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## G = He3⋊C42order 432 = 24·33

### The semidirect product of He3 and C42 acting via C42/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊C42
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C22×He3 — C2×C32⋊C12 — He3⋊C42
 Lower central He3 — He3⋊C42
 Upper central C1 — C22

Generators and relations for He3⋊C42
G = < a,b,c,d,e | a3=b3=c3=d4=e4=1, ab=ba, cac-1=ab-1, dad-1=a-1, ae=ea, bc=cb, dbd-1=ebe-1=b-1, cd=dc, ece-1=c-1, de=ed >

Subgroups: 495 in 121 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C32, C32, Dic3, C12, C2×C6, C2×C6, C42, C3×C6, C3×C6, C2×Dic3, C2×C12, He3, C3×Dic3, C3⋊Dic3, C62, C62, C4×Dic3, C2×He3, C2×He3, C6×Dic3, C2×C3⋊Dic3, C32⋊C12, He33C4, C22×He3, Dic32, C2×C32⋊C12, C2×He33C4, He3⋊C42
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, C4×S3, C2×Dic3, S32, C4×Dic3, S3×Dic3, C6.D6, C32⋊D6, Dic32, C6.S32, He3⋊(C2×C4), He3⋊C42

Smallest permutation representation of He3⋊C42
On 144 points
Generators in S144
(1 51 117)(2 118 52)(3 49 119)(4 120 50)(5 33 64)(6 61 34)(7 35 62)(8 63 36)(9 69 40)(10 37 70)(11 71 38)(12 39 72)(13 42 20)(14 17 43)(15 44 18)(16 19 41)(21 55 129)(22 130 56)(23 53 131)(24 132 54)(25 68 133)(26 134 65)(27 66 135)(28 136 67)(29 60 123)(30 124 57)(31 58 121)(32 122 59)(45 74 103)(46 104 75)(47 76 101)(48 102 73)(77 113 86)(78 87 114)(79 115 88)(80 85 116)(81 141 110)(82 111 142)(83 143 112)(84 109 144)(89 94 126)(90 127 95)(91 96 128)(92 125 93)(97 106 140)(98 137 107)(99 108 138)(100 139 105)
(1 114 15)(2 16 115)(3 116 13)(4 14 113)(5 136 139)(6 140 133)(7 134 137)(8 138 135)(9 141 47)(10 48 142)(11 143 45)(12 46 144)(17 86 120)(18 117 87)(19 88 118)(20 119 85)(21 125 124)(22 121 126)(23 127 122)(24 123 128)(25 61 97)(26 98 62)(27 63 99)(28 100 64)(29 91 132)(30 129 92)(31 89 130)(32 131 90)(33 67 105)(34 106 68)(35 65 107)(36 108 66)(37 102 82)(38 83 103)(39 104 84)(40 81 101)(41 79 52)(42 49 80)(43 77 50)(44 51 78)(53 95 59)(54 60 96)(55 93 57)(56 58 94)(69 110 76)(70 73 111)(71 112 74)(72 75 109)
(5 136 139)(6 133 140)(7 134 137)(8 135 138)(9 47 141)(10 48 142)(11 45 143)(12 46 144)(17 86 120)(18 87 117)(19 88 118)(20 85 119)(29 91 132)(30 92 129)(31 89 130)(32 90 131)(33 105 67)(34 106 68)(35 107 65)(36 108 66)(37 82 102)(38 83 103)(39 84 104)(40 81 101)(41 52 79)(42 49 80)(43 50 77)(44 51 78)(53 95 59)(54 96 60)(55 93 57)(56 94 58)
(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 76 126 26)(2 73 127 27)(3 74 128 28)(4 75 125 25)(5 80 83 132)(6 77 84 129)(7 78 81 130)(8 79 82 131)(9 58 107 18)(10 59 108 19)(11 60 105 20)(12 57 106 17)(13 71 123 100)(14 72 124 97)(15 69 121 98)(16 70 122 99)(21 61 113 109)(22 62 114 110)(23 63 115 111)(24 64 116 112)(29 139 42 38)(30 140 43 39)(31 137 44 40)(32 138 41 37)(33 85 143 54)(34 86 144 55)(35 87 141 56)(36 88 142 53)(45 96 67 119)(46 93 68 120)(47 94 65 117)(48 95 66 118)(49 103 91 136)(50 104 92 133)(51 101 89 134)(52 102 90 135)

G:=sub<Sym(144)| (1,51,117)(2,118,52)(3,49,119)(4,120,50)(5,33,64)(6,61,34)(7,35,62)(8,63,36)(9,69,40)(10,37,70)(11,71,38)(12,39,72)(13,42,20)(14,17,43)(15,44,18)(16,19,41)(21,55,129)(22,130,56)(23,53,131)(24,132,54)(25,68,133)(26,134,65)(27,66,135)(28,136,67)(29,60,123)(30,124,57)(31,58,121)(32,122,59)(45,74,103)(46,104,75)(47,76,101)(48,102,73)(77,113,86)(78,87,114)(79,115,88)(80,85,116)(81,141,110)(82,111,142)(83,143,112)(84,109,144)(89,94,126)(90,127,95)(91,96,128)(92,125,93)(97,106,140)(98,137,107)(99,108,138)(100,139,105), (1,114,15)(2,16,115)(3,116,13)(4,14,113)(5,136,139)(6,140,133)(7,134,137)(8,138,135)(9,141,47)(10,48,142)(11,143,45)(12,46,144)(17,86,120)(18,117,87)(19,88,118)(20,119,85)(21,125,124)(22,121,126)(23,127,122)(24,123,128)(25,61,97)(26,98,62)(27,63,99)(28,100,64)(29,91,132)(30,129,92)(31,89,130)(32,131,90)(33,67,105)(34,106,68)(35,65,107)(36,108,66)(37,102,82)(38,83,103)(39,104,84)(40,81,101)(41,79,52)(42,49,80)(43,77,50)(44,51,78)(53,95,59)(54,60,96)(55,93,57)(56,58,94)(69,110,76)(70,73,111)(71,112,74)(72,75,109), (5,136,139)(6,133,140)(7,134,137)(8,135,138)(9,47,141)(10,48,142)(11,45,143)(12,46,144)(17,86,120)(18,87,117)(19,88,118)(20,85,119)(29,91,132)(30,92,129)(31,89,130)(32,90,131)(33,105,67)(34,106,68)(35,107,65)(36,108,66)(37,82,102)(38,83,103)(39,84,104)(40,81,101)(41,52,79)(42,49,80)(43,50,77)(44,51,78)(53,95,59)(54,96,60)(55,93,57)(56,94,58), (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), (1,76,126,26)(2,73,127,27)(3,74,128,28)(4,75,125,25)(5,80,83,132)(6,77,84,129)(7,78,81,130)(8,79,82,131)(9,58,107,18)(10,59,108,19)(11,60,105,20)(12,57,106,17)(13,71,123,100)(14,72,124,97)(15,69,121,98)(16,70,122,99)(21,61,113,109)(22,62,114,110)(23,63,115,111)(24,64,116,112)(29,139,42,38)(30,140,43,39)(31,137,44,40)(32,138,41,37)(33,85,143,54)(34,86,144,55)(35,87,141,56)(36,88,142,53)(45,96,67,119)(46,93,68,120)(47,94,65,117)(48,95,66,118)(49,103,91,136)(50,104,92,133)(51,101,89,134)(52,102,90,135)>;

G:=Group( (1,51,117)(2,118,52)(3,49,119)(4,120,50)(5,33,64)(6,61,34)(7,35,62)(8,63,36)(9,69,40)(10,37,70)(11,71,38)(12,39,72)(13,42,20)(14,17,43)(15,44,18)(16,19,41)(21,55,129)(22,130,56)(23,53,131)(24,132,54)(25,68,133)(26,134,65)(27,66,135)(28,136,67)(29,60,123)(30,124,57)(31,58,121)(32,122,59)(45,74,103)(46,104,75)(47,76,101)(48,102,73)(77,113,86)(78,87,114)(79,115,88)(80,85,116)(81,141,110)(82,111,142)(83,143,112)(84,109,144)(89,94,126)(90,127,95)(91,96,128)(92,125,93)(97,106,140)(98,137,107)(99,108,138)(100,139,105), (1,114,15)(2,16,115)(3,116,13)(4,14,113)(5,136,139)(6,140,133)(7,134,137)(8,138,135)(9,141,47)(10,48,142)(11,143,45)(12,46,144)(17,86,120)(18,117,87)(19,88,118)(20,119,85)(21,125,124)(22,121,126)(23,127,122)(24,123,128)(25,61,97)(26,98,62)(27,63,99)(28,100,64)(29,91,132)(30,129,92)(31,89,130)(32,131,90)(33,67,105)(34,106,68)(35,65,107)(36,108,66)(37,102,82)(38,83,103)(39,104,84)(40,81,101)(41,79,52)(42,49,80)(43,77,50)(44,51,78)(53,95,59)(54,60,96)(55,93,57)(56,58,94)(69,110,76)(70,73,111)(71,112,74)(72,75,109), (5,136,139)(6,133,140)(7,134,137)(8,135,138)(9,47,141)(10,48,142)(11,45,143)(12,46,144)(17,86,120)(18,87,117)(19,88,118)(20,85,119)(29,91,132)(30,92,129)(31,89,130)(32,90,131)(33,105,67)(34,106,68)(35,107,65)(36,108,66)(37,82,102)(38,83,103)(39,84,104)(40,81,101)(41,52,79)(42,49,80)(43,50,77)(44,51,78)(53,95,59)(54,96,60)(55,93,57)(56,94,58), (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), (1,76,126,26)(2,73,127,27)(3,74,128,28)(4,75,125,25)(5,80,83,132)(6,77,84,129)(7,78,81,130)(8,79,82,131)(9,58,107,18)(10,59,108,19)(11,60,105,20)(12,57,106,17)(13,71,123,100)(14,72,124,97)(15,69,121,98)(16,70,122,99)(21,61,113,109)(22,62,114,110)(23,63,115,111)(24,64,116,112)(29,139,42,38)(30,140,43,39)(31,137,44,40)(32,138,41,37)(33,85,143,54)(34,86,144,55)(35,87,141,56)(36,88,142,53)(45,96,67,119)(46,93,68,120)(47,94,65,117)(48,95,66,118)(49,103,91,136)(50,104,92,133)(51,101,89,134)(52,102,90,135) );

G=PermutationGroup([[(1,51,117),(2,118,52),(3,49,119),(4,120,50),(5,33,64),(6,61,34),(7,35,62),(8,63,36),(9,69,40),(10,37,70),(11,71,38),(12,39,72),(13,42,20),(14,17,43),(15,44,18),(16,19,41),(21,55,129),(22,130,56),(23,53,131),(24,132,54),(25,68,133),(26,134,65),(27,66,135),(28,136,67),(29,60,123),(30,124,57),(31,58,121),(32,122,59),(45,74,103),(46,104,75),(47,76,101),(48,102,73),(77,113,86),(78,87,114),(79,115,88),(80,85,116),(81,141,110),(82,111,142),(83,143,112),(84,109,144),(89,94,126),(90,127,95),(91,96,128),(92,125,93),(97,106,140),(98,137,107),(99,108,138),(100,139,105)], [(1,114,15),(2,16,115),(3,116,13),(4,14,113),(5,136,139),(6,140,133),(7,134,137),(8,138,135),(9,141,47),(10,48,142),(11,143,45),(12,46,144),(17,86,120),(18,117,87),(19,88,118),(20,119,85),(21,125,124),(22,121,126),(23,127,122),(24,123,128),(25,61,97),(26,98,62),(27,63,99),(28,100,64),(29,91,132),(30,129,92),(31,89,130),(32,131,90),(33,67,105),(34,106,68),(35,65,107),(36,108,66),(37,102,82),(38,83,103),(39,104,84),(40,81,101),(41,79,52),(42,49,80),(43,77,50),(44,51,78),(53,95,59),(54,60,96),(55,93,57),(56,58,94),(69,110,76),(70,73,111),(71,112,74),(72,75,109)], [(5,136,139),(6,133,140),(7,134,137),(8,135,138),(9,47,141),(10,48,142),(11,45,143),(12,46,144),(17,86,120),(18,87,117),(19,88,118),(20,85,119),(29,91,132),(30,92,129),(31,89,130),(32,90,131),(33,105,67),(34,106,68),(35,107,65),(36,108,66),(37,82,102),(38,83,103),(39,84,104),(40,81,101),(41,52,79),(42,49,80),(43,50,77),(44,51,78),(53,95,59),(54,96,60),(55,93,57),(56,94,58)], [(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)], [(1,76,126,26),(2,73,127,27),(3,74,128,28),(4,75,125,25),(5,80,83,132),(6,77,84,129),(7,78,81,130),(8,79,82,131),(9,58,107,18),(10,59,108,19),(11,60,105,20),(12,57,106,17),(13,71,123,100),(14,72,124,97),(15,69,121,98),(16,70,122,99),(21,61,113,109),(22,62,114,110),(23,63,115,111),(24,64,116,112),(29,139,42,38),(30,140,43,39),(31,137,44,40),(32,138,41,37),(33,85,143,54),(34,86,144,55),(35,87,141,56),(36,88,142,53),(45,96,67,119),(46,93,68,120),(47,94,65,117),(48,95,66,118),(49,103,91,136),(50,104,92,133),(51,101,89,134),(52,102,90,135)]])

44 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A ··· 4L 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 9 ··· 9 2 2 2 6 ··· 6 12 12 12 18 ··· 18

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 6 6 6 type + + + + - + + - + + - image C1 C2 C2 C4 C4 S3 Dic3 D6 C4×S3 S32 S3×Dic3 C6.D6 C32⋊D6 C6.S32 He3⋊(C2×C4) kernel He3⋊C42 C2×C32⋊C12 C2×He3⋊3C4 C32⋊C12 He3⋊3C4 C2×C3⋊Dic3 C3⋊Dic3 C62 C3×C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 8 4 2 4 2 8 1 2 1 2 4 2

Matrix representation of He3⋊C42 in GL10(𝔽13)

 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 2 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 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 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 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 1 0 0 3 0 0 0 0 0 0 1 2 0 0 3 0 0 0 0 0 1 0 2 0 0 3
,
 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 3 0 0 0 0 0 0 0 0 4 0 9 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 11 0 0 9 0 0 0 0 0 1 0 2 0 0 3
,
 5 0 8 0 0 0 0 0 0 0 0 5 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 9 0 0 11 0 0 0 0 0 0 0 0 9 1 0 12 0 0 0 0 0 9 0 1 12 0 0 0 0 0 2 0 0 4 0 0 0 0 0 0 2 0 4 0 0 4 0 0 0 0 2 4 0 0 4 0
,
 1 0 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 4 0 0 2 0 0 0 0 0 0 0 4 0 12 1 0 0 0 0 0 0 0 4 12 0 1 0 0 0 0 11 0 0 9 0 0 0 0 0 0 11 9 0 0 9 0 0 0 0 0 11 0 9 0 0 9

G:=sub<GL(10,GF(13))| [12,0,12,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,9,0,0,1,1,1,0,0,0,0,0,9,0,0,2,0,0,0,0,0,0,0,9,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,1,4,0,12,1,0,0,0,0,0,3,0,0,11,0,0,0,0,0,0,0,9,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,3],[5,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,8,0,8,0,0,0,0,0,0,0,0,8,0,8,0,0,0,0,0,0,0,0,0,0,9,0,0,2,2,2,0,0,0,0,0,0,9,0,0,4,0,0,0,0,0,9,0,0,4,0,0,0,0,0,11,1,1,4,0,0,0,0,0,0,0,0,12,0,0,4,0,0,0,0,0,12,0,0,4,0],[1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,4,0,0,11,11,11,0,0,0,0,0,4,0,0,9,0,0,0,0,0,0,0,4,0,0,9,0,0,0,0,2,12,12,9,0,0,0,0,0,0,0,1,0,0,9,0,0,0,0,0,0,0,1,0,0,9] >;

He3⋊C42 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes C_4^2
% in TeX

G:=Group("He3:C4^2");
// GroupNames label

G:=SmallGroup(432,94);
// by ID

G=gap.SmallGroup(432,94);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,36,571,4037,537,14118,7069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^4=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e^-1=b^-1,c*d=d*c,e*c*e^-1=c^-1,d*e=e*d>;
// generators/relations

׿
×
𝔽