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## G = C2×C32⋊7D8order 288 = 25·32

### Direct product of C2 and C32⋊7D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C2×C32⋊7D8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C12⋊S3 — C2×C12⋊S3 — C2×C32⋊7D8
 Lower central C32 — C3×C6 — C3×C12 — C2×C32⋊7D8
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for C2×C327D8
G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 980 in 228 conjugacy classes, 77 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×4], C4 [×2], C22, C22 [×8], S3 [×8], C6 [×12], C6 [×8], C8 [×2], C2×C4, D4 [×2], D4 [×4], C23 [×2], C32, C12 [×8], D6 [×16], C2×C6 [×4], C2×C6 [×16], C2×C8, D8 [×4], C2×D4, C2×D4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C3⋊C8 [×8], D12 [×12], C2×C12 [×4], C3×D4 [×8], C3×D4 [×4], C22×S3 [×4], C22×C6 [×4], C2×D8, C3×C12 [×2], C2×C3⋊S3 [×4], C62, C62 [×4], C2×C3⋊C8 [×4], D4⋊S3 [×16], C2×D12 [×4], C6×D4 [×4], C324C8 [×2], C12⋊S3 [×2], C12⋊S3, C6×C12, D4×C32 [×2], D4×C32, C22×C3⋊S3, C2×C62, C2×D4⋊S3 [×4], C2×C324C8, C327D8 [×4], C2×C12⋊S3, D4×C3×C6, C2×C327D8
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], D8 [×2], C2×D4, C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C2×D8, C2×C3⋊S3 [×3], D4⋊S3 [×8], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, C2×D4⋊S3 [×4], C327D8 [×2], C2×C327D4, C2×C327D8

Smallest permutation representation of C2×C327D8
On 144 points
Generators in S144
(1 60)(2 61)(3 62)(4 63)(5 64)(6 57)(7 58)(8 59)(9 76)(10 77)(11 78)(12 79)(13 80)(14 73)(15 74)(16 75)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(25 138)(26 139)(27 140)(28 141)(29 142)(30 143)(31 144)(32 137)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 97)(56 98)(81 127)(82 128)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(105 119)(106 120)(107 113)(108 114)(109 115)(110 116)(111 117)(112 118)
(1 27 133)(2 134 28)(3 29 135)(4 136 30)(5 31 129)(6 130 32)(7 25 131)(8 132 26)(9 70 110)(10 111 71)(11 72 112)(12 105 65)(13 66 106)(14 107 67)(15 68 108)(16 109 69)(17 84 55)(18 56 85)(19 86 49)(20 50 87)(21 88 51)(22 52 81)(23 82 53)(24 54 83)(33 79 119)(34 120 80)(35 73 113)(36 114 74)(37 75 115)(38 116 76)(39 77 117)(40 118 78)(41 128 103)(42 104 121)(43 122 97)(44 98 123)(45 124 99)(46 100 125)(47 126 101)(48 102 127)(57 90 137)(58 138 91)(59 92 139)(60 140 93)(61 94 141)(62 142 95)(63 96 143)(64 144 89)
(1 20 13)(2 14 21)(3 22 15)(4 16 23)(5 24 9)(6 10 17)(7 18 11)(8 12 19)(25 56 72)(26 65 49)(27 50 66)(28 67 51)(29 52 68)(30 69 53)(31 54 70)(32 71 55)(33 99 139)(34 140 100)(35 101 141)(36 142 102)(37 103 143)(38 144 104)(39 97 137)(40 138 98)(41 63 75)(42 76 64)(43 57 77)(44 78 58)(45 59 79)(46 80 60)(47 61 73)(48 74 62)(81 108 135)(82 136 109)(83 110 129)(84 130 111)(85 112 131)(86 132 105)(87 106 133)(88 134 107)(89 121 116)(90 117 122)(91 123 118)(92 119 124)(93 125 120)(94 113 126)(95 127 114)(96 115 128)
(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)
(2 8)(3 7)(4 6)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)(25 135)(26 134)(27 133)(28 132)(29 131)(30 130)(31 129)(32 136)(33 126)(34 125)(35 124)(36 123)(37 122)(38 121)(39 128)(40 127)(41 77)(42 76)(43 75)(44 74)(45 73)(46 80)(47 79)(48 78)(49 107)(50 106)(51 105)(52 112)(53 111)(54 110)(55 109)(56 108)(57 63)(58 62)(59 61)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(89 144)(90 143)(91 142)(92 141)(93 140)(94 139)(95 138)(96 137)(97 115)(98 114)(99 113)(100 120)(101 119)(102 118)(103 117)(104 116)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 6A ··· 6L 6M ··· 6AB 8A 8B 8C 8D 12A ··· 12H order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 1 1 4 4 36 36 2 2 2 2 2 2 2 ··· 2 4 ··· 4 18 18 18 18 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 D8 C3⋊D4 C3⋊D4 D4⋊S3 kernel C2×C32⋊7D8 C2×C32⋊4C8 C32⋊7D8 C2×C12⋊S3 D4×C3×C6 C6×D4 C3×C12 C62 C2×C12 C3×D4 C3×C6 C12 C2×C6 C6 # reps 1 1 4 1 1 4 1 1 4 8 4 8 8 8

Matrix representation of C2×C327D8 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 72 0 0 0 0 1 0 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
,
 0 1 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 72 72 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 13 43 0 0 0 0 30 60 0 0 0 0 0 0 41 16 0 0 0 0 41 0
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 2 72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,13,30,0,0,0,0,43,60,0,0,0,0,0,0,41,41,0,0,0,0,16,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,2,0,0,0,0,0,72] >;

C2×C327D8 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_7D_8
% in TeX

G:=Group("C2xC3^2:7D8");
// GroupNames label

G:=SmallGroup(288,788);
// by ID

G=gap.SmallGroup(288,788);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,675,185,80,2693,9414]);
// Polycyclic

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

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