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## G = Q16×C3⋊S3order 288 = 25·32

### Direct product of Q16 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — Q16×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×C3⋊S3 — Q8×C3⋊S3 — Q16×C3⋊S3
 Lower central C32 — C3×C6 — C3×C12 — Q16×C3⋊S3
 Upper central C1 — C2 — C4 — Q16

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 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12L 24A ··· 24H order 1 2 2 2 3 3 3 3 4 4 4 4 4 4 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 9 9 2 2 2 2 2 4 4 18 36 36 2 2 2 2 2 2 18 18 4 4 4 4 8 ··· 8 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 Q16 S3×D4 S3×Q16 kernel Q16×C3⋊S3 C8×C3⋊S3 C32⋊5Q16 C32⋊7Q16 C32×Q16 Q8×C3⋊S3 C3×Q16 C3⋊Dic3 C2×C3⋊S3 C24 C3×Q8 C3⋊S3 C6 C3 # reps 1 1 1 2 1 2 4 1 1 4 8 4 4 8

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

 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 35 0 0 0 0 25 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 43 0 0 0 0 69 5
,
 0 1 0 0 0 0 72 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 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72

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

׿
×
𝔽