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## G = He3⋊4Q16order 432 = 24·33

### 1st semidirect product of He3 and Q16 acting via Q16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — He3⋊4Q16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×He3 — He3⋊3Q8 — He3⋊4Q16
 Lower central C32 — C3×C6 — C3×C12 — He3⋊4Q16
 Upper central C1 — C2 — C4 — C8

Generators and relations for He34Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1, bc=cb, bd=db, ebe-1=b-1, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 321 in 71 conjugacy classes, 26 normal (22 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C32, C32, Dic3, C12, C12, Q16, C3×C6, C3×C6, C24, C24, Dic6, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, Dic12, C3×Q16, C2×He3, C3×C24, C3×C24, C3×Dic6, C324Q8, C32⋊C12, C4×He3, C3×Dic12, C325Q16, C8×He3, He33Q8, He34Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, C3×S3, D12, C3×D4, S3×C6, Dic12, C3×Q16, C32⋊C6, C3×D12, C2×C32⋊C6, C3×Dic12, He34D4, He34Q16

Smallest permutation representation of He34Q16
On 144 points
Generators in S144
(9 131 122)(10 132 123)(11 133 124)(12 134 125)(13 135 126)(14 136 127)(15 129 128)(16 130 121)(25 51 78)(26 52 79)(27 53 80)(28 54 73)(29 55 74)(30 56 75)(31 49 76)(32 50 77)(33 46 105)(34 47 106)(35 48 107)(36 41 108)(37 42 109)(38 43 110)(39 44 111)(40 45 112)(57 89 116)(58 90 117)(59 91 118)(60 92 119)(61 93 120)(62 94 113)(63 95 114)(64 96 115)
(1 67 17)(2 68 18)(3 69 19)(4 70 20)(5 71 21)(6 72 22)(7 65 23)(8 66 24)(9 131 122)(10 132 123)(11 133 124)(12 134 125)(13 135 126)(14 136 127)(15 129 128)(16 130 121)(25 51 78)(26 52 79)(27 53 80)(28 54 73)(29 55 74)(30 56 75)(31 49 76)(32 50 77)(33 105 46)(34 106 47)(35 107 48)(36 108 41)(37 109 42)(38 110 43)(39 111 44)(40 112 45)(57 116 89)(58 117 90)(59 118 91)(60 119 92)(61 120 93)(62 113 94)(63 114 95)(64 115 96)(81 142 98)(82 143 99)(83 144 100)(84 137 101)(85 138 102)(86 139 103)(87 140 104)(88 141 97)
(1 117 32)(2 118 25)(3 119 26)(4 120 27)(5 113 28)(6 114 29)(7 115 30)(8 116 31)(9 86 105)(10 87 106)(11 88 107)(12 81 108)(13 82 109)(14 83 110)(15 84 111)(16 85 112)(17 58 77)(18 59 78)(19 60 79)(20 61 80)(21 62 73)(22 63 74)(23 64 75)(24 57 76)(33 122 103)(34 123 104)(35 124 97)(36 125 98)(37 126 99)(38 127 100)(39 128 101)(40 121 102)(41 134 142)(42 135 143)(43 136 144)(44 129 137)(45 130 138)(46 131 139)(47 132 140)(48 133 141)(49 66 89)(50 67 90)(51 68 91)(52 69 92)(53 70 93)(54 71 94)(55 72 95)(56 65 96)
(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)
(1 101 5 97)(2 100 6 104)(3 99 7 103)(4 98 8 102)(9 79 13 75)(10 78 14 74)(11 77 15 73)(12 76 16 80)(17 84 21 88)(18 83 22 87)(19 82 23 86)(20 81 24 85)(25 127 29 123)(26 126 30 122)(27 125 31 121)(28 124 32 128)(33 119 37 115)(34 118 38 114)(35 117 39 113)(36 116 40 120)(41 89 45 93)(42 96 46 92)(43 95 47 91)(44 94 48 90)(49 130 53 134)(50 129 54 133)(51 136 55 132)(52 135 56 131)(57 112 61 108)(58 111 62 107)(59 110 63 106)(60 109 64 105)(65 139 69 143)(66 138 70 142)(67 137 71 141)(68 144 72 140)

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

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

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

53 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 4A 4B 4C 6A 6B 6C 6D 6E 6F 8A 8B 12A 12B 12C ··· 12J 12K 12L 12M 12N 24A 24B 24C 24D 24E ··· 24T order 1 2 3 3 3 3 3 3 4 4 4 6 6 6 6 6 6 8 8 12 12 12 ··· 12 12 12 12 12 24 24 24 24 24 ··· 24 size 1 1 2 3 3 6 6 6 2 36 36 2 3 3 6 6 6 2 2 2 2 6 ··· 6 36 36 36 36 2 2 2 2 6 ··· 6

53 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6 type + + + + + + - + - + + + - image C1 C2 C2 C3 C6 C6 S3 D4 D6 Q16 C3×S3 D12 C3×D4 S3×C6 Dic12 C3×Q16 C3×D12 C3×Dic12 C32⋊C6 C2×C32⋊C6 He3⋊4D4 He3⋊4Q16 kernel He3⋊4Q16 C8×He3 He3⋊3Q8 C32⋊5Q16 C3×C24 C32⋊4Q8 C3×C24 C2×He3 C3×C12 He3 C24 C3×C6 C3×C6 C12 C32 C32 C6 C3 C8 C4 C2 C1 # reps 1 1 2 2 2 4 1 1 1 2 2 2 2 2 4 4 4 8 1 1 2 4

Matrix representation of He34Q16 in GL8(𝔽73)

 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0
,
 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 18 5 0 0 0 0 0 0 68 23 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72
,
 22 63 0 0 0 0 0 0 12 51 0 0 0 0 0 0 0 0 14 66 0 0 0 0 0 0 7 59 0 0 0 0 0 0 0 0 14 66 0 0 0 0 0 0 7 59 0 0 0 0 0 0 0 0 14 66 0 0 0 0 0 0 7 59

G:=sub<GL(8,GF(73))| [0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[18,68,0,0,0,0,0,0,5,23,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[22,12,0,0,0,0,0,0,63,51,0,0,0,0,0,0,0,0,14,7,0,0,0,0,0,0,66,59,0,0,0,0,0,0,0,0,14,7,0,0,0,0,0,0,66,59,0,0,0,0,0,0,0,0,14,7,0,0,0,0,0,0,66,59] >;

He34Q16 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_4Q_{16}
% in TeX

G:=Group("He3:4Q16");
// GroupNames label

G:=SmallGroup(432,114);
// by ID

G=gap.SmallGroup(432,114);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,260,1011,80,4037,2035,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e^-1=a^-1,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽