,134)(80,128,135)(97,143,120)(98,144,117)(99,141,118)(100,142,119), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,45,26)(6,46,27)(7,47,28)(8,48,25)(9,31,61)(10,32,62)(11,29,63)(12,30,64)(21,70,37)(22,71,38)(23,72,39)(24,69,40)(33,66,49)(34,67,50)(35,68,51)(36,65,52)(41,110,94)(42,111,95)(43,112,96)(44,109,93)(53,107,58)(54,108,59)(55,105,60)(56,106,57)(73,139,132)(74,140,129)(75,137,130)(76,138,131)(77,125,136)(78,126,133)(79,127,134)(80,128,135)(81,124,101)(82,121,102)(83,122,103)(84,123,104)(85,90,115)(86,91,116)(87,92,113)(88,89,114)(97,143,120)(98,144,117)(99,141,118)(100,142,119), (1,42,71)(2,43,72)(3,44,69)(4,41,70)(5,66,32)(6,67,29)(7,68,30)(8,65,31)(9,25,36)(10,26,33)(11,27,34)(12,28,35)(13,96,23)(14,93,24)(15,94,21)(16,95,22)(17,111,38)(18,112,39)(19,109,40)(20,110,37)(45,49,62)(46,50,63)(47,51,64)(48,52,61)(53,98,92)(54,99,89)(55,100,90)(56,97,91)(57,120,86)(58,117,87)(59,118,88)(60,119,85)(73,80,101)(74,77,102)(75,78,103)(76,79,104)(81,139,128)(82,140,125)(83,137,126)(84,138,127)(105,142,115)(106,143,116)(107,144,113)(108,141,114)(121,129,136)(122,130,133)(123,131,134)(124,132,135) );
G=PermutationGroup([[(1,8),(2,5),(3,6),(4,7),(9,22),(10,23),(11,24),(12,21),(13,26),(14,27),(15,28),(16,25),(17,48),(18,45),(19,46),(20,47),(29,69),(30,70),(31,71),(32,72),(33,96),(34,93),(35,94),(36,95),(37,64),(38,61),(39,62),(40,63),(41,68),(42,65),(43,66),(44,67),(49,112),(50,109),(51,110),(52,111),(53,76),(54,73),(55,74),(56,75),(57,130),(58,131),(59,132),(60,129),(77,100),(78,97),(79,98),(80,99),(81,114),(82,115),(83,116),(84,113),(85,121),(86,122),(87,123),(88,124),(89,101),(90,102),(91,103),(92,104),(105,140),(106,137),(107,138),(108,139),(117,134),(118,135),(119,136),(120,133),(125,142),(126,143),(127,144),(128,141)], [(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)], [(1,81,3,83),(2,84,4,82),(5,113,7,115),(6,116,8,114),(9,99,11,97),(10,98,12,100),(13,104,15,102),(14,103,16,101),(17,124,19,122),(18,123,20,121),(21,77,23,79),(22,80,24,78),(25,89,27,91),(26,92,28,90),(29,143,31,141),(30,142,32,144),(33,53,35,55),(34,56,36,54),(37,136,39,134),(38,135,40,133),(41,140,43,138),(42,139,44,137),(45,87,47,85),(46,86,48,88),(49,58,51,60),(50,57,52,59),(61,118,63,120),(62,117,64,119),(65,108,67,106),(66,107,68,105),(69,126,71,128),(70,125,72,127),(73,93,75,95),(74,96,76,94),(109,130,111,132),(110,129,112,131)], [(9,31,61),(10,32,62),(11,29,63),(12,30,64),(21,70,37),(22,71,38),(23,72,39),(24,69,40),(33,49,66),(34,50,67),(35,51,68),(36,52,65),(41,94,110),(42,95,111),(43,96,112),(44,93,109),(53,58,107),(54,59,108),(55,60,105),(56,57,106),(73,132,139),(74,129,140),(75,130,137),(76,131,138),(77,125,136),(78,126,133),(79,127,134),(80,128,135),(97,143,120),(98,144,117),(99,141,118),(100,142,119)], [(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,45,26),(6,46,27),(7,47,28),(8,48,25),(9,31,61),(10,32,62),(11,29,63),(12,30,64),(21,70,37),(22,71,38),(23,72,39),(24,69,40),(33,66,49),(34,67,50),(35,68,51),(36,65,52),(41,110,94),(42,111,95),(43,112,96),(44,109,93),(53,107,58),(54,108,59),(55,105,60),(56,106,57),(73,139,132),(74,140,129),(75,137,130),(76,138,131),(77,125,136),(78,126,133),(79,127,134),(80,128,135),(81,124,101),(82,121,102),(83,122,103),(84,123,104),(85,90,115),(86,91,116),(87,92,113),(88,89,114),(97,143,120),(98,144,117),(99,141,118),(100,142,119)], [(1,42,71),(2,43,72),(3,44,69),(4,41,70),(5,66,32),(6,67,29),(7,68,30),(8,65,31),(9,25,36),(10,26,33),(11,27,34),(12,28,35),(13,96,23),(14,93,24),(15,94,21),(16,95,22),(17,111,38),(18,112,39),(19,109,40),(20,110,37),(45,49,62),(46,50,63),(47,51,64),(48,52,61),(53,98,92),(54,99,89),(55,100,90),(56,97,91),(57,120,86),(58,117,87),(59,118,88),(60,119,85),(73,80,101),(74,77,102),(75,78,103),(76,79,104),(81,139,128),(82,140,125),(83,137,126),(84,138,127),(105,142,115),(106,143,116),(107,144,113),(108,141,114),(121,129,136),(122,130,133),(123,131,134),(124,132,135)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3J | 4A | ··· | 4F | 6A | ··· | 6F | 6G | ··· | 6AD | 12A | ··· | 12L | 12M | ··· | 12BH |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 6 | ··· | 6 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 6 |
| type | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | Q8 | C3×Q8 | He3 | C2×He3 | C2×He3 | Q8×He3 |
| kernel | C2×Q8×He3 | C2×C4×He3 | Q8×He3 | Q8×C3×C6 | C6×C12 | Q8×C32 | C2×He3 | C3×C6 | C2×Q8 | C2×C4 | Q8 | C2 |
| # reps | 1 | 3 | 4 | 8 | 24 | 32 | 2 | 16 | 2 | 6 | 8 | 4 |
Matrix representation of C2×Q8×He3 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 2 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 |
| 8 | 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 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 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,12,0,0,0,2,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[8,8,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,9],[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],[3,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;
C2×Q8×He3 in GAP, Magma, Sage, TeX
C_2\times Q_8\times {\rm He}_3 % in TeX
G:=Group("C2xQ8xHe3"); // GroupNames label
G:=SmallGroup(432,407);
// by ID
G=gap.SmallGroup(432,407);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,504,1037,512,760]);
// 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,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f^-1=d*e^-1,e*f=f*e>;
// generators/relations