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G = He3⋊Q16order 432 = 24·33

The semidirect product of He3 and Q16 acting via Q16/C2=D4

non-abelian, soluble

Aliases: He3⋊Q16, C6.3S3≀C2, He32Q8.C2, C3.(C32⋊Q16), C2.5(He3⋊D4), (C2×He3).3D4, He32C8.1C2, He33C4.3C22, SmallGroup(432,236)

Series: Derived Chief Lower central Upper central

C1C3He3He33C4 — He3⋊Q16
C1C3He3C2×He3He33C4He32Q8 — He3⋊Q16
He3C2×He3He33C4 — He3⋊Q16
C1C2

Generators and relations for He3⋊Q16
 G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, cac-1=dcd-1=ece-1=ab-1, dad-1=bc-1, eae-1=b-1c, bc=cb, bd=db, ebe-1=b-1, ede-1=d-1 >

6C3
6C3
9C4
18C4
18C4
6C6
6C6
2C32
2C32
9C8
27Q8
27Q8
6Dic3
6Dic3
6Dic3
6Dic3
9C12
18C12
18Dic3
18Dic3
18C12
2C3×C6
2C3×C6
27Q16
9C24
9Dic6
9Dic6
18Dic6
18Dic6
2C3⋊Dic3
2C3⋊Dic3
6C3×Dic3
6C3×Dic3
6C3×Dic3
6C3×Dic3
9Dic12
6C322Q8
6C322Q8
2C32⋊C12
2C32⋊C12

Character table of He3⋊Q16

 class 123A3B3C4A4B4C6A6B6C8A8B12A12B12C12D12E12F24A24B24C24D
 size 112121218363621212181818183636363618181818
ρ111111111111111111111111    trivial
ρ21111111-1111-1-111-1-111-1-1-1-1    linear of order 2
ρ3111111-11111-1-11111-1-1-1-1-1-1    linear of order 2
ρ4111111-1-11111111-1-1-1-11111    linear of order 2
ρ522222-20022200-2-200000000    orthogonal lifted from D4
ρ62-2222000-2-2-22-2000000-2-222    symplectic lifted from Q16, Schur index 2
ρ72-2222000-2-2-2-2200000022-2-2    symplectic lifted from Q16, Schur index 2
ρ84441-20204-21000000-1-10000    orthogonal lifted from S3≀C2
ρ9444-2100241-20000-1-1000000    orthogonal lifted from S3≀C2
ρ10444-2100-241-2000011000000    orthogonal lifted from S3≀C2
ρ114441-20-204-21000000110000    orthogonal lifted from S3≀C2
ρ124-441-2000-42-1000000-330000    symplectic lifted from C32⋊Q16, Schur index 2
ρ134-44-21000-4-120000-33000000    symplectic lifted from C32⋊Q16, Schur index 2
ρ144-44-21000-4-1200003-3000000    symplectic lifted from C32⋊Q16, Schur index 2
ρ154-441-2000-42-10000003-30000    symplectic lifted from C32⋊Q16, Schur index 2
ρ1666-300-200-30022110000-1-1-1-1    orthogonal lifted from He3⋊D4
ρ1766-300-200-300-2-21100001111    orthogonal lifted from He3⋊D4
ρ1866-300200-30000-1-100003-3-33    orthogonal lifted from He3⋊D4
ρ1966-300200-30000-1-10000-333-3    orthogonal lifted from He3⋊D4
ρ206-6-300000300-223-30000ζ87ζ385ζ385ζ83ζ3838ζ3ζ83ζ328ζ328ζ87ζ328785ζ32    symplectic faithful, Schur index 2
ρ216-6-3000003002-23-30000ζ87ζ328785ζ32ζ83ζ328ζ328ζ83ζ3838ζ3ζ87ζ385ζ385    symplectic faithful, Schur index 2
ρ226-6-3000003002-2-330000ζ83ζ328ζ328ζ87ζ328785ζ32ζ87ζ385ζ385ζ83ζ3838ζ3    symplectic faithful, Schur index 2
ρ236-6-300000300-22-330000ζ83ζ3838ζ3ζ87ζ385ζ385ζ87ζ328785ζ32ζ83ζ328ζ328    symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of He3⋊Q16 in GL6(𝔽73)

000100
00727200
000001
00007272
010000
72720000
,
010000
72720000
000100
00727200
000001
00007272
,
001000
000100
00007272
000010
010000
72720000
,
155338583858
203515531553
385815533858
155320351553
155315532035
203520353858
,
61510000
63120000
00615100
00631200
00006312
00002210

G:=sub<GL(6,GF(73))| [0,0,0,0,0,72,0,0,0,0,1,72,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[0,0,0,0,0,72,0,0,0,0,1,72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0],[15,20,38,15,15,20,53,35,58,53,53,35,38,15,15,20,15,20,58,53,53,35,53,35,38,15,38,15,20,38,58,53,58,53,35,58],[61,63,0,0,0,0,51,12,0,0,0,0,0,0,61,63,0,0,0,0,51,12,0,0,0,0,0,0,63,22,0,0,0,0,12,10] >;

He3⋊Q16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(432,236);
// by ID

G=gap.SmallGroup(432,236);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,64,254,135,58,1124,851,298,348,1027,537,14118,7069]);
// 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=d*c*d^-1=e*c*e^-1=a*b^-1,d*a*d^-1=b*c^-1,e*a*e^-1=b^-1*c,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,e*d*e^-1=d^-1>;
// generators/relations

Export

Subgroup lattice of He3⋊Q16 in TeX
Character table of He3⋊Q16 in TeX

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