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## G = He3⋊5Q16order 432 = 24·33

### 2nd semidirect product of He3 and Q16 acting via Q16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C4×He3 — He3⋊5Q16
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊4Q8 — He3⋊5Q16
 Lower central He3 — C2×He3 — C4×He3 — He3⋊5Q16
 Upper central C1 — C6 — C12 — C24

Generators and relations for He35Q16
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, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 377 in 99 conjugacy classes, 31 normal (13 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C32, Dic3, C12, C12, Q16, C3×C6, C24, C24, Dic6, C3×Q8, He3, C3×Dic3, C3×C12, Dic12, C3×Q16, C2×He3, C3×C24, C3×Dic6, He33C4, C4×He3, C3×Dic12, C8×He3, He34Q8, He35Q16
Quotients: C1, C2, C22, S3, D4, D6, Q16, C3⋊S3, D12, C2×C3⋊S3, Dic12, He3⋊C2, C12⋊S3, C2×He3⋊C2, C325Q16, He35D4, He35Q16

Smallest permutation representation of He35Q16
On 144 points
Generators in S144
(25 38 62)(26 39 63)(27 40 64)(28 33 57)(29 34 58)(30 35 59)(31 36 60)(32 37 61)(49 95 137)(50 96 138)(51 89 139)(52 90 140)(53 91 141)(54 92 142)(55 93 143)(56 94 144)(65 123 104)(66 124 97)(67 125 98)(68 126 99)(69 127 100)(70 128 101)(71 121 102)(72 122 103)(73 112 120)(74 105 113)(75 106 114)(76 107 115)(77 108 116)(78 109 117)(79 110 118)(80 111 119)
(1 86 134)(2 87 135)(3 88 136)(4 81 129)(5 82 130)(6 83 131)(7 84 132)(8 85 133)(9 43 20)(10 44 21)(11 45 22)(12 46 23)(13 47 24)(14 48 17)(15 41 18)(16 42 19)(25 62 38)(26 63 39)(27 64 40)(28 57 33)(29 58 34)(30 59 35)(31 60 36)(32 61 37)(49 95 137)(50 96 138)(51 89 139)(52 90 140)(53 91 141)(54 92 142)(55 93 143)(56 94 144)(65 104 123)(66 97 124)(67 98 125)(68 99 126)(69 100 127)(70 101 128)(71 102 121)(72 103 122)(73 112 120)(74 105 113)(75 106 114)(76 107 115)(77 108 116)(78 109 117)(79 110 118)(80 111 119)
(1 102 118)(2 103 119)(3 104 120)(4 97 113)(5 98 114)(6 99 115)(7 100 116)(8 101 117)(9 35 56)(10 36 49)(11 37 50)(12 38 51)(13 39 52)(14 40 53)(15 33 54)(16 34 55)(17 64 141)(18 57 142)(19 58 143)(20 59 144)(21 60 137)(22 61 138)(23 62 139)(24 63 140)(25 89 46)(26 90 47)(27 91 48)(28 92 41)(29 93 42)(30 94 43)(31 95 44)(32 96 45)(65 112 136)(66 105 129)(67 106 130)(68 107 131)(69 108 132)(70 109 133)(71 110 134)(72 111 135)(73 88 123)(74 81 124)(75 82 125)(76 83 126)(77 84 127)(78 85 128)(79 86 121)(80 87 122)
(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 20 5 24)(2 19 6 23)(3 18 7 22)(4 17 8 21)(9 82 13 86)(10 81 14 85)(11 88 15 84)(12 87 16 83)(25 111 29 107)(26 110 30 106)(27 109 31 105)(28 108 32 112)(33 77 37 73)(34 76 38 80)(35 75 39 79)(36 74 40 78)(41 132 45 136)(42 131 46 135)(43 130 47 134)(44 129 48 133)(49 124 53 128)(50 123 54 127)(51 122 55 126)(52 121 56 125)(57 116 61 120)(58 115 62 119)(59 114 63 118)(60 113 64 117)(65 92 69 96)(66 91 70 95)(67 90 71 94)(68 89 72 93)(97 141 101 137)(98 140 102 144)(99 139 103 143)(100 138 104 142)

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

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

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

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 1 1 6 6 6 6 2 36 36 1 1 6 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 2 2 2 2 2 2 3 3 6 6 type + + + + + + - + - image C1 C2 C2 S3 D4 D6 Q16 D12 Dic12 He3⋊C2 C2×He3⋊C2 He3⋊5D4 He3⋊5Q16 kernel He3⋊5Q16 C8×He3 He3⋊4Q8 C3×C24 C2×He3 C3×C12 He3 C3×C6 C32 C8 C4 C2 C1 # reps 1 1 2 4 1 4 2 8 16 4 4 2 4

Matrix representation of He35Q16 in GL5(𝔽73)

 0 72 0 0 0 1 72 0 0 0 0 0 1 0 0 0 0 1 64 0 0 0 65 0 8
,
 1 0 0 0 0 0 1 0 0 0 0 0 64 0 0 0 0 0 64 0 0 0 0 0 64
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 63 0 0 0 0 72 1 0 0 0 72 0
,
 23 68 0 0 0 5 18 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 34 65 0 0 0 26 39 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 72 0

G:=sub<GL(5,GF(73))| [0,1,0,0,0,72,72,0,0,0,0,0,1,1,65,0,0,0,64,0,0,0,0,0,8],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,63,72,72,0,0,0,1,0],[23,5,0,0,0,68,18,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[34,26,0,0,0,65,39,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,72,0] >;

He35Q16 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_5Q_{16}
% in TeX

G:=Group("He3:5Q16");
// GroupNames label

G:=SmallGroup(432,177);
// by ID

G=gap.SmallGroup(432,177);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,92,254,58,1124,4037,537]);
// 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,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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