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## G = C2×Q8×He3order 432 = 24·33

### Direct product of C2, Q8 and He3

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×Q8×He3
 Chief series C1 — C3 — C6 — C3×C6 — C2×He3 — C4×He3 — Q8×He3 — C2×Q8×He3
 Lower central C1 — C6 — C2×Q8×He3
 Upper central C1 — C2×C6 — C2×Q8×He3

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

Smallest permutation representation of 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

׿
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𝔽