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## G = He3⋊6Q16order 432 = 24·33

### 1st semidirect product of He3 and Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — He3⋊6Q16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×He3 — He3⋊3Q8 — He3⋊6Q16
 Lower central C32 — C3×C6 — C3×C12 — He3⋊6Q16
 Upper central C1 — C2 — C4 — Q8

Generators and relations for He36Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, cac-1=ab-1, dad-1=a-1, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 289 in 75 conjugacy classes, 26 normal (all characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, Q8, C32, C32, Dic3, C12, C12, Q16, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3⋊Q16, C3×Q16, C2×He3, C3×C3⋊C8, C324C8, C3×Dic6, C324Q8, Q8×C32, Q8×C32, C32⋊C12, C4×He3, C4×He3, C3×C3⋊Q16, C327Q16, He33C8, He33Q8, Q8×He3, He36Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, C3×S3, C3⋊D4, C3×D4, S3×C6, C3⋊Q16, C3×Q16, C32⋊C6, C3×C3⋊D4, C2×C32⋊C6, C3×C3⋊Q16, He36D4, He36Q16

Smallest permutation representation of He36Q16
On 144 points
Generators in S144
(9 39 120)(10 113 40)(11 33 114)(12 115 34)(13 35 116)(14 117 36)(15 37 118)(16 119 38)(25 64 121)(26 122 57)(27 58 123)(28 124 59)(29 60 125)(30 126 61)(31 62 127)(32 128 63)(41 55 106)(42 107 56)(43 49 108)(44 109 50)(45 51 110)(46 111 52)(47 53 112)(48 105 54)(65 132 100)(66 101 133)(67 134 102)(68 103 135)(69 136 104)(70 97 129)(71 130 98)(72 99 131)
(1 139 77)(2 78 140)(3 141 79)(4 80 142)(5 143 73)(6 74 144)(7 137 75)(8 76 138)(9 120 39)(10 40 113)(11 114 33)(12 34 115)(13 116 35)(14 36 117)(15 118 37)(16 38 119)(17 81 92)(18 93 82)(19 83 94)(20 95 84)(21 85 96)(22 89 86)(23 87 90)(24 91 88)(25 64 121)(26 122 57)(27 58 123)(28 124 59)(29 60 125)(30 126 61)(31 62 127)(32 128 63)(41 55 106)(42 107 56)(43 49 108)(44 109 50)(45 51 110)(46 111 52)(47 53 112)(48 105 54)(65 100 132)(66 133 101)(67 102 134)(68 135 103)(69 104 136)(70 129 97)(71 98 130)(72 131 99)
(1 15 43)(2 16 44)(3 9 45)(4 10 46)(5 11 47)(6 12 48)(7 13 41)(8 14 42)(17 132 62)(18 133 63)(19 134 64)(20 135 57)(21 136 58)(22 129 59)(23 130 60)(24 131 61)(25 94 102)(26 95 103)(27 96 104)(28 89 97)(29 90 98)(30 91 99)(31 92 100)(32 93 101)(33 112 73)(34 105 74)(35 106 75)(36 107 76)(37 108 77)(38 109 78)(39 110 79)(40 111 80)(49 139 118)(50 140 119)(51 141 120)(52 142 113)(53 143 114)(54 144 115)(55 137 116)(56 138 117)(65 127 81)(66 128 82)(67 121 83)(68 122 84)(69 123 85)(70 124 86)(71 125 87)(72 126 88)
(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 87 5 83)(2 86 6 82)(3 85 7 81)(4 84 8 88)(9 69 13 65)(10 68 14 72)(11 67 15 71)(12 66 16 70)(17 79 21 75)(18 78 22 74)(19 77 23 73)(20 76 24 80)(25 49 29 53)(26 56 30 52)(27 55 31 51)(28 54 32 50)(33 134 37 130)(34 133 38 129)(35 132 39 136)(36 131 40 135)(41 127 45 123)(42 126 46 122)(43 125 47 121)(44 124 48 128)(57 107 61 111)(58 106 62 110)(59 105 63 109)(60 112 64 108)(89 144 93 140)(90 143 94 139)(91 142 95 138)(92 141 96 137)(97 115 101 119)(98 114 102 118)(99 113 103 117)(100 120 104 116)

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

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

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

41 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 12D 12E 12F ··· 12P 12Q 12R 24A 24B 24C 24D 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 12 24 24 24 24 size 1 1 2 3 3 6 6 6 2 4 36 2 3 3 6 6 6 18 18 4 4 4 6 6 12 ··· 12 36 36 18 18 18 18

41 irreducible representations

 dim 1 1 1 1 1 1 1 1 12 2 2 2 2 2 2 2 2 2 2 4 4 6 6 6 type + + + + - + + + - - + + image C1 C2 C2 C2 C3 C6 C6 C6 He3⋊6Q16 S3 D4 D6 Q16 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×Q16 C3×C3⋊D4 C3⋊Q16 C3×C3⋊Q16 C32⋊C6 C2×C32⋊C6 He3⋊6D4 kernel He3⋊6Q16 He3⋊3C8 He3⋊3Q8 Q8×He3 C32⋊7Q16 C32⋊4C8 C32⋊4Q8 Q8×C32 C1 Q8×C32 C2×He3 C3×C12 He3 C3×Q8 C3×C6 C3×C6 C12 C32 C6 C32 C3 Q8 C4 C2 # reps 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 4 4 1 2 1 1 2

Matrix representation of He36Q16 in GL10(𝔽73)

 0 72 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 72 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 72 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 72
,
 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 72 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 72 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 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 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 0 0 0 32 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 57 0 32 0 0 0 0 0 0 57 0 32 0 0 0 0 0 0 0 0 0 0 0 31 45 0 0 0 0 0 0 0 0 3 42 0 0 0 0 0 0 0 0 0 0 31 45 0 0 0 0 0 0 0 0 3 42 0 0 0 0 0 0 0 0 0 0 31 45 0 0 0 0 0 0 0 0 3 42
,
 60 0 2 0 0 0 0 0 0 0 0 60 0 2 0 0 0 0 0 0 61 0 13 0 0 0 0 0 0 0 0 61 0 13 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72

G:=sub<GL(10,GF(73))| [0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,57,0,0,0,0,0,0,0,0,57,0,0,0,0,0,0,0,0,32,0,32,0,0,0,0,0,0,32,0,32,0,0,0,0,0,0,0,0,0,0,0,31,3,0,0,0,0,0,0,0,0,45,42,0,0,0,0,0,0,0,0,0,0,31,3,0,0,0,0,0,0,0,0,45,42,0,0,0,0,0,0,0,0,0,0,31,3,0,0,0,0,0,0,0,0,45,42],[60,0,61,0,0,0,0,0,0,0,0,60,0,61,0,0,0,0,0,0,2,0,13,0,0,0,0,0,0,0,0,2,0,13,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72] >;

He36Q16 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_6Q_{16}
% in TeX

G:=Group("He3:6Q16");
// GroupNames label

G:=SmallGroup(432,160);
// by ID

G=gap.SmallGroup(432,160);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,176,1011,514,80,4037,2035,14118]);
// 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,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
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