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## G = Q16×He3order 432 = 24·33

### Direct product of Q16 and He3

direct product, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q16×He3
G = < a,b,c,d,e | a8=c3=d3=e3=1, b2=a4, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 209 in 99 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C32, C12, C12, Q16, C3×C6, C24, C24, C3×Q8, C3×Q8, He3, C3×C12, C3×C12, C3×Q16, C3×Q16, C2×He3, C3×C24, Q8×C32, C4×He3, C4×He3, C32×Q16, C8×He3, Q8×He3, Q16×He3
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, Q16, C3×C6, C3×D4, He3, C62, C3×Q16, C2×He3, D4×C32, C22×He3, C32×Q16, D4×He3, Q16×He3

Smallest permutation representation of Q16×He3
On 144 points
Generators in S144
(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 12 5 16)(2 11 6 15)(3 10 7 14)(4 9 8 13)(17 43 21 47)(18 42 22 46)(19 41 23 45)(20 48 24 44)(25 54 29 50)(26 53 30 49)(27 52 31 56)(28 51 32 55)(33 96 37 92)(34 95 38 91)(35 94 39 90)(36 93 40 89)(57 141 61 137)(58 140 62 144)(59 139 63 143)(60 138 64 142)(65 105 69 109)(66 112 70 108)(67 111 71 107)(68 110 72 106)(73 128 77 124)(74 127 78 123)(75 126 79 122)(76 125 80 121)(81 136 85 132)(82 135 86 131)(83 134 87 130)(84 133 88 129)(97 115 101 119)(98 114 102 118)(99 113 103 117)(100 120 104 116)
(1 82 139)(2 83 140)(3 84 141)(4 85 142)(5 86 143)(6 87 144)(7 88 137)(8 81 138)(9 132 60)(10 133 61)(11 134 62)(12 135 63)(13 136 64)(14 129 57)(15 130 58)(16 131 59)(17 53 67)(18 54 68)(19 55 69)(20 56 70)(21 49 71)(22 50 72)(23 51 65)(24 52 66)(25 106 46)(26 107 47)(27 108 48)(28 109 41)(29 110 42)(30 111 43)(31 112 44)(32 105 45)(33 120 76)(34 113 77)(35 114 78)(36 115 79)(37 116 80)(38 117 73)(39 118 74)(40 119 75)(89 97 126)(90 98 127)(91 99 128)(92 100 121)(93 101 122)(94 102 123)(95 103 124)(96 104 125)
(1 39 26)(2 40 27)(3 33 28)(4 34 29)(5 35 30)(6 36 31)(7 37 32)(8 38 25)(9 95 50)(10 96 51)(11 89 52)(12 90 53)(13 91 54)(14 92 55)(15 93 56)(16 94 49)(17 63 127)(18 64 128)(19 57 121)(20 58 122)(21 59 123)(22 60 124)(23 61 125)(24 62 126)(41 141 76)(42 142 77)(43 143 78)(44 144 79)(45 137 80)(46 138 73)(47 139 74)(48 140 75)(65 133 104)(66 134 97)(67 135 98)(68 136 99)(69 129 100)(70 130 101)(71 131 102)(72 132 103)(81 117 106)(82 118 107)(83 119 108)(84 120 109)(85 113 110)(86 114 111)(87 115 112)(88 116 105)
(1 139 107)(2 140 108)(3 141 109)(4 142 110)(5 143 111)(6 144 112)(7 137 105)(8 138 106)(9 60 72)(10 61 65)(11 62 66)(12 63 67)(13 64 68)(14 57 69)(15 58 70)(16 59 71)(17 98 53)(18 99 54)(19 100 55)(20 101 56)(21 102 49)(22 103 50)(23 104 51)(24 97 52)(25 46 117)(26 47 118)(27 48 119)(28 41 120)(29 42 113)(30 43 114)(31 44 115)(32 45 116)(33 76 84)(34 77 85)(35 78 86)(36 79 87)(37 80 88)(38 73 81)(39 74 82)(40 75 83)(89 126 134)(90 127 135)(91 128 136)(92 121 129)(93 122 130)(94 123 131)(95 124 132)(96 125 133)

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

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

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

77 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 4A 4B 4C 6A 6B 6C ··· 6J 8A 8B 12A 12B 12C 12D 12E 12F 12G ··· 12N 12O ··· 12AD 24A 24B 24C 24D 24E ··· 24T order 1 2 3 3 3 ··· 3 4 4 4 6 6 6 ··· 6 8 8 12 12 12 12 12 12 12 ··· 12 12 ··· 12 24 24 24 24 24 ··· 24 size 1 1 1 1 3 ··· 3 2 4 4 1 1 3 ··· 3 2 2 2 2 4 4 4 4 6 ··· 6 12 ··· 12 2 2 2 2 6 ··· 6

77 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + + - image C1 C2 C2 C3 C6 C6 D4 Q16 C3×D4 C3×Q16 He3 C2×He3 C2×He3 D4×He3 Q16×He3 kernel Q16×He3 C8×He3 Q8×He3 C32×Q16 C3×C24 Q8×C32 C2×He3 He3 C3×C6 C32 Q16 C8 Q8 C2 C1 # reps 1 1 2 8 8 16 1 2 8 16 2 2 4 2 4

Matrix representation of Q16×He3 in GL5(𝔽73)

 57 16 0 0 0 57 57 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 60 66 0 0 0 66 13 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 1 0 0 0 0 0 64 63 0 0 0 0 9 1 0 0 0 65 0
,
 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
,
 64 0 0 0 0 0 64 0 0 0 0 0 1 0 17 0 0 72 0 65 0 0 8 8 72

G:=sub<GL(5,GF(73))| [57,57,0,0,0,16,57,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[60,66,0,0,0,66,13,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,63,9,65,0,0,0,1,0],[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],[64,0,0,0,0,0,64,0,0,0,0,0,1,72,8,0,0,0,0,8,0,0,17,65,72] >;

Q16×He3 in GAP, Magma, Sage, TeX

Q_{16}\times {\rm He}_3
% in TeX

G:=Group("Q16xHe3");
// GroupNames label

G:=SmallGroup(432,222);
// by ID

G=gap.SmallGroup(432,222);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,1512,533,1520,605,8824,4421,242]);
// Polycyclic

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

׿
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