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G = Q16×C3⋊S3order 288 = 25·32

Direct product of Q16 and C3⋊S3

direct product, metabelian, supersoluble, monomial

Aliases: Q16×C3⋊S3, C24.27D6, C34(S3×Q16), (C3×Q16)⋊2S3, C6.125(S3×D4), (C3×Q8).38D6, C3212(C2×Q16), C325Q169C2, C3⋊Dic3.50D4, (C32×Q16)⋊5C2, C327Q167C2, C12.94(C22×S3), (C3×C12).98C23, (C3×C24).30C22, C324C8.28C22, (Q8×C32).18C22, C324Q8.19C22, C8.9(C2×C3⋊S3), (C8×C3⋊S3).2C2, C2.22(D4×C3⋊S3), (Q8×C3⋊S3).3C2, Q8.8(C2×C3⋊S3), (C2×C3⋊S3).74D4, C4.8(C22×C3⋊S3), (C3×C6).246(C2×D4), (C4×C3⋊S3).74C22, SmallGroup(288,774)

Series: Derived Chief Lower central Upper central

C1C3×C12 — Q16×C3⋊S3
C1C3C32C3×C6C3×C12C4×C3⋊S3Q8×C3⋊S3 — Q16×C3⋊S3
C32C3×C6C3×C12 — Q16×C3⋊S3
C1C2C4Q16

Generators and relations for Q16×C3⋊S3
 G = < a,b,c,d,e | a8=c3=d3=e2=1, b2=a4, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 660 in 180 conjugacy classes, 57 normal (17 characteristic)
C1, C2, C2 [×2], C3 [×4], C4, C4 [×5], C22, S3 [×8], C6 [×4], C8, C8, C2×C4 [×3], Q8 [×2], Q8 [×4], C32, Dic3 [×12], C12 [×4], C12 [×8], D6 [×4], C2×C8, Q16, Q16 [×3], C2×Q8 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8 [×4], C24 [×4], Dic6 [×16], C4×S3 [×12], C3×Q8 [×8], C2×Q16, C3⋊Dic3, C3⋊Dic3 [×2], C3×C12, C3×C12 [×2], C2×C3⋊S3, S3×C8 [×4], Dic12 [×4], C3⋊Q16 [×8], C3×Q16 [×4], S3×Q8 [×8], C324C8, C3×C24, C324Q8 [×2], C324Q8 [×2], C4×C3⋊S3, C4×C3⋊S3 [×2], Q8×C32 [×2], S3×Q16 [×4], C8×C3⋊S3, C325Q16, C327Q16 [×2], C32×Q16, Q8×C3⋊S3 [×2], Q16×C3⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], Q16 [×2], C2×D4, C3⋊S3, C22×S3 [×4], C2×Q16, C2×C3⋊S3 [×3], S3×D4 [×4], C22×C3⋊S3, S3×Q16 [×4], D4×C3⋊S3, Q16×C3⋊S3

Smallest permutation representation of Q16×C3⋊S3
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 61 5 57)(2 60 6 64)(3 59 7 63)(4 58 8 62)(9 31 13 27)(10 30 14 26)(11 29 15 25)(12 28 16 32)(17 94 21 90)(18 93 22 89)(19 92 23 96)(20 91 24 95)(33 44 37 48)(34 43 38 47)(35 42 39 46)(36 41 40 45)(49 138 53 142)(50 137 54 141)(51 144 55 140)(52 143 56 139)(65 114 69 118)(66 113 70 117)(67 120 71 116)(68 119 72 115)(73 106 77 110)(74 105 78 109)(75 112 79 108)(76 111 80 107)(81 134 85 130)(82 133 86 129)(83 132 87 136)(84 131 88 135)(97 127 101 123)(98 126 102 122)(99 125 103 121)(100 124 104 128)
(1 133 18)(2 134 19)(3 135 20)(4 136 21)(5 129 22)(6 130 23)(7 131 24)(8 132 17)(9 115 79)(10 116 80)(11 117 73)(12 118 74)(13 119 75)(14 120 76)(15 113 77)(16 114 78)(25 70 110)(26 71 111)(27 72 112)(28 65 105)(29 66 106)(30 67 107)(31 68 108)(32 69 109)(33 137 102)(34 138 103)(35 139 104)(36 140 97)(37 141 98)(38 142 99)(39 143 100)(40 144 101)(41 51 127)(42 52 128)(43 53 121)(44 54 122)(45 55 123)(46 56 124)(47 49 125)(48 50 126)(57 82 89)(58 83 90)(59 84 91)(60 85 92)(61 86 93)(62 87 94)(63 88 95)(64 81 96)
(1 119 54)(2 120 55)(3 113 56)(4 114 49)(5 115 50)(6 116 51)(7 117 52)(8 118 53)(9 48 22)(10 41 23)(11 42 24)(12 43 17)(13 44 18)(14 45 19)(15 46 20)(16 47 21)(25 35 91)(26 36 92)(27 37 93)(28 38 94)(29 39 95)(30 40 96)(31 33 89)(32 34 90)(57 68 137)(58 69 138)(59 70 139)(60 71 140)(61 72 141)(62 65 142)(63 66 143)(64 67 144)(73 128 131)(74 121 132)(75 122 133)(76 123 134)(77 124 135)(78 125 136)(79 126 129)(80 127 130)(81 107 101)(82 108 102)(83 109 103)(84 110 104)(85 111 97)(86 112 98)(87 105 99)(88 106 100)
(1 5)(2 6)(3 7)(4 8)(9 122)(10 123)(11 124)(12 125)(13 126)(14 127)(15 128)(16 121)(17 136)(18 129)(19 130)(20 131)(21 132)(22 133)(23 134)(24 135)(25 100)(26 101)(27 102)(28 103)(29 104)(30 97)(31 98)(32 99)(33 112)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 76)(42 77)(43 78)(44 79)(45 80)(46 73)(47 74)(48 75)(49 118)(50 119)(51 120)(52 113)(53 114)(54 115)(55 116)(56 117)(57 61)(58 62)(59 63)(60 64)(65 138)(66 139)(67 140)(68 141)(69 142)(70 143)(71 144)(72 137)(81 92)(82 93)(83 94)(84 95)(85 96)(86 89)(87 90)(88 91)

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

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

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

42 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E4F6A6B6C6D8A8B8C8D12A12B12C12D12E···12L24A···24H
order12223333444444666688881212121212···1224···24
size11992222244183636222222181844448···84···4

42 irreducible representations

dim11111122222244
type+++++++++++-+-
imageC1C2C2C2C2C2S3D4D4D6D6Q16S3×D4S3×Q16
kernelQ16×C3⋊S3C8×C3⋊S3C325Q16C327Q16C32×Q16Q8×C3⋊S3C3×Q16C3⋊Dic3C2×C3⋊S3C24C3×Q8C3⋊S3C6C3
# reps11121241148448

Matrix representation of Q16×C3⋊S3 in GL6(𝔽73)

100000
010000
001000
000100
0000035
00002532
,
7200000
0720000
001000
000100
00006843
0000695
,
010000
72720000
001000
000100
000010
000001
,
100000
010000
00727200
001000
000010
000001
,
100000
72720000
001000
00727200
0000720
0000072

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,25,0,0,0,0,35,32],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,68,69,0,0,0,0,43,5],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

Q16×C3⋊S3 in GAP, Magma, Sage, TeX

Q_{16}\times C_3\rtimes S_3
% in TeX

G:=Group("Q16xC3:S3");
// GroupNames label

G:=SmallGroup(288,774);
// by ID

G=gap.SmallGroup(288,774);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,135,100,346,185,80,2693,9414]);
// Polycyclic

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

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×
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