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## G = C2×C8×C3⋊S3order 288 = 25·32

### Direct product of C2×C8 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C8×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×C3⋊S3 — C2×C4×C3⋊S3 — C2×C8×C3⋊S3
 Lower central C32 — C2×C8×C3⋊S3
 Upper central C1 — C2×C8

Generators and relations for C2×C8×C3⋊S3
G = < a,b,c,d,e | a2=b8=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 660 in 228 conjugacy classes, 93 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×6], S3 [×16], C6 [×12], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×8], C12 [×8], D6 [×24], C2×C6 [×4], C2×C8, C2×C8 [×5], C22×C4, C3⋊S3 [×4], C3×C6, C3×C6 [×2], C3⋊C8 [×8], C24 [×8], C4×S3 [×16], C2×Dic3 [×4], C2×C12 [×4], C22×S3 [×4], C22×C8, C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, S3×C8 [×16], C2×C3⋊C8 [×4], C2×C24 [×4], S3×C2×C4 [×4], C324C8 [×2], C3×C24 [×2], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×C2×C8 [×4], C8×C3⋊S3 [×4], C2×C324C8, C6×C24, C2×C4×C3⋊S3, C2×C8×C3⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C8 [×4], C2×C4 [×6], C23, D6 [×12], C2×C8 [×6], C22×C4, C3⋊S3, C4×S3 [×8], C22×S3 [×4], C22×C8, C2×C3⋊S3 [×3], S3×C8 [×8], S3×C2×C4 [×4], C4×C3⋊S3 [×2], C22×C3⋊S3, S3×C2×C8 [×4], C8×C3⋊S3 [×2], C2×C4×C3⋊S3, C2×C8×C3⋊S3

Smallest permutation representation of C2×C8×C3⋊S3
On 144 points
Generators in S144
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 87)(18 88)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 104)(26 97)(27 98)(28 99)(29 100)(30 101)(31 102)(32 103)(33 77)(34 78)(35 79)(36 80)(37 73)(38 74)(39 75)(40 76)(41 120)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 92)(50 93)(51 94)(52 95)(53 96)(54 89)(55 90)(56 91)(57 136)(58 129)(59 130)(60 131)(61 132)(62 133)(63 134)(64 135)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)(105 122)(106 123)(107 124)(108 125)(109 126)(110 127)(111 128)(112 121)
(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 95 102)(2 96 103)(3 89 104)(4 90 97)(5 91 98)(6 92 99)(7 93 100)(8 94 101)(9 52 31)(10 53 32)(11 54 25)(12 55 26)(13 56 27)(14 49 28)(15 50 29)(16 51 30)(17 47 39)(18 48 40)(19 41 33)(20 42 34)(21 43 35)(22 44 36)(23 45 37)(24 46 38)(57 110 142)(58 111 143)(59 112 144)(60 105 137)(61 106 138)(62 107 139)(63 108 140)(64 109 141)(65 131 122)(66 132 123)(67 133 124)(68 134 125)(69 135 126)(70 136 127)(71 129 128)(72 130 121)(73 85 116)(74 86 117)(75 87 118)(76 88 119)(77 81 120)(78 82 113)(79 83 114)(80 84 115)
(1 118 68)(2 119 69)(3 120 70)(4 113 71)(5 114 72)(6 115 65)(7 116 66)(8 117 67)(9 47 140)(10 48 141)(11 41 142)(12 42 143)(13 43 144)(14 44 137)(15 45 138)(16 46 139)(17 108 31)(18 109 32)(19 110 25)(20 111 26)(21 112 27)(22 105 28)(23 106 29)(24 107 30)(33 57 54)(34 58 55)(35 59 56)(36 60 49)(37 61 50)(38 62 51)(39 63 52)(40 64 53)(73 132 93)(74 133 94)(75 134 95)(76 135 96)(77 136 89)(78 129 90)(79 130 91)(80 131 92)(81 127 104)(82 128 97)(83 121 98)(84 122 99)(85 123 100)(86 124 101)(87 125 102)(88 126 103)
(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 54)(26 55)(27 56)(28 49)(29 50)(30 51)(31 52)(32 53)(33 110)(34 111)(35 112)(36 105)(37 106)(38 107)(39 108)(40 109)(41 142)(42 143)(43 144)(44 137)(45 138)(46 139)(47 140)(48 141)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 113)(72 114)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 121)(80 122)(81 136)(82 129)(83 130)(84 131)(85 132)(86 133)(87 134)(88 135)(89 104)(90 97)(91 98)(92 99)(93 100)(94 101)(95 102)(96 103)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6L 8A ··· 8H 8I ··· 8P 12A ··· 12P 24A ··· 24AF order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 1 1 9 9 9 9 2 2 2 2 1 1 1 1 9 9 9 9 2 ··· 2 1 ··· 1 9 ··· 9 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 S3 D6 D6 C4×S3 C4×S3 S3×C8 kernel C2×C8×C3⋊S3 C8×C3⋊S3 C2×C32⋊4C8 C6×C24 C2×C4×C3⋊S3 C4×C3⋊S3 C2×C3⋊Dic3 C22×C3⋊S3 C2×C3⋊S3 C2×C24 C24 C2×C12 C12 C2×C6 C6 # reps 1 4 1 1 1 4 2 2 16 4 8 4 8 8 32

Matrix representation of C2×C8×C3⋊S3 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 72 0 0 0 0 72
,
 10 0 0 0 0 10 0 0 0 0 63 0 0 0 0 63
,
 1 0 0 0 0 1 0 0 0 0 72 1 0 0 72 0
,
 72 72 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 72 72 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[10,0,0,0,0,10,0,0,0,0,63,0,0,0,0,63],[1,0,0,0,0,1,0,0,0,0,72,72,0,0,1,0],[72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[1,72,0,0,0,72,0,0,0,0,0,1,0,0,1,0] >;

C2×C8×C3⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_8\times C_3\rtimes S_3
% in TeX

G:=Group("C2xC8xC3:S3");
// GroupNames label

G:=SmallGroup(288,756);
// by ID

G=gap.SmallGroup(288,756);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,58,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=c^3=d^3=e^2=1,a*b=b*a,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