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## G = He3⋊3C16order 432 = 24·33

### 1st semidirect product of He3 and C16 acting via C16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — He3⋊3C16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C8×He3 — He3⋊3C16
 Lower central C32 — He3⋊3C16
 Upper central C1 — C8

Generators and relations for He33C16
G = < a,b,c,d | a3=b3=c3=d16=1, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, dbd-1=b-1, cd=dc >

Smallest permutation representation of He33C16
On 144 points
Generators in S144
(1 51 39)(2 40 52)(3 53 41)(4 42 54)(5 55 43)(6 44 56)(7 57 45)(8 46 58)(9 59 47)(10 48 60)(11 61 33)(12 34 62)(13 63 35)(14 36 64)(15 49 37)(16 38 50)(17 116 77)(18 78 117)(19 118 79)(20 80 119)(21 120 65)(22 66 121)(23 122 67)(24 68 123)(25 124 69)(26 70 125)(27 126 71)(28 72 127)(29 128 73)(30 74 113)(31 114 75)(32 76 115)(81 106 135)(82 136 107)(83 108 137)(84 138 109)(85 110 139)(86 140 111)(87 112 141)(88 142 97)(89 98 143)(90 144 99)(91 100 129)(92 130 101)(93 102 131)(94 132 103)(95 104 133)(96 134 105)
(1 97 76)(2 77 98)(3 99 78)(4 79 100)(5 101 80)(6 65 102)(7 103 66)(8 67 104)(9 105 68)(10 69 106)(11 107 70)(12 71 108)(13 109 72)(14 73 110)(15 111 74)(16 75 112)(17 143 40)(18 41 144)(19 129 42)(20 43 130)(21 131 44)(22 45 132)(23 133 46)(24 47 134)(25 135 48)(26 33 136)(27 137 34)(28 35 138)(29 139 36)(30 37 140)(31 141 38)(32 39 142)(49 86 113)(50 114 87)(51 88 115)(52 116 89)(53 90 117)(54 118 91)(55 92 119)(56 120 93)(57 94 121)(58 122 95)(59 96 123)(60 124 81)(61 82 125)(62 126 83)(63 84 127)(64 128 85)
(17 143 40)(18 144 41)(19 129 42)(20 130 43)(21 131 44)(22 132 45)(23 133 46)(24 134 47)(25 135 48)(26 136 33)(27 137 34)(28 138 35)(29 139 36)(30 140 37)(31 141 38)(32 142 39)(49 86 113)(50 87 114)(51 88 115)(52 89 116)(53 90 117)(54 91 118)(55 92 119)(56 93 120)(57 94 121)(58 95 122)(59 96 123)(60 81 124)(61 82 125)(62 83 126)(63 84 127)(64 85 128)
(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)

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

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

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

80 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 4A 4B 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G ··· 12L 16A ··· 16H 24A 24B 24C 24D 24E ··· 24L 24M ··· 24X 48A ··· 48P order 1 2 3 3 3 3 3 3 4 4 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 ··· 12 16 ··· 16 24 24 24 24 24 ··· 24 24 ··· 24 48 ··· 48 size 1 1 2 3 3 6 6 6 1 1 2 3 3 6 6 6 1 1 1 1 2 2 3 3 3 3 6 ··· 6 9 ··· 9 2 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 6 6 6 6 type + + + - + - image C1 C2 C3 C4 C6 C8 C12 C16 C24 C48 S3 Dic3 C3×S3 C3⋊C8 C3×Dic3 C3⋊C16 C3×C3⋊C8 C3×C3⋊C16 C32⋊C6 C32⋊C12 He3⋊3C8 He3⋊3C16 kernel He3⋊3C16 C8×He3 C24.S3 C4×He3 C3×C24 C2×He3 C3×C12 He3 C3×C6 C32 C3×C24 C3×C12 C24 C3×C6 C12 C32 C6 C3 C8 C4 C2 C1 # reps 1 1 2 2 2 4 4 8 8 16 1 1 2 2 2 4 4 8 1 1 2 4

Matrix representation of He33C16 in GL6(𝔽97)

 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0
,
 96 1 0 0 0 0 96 0 0 0 0 0 0 0 96 1 0 0 0 0 96 0 0 0 0 0 0 0 96 1 0 0 0 0 96 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 96 0 0 0 0 1 96 0 0 0 0 0 0 96 1 0 0 0 0 96 0
,
 89 0 0 0 0 0 89 8 0 0 0 0 0 0 0 0 89 0 0 0 0 0 89 8 0 0 89 0 0 0 0 0 89 8 0 0

G:=sub<GL(6,GF(97))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[96,96,0,0,0,0,1,0,0,0,0,0,0,0,96,96,0,0,0,0,1,0,0,0,0,0,0,0,96,96,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96,0,0,0,0,0,0,96,96,0,0,0,0,1,0],[89,89,0,0,0,0,0,8,0,0,0,0,0,0,0,0,89,89,0,0,0,0,0,8,0,0,89,89,0,0,0,0,0,8,0,0] >;

He33C16 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_3C_{16}
% in TeX

G:=Group("He3:3C16");
// GroupNames label

G:=SmallGroup(432,30);
// by ID

G=gap.SmallGroup(432,30);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,-3,-3,42,58,80,4037,4044,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^16=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,c*d=d*c>;
// generators/relations

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