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## G = C2×Q8⋊2D9order 288 = 25·32

### Direct product of C2 and Q8⋊2D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C2×Q8⋊2D9
 Chief series C1 — C3 — C9 — C18 — C36 — D36 — C2×D36 — C2×Q8⋊2D9
 Lower central C9 — C18 — C36 — C2×Q8⋊2D9
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for C2×Q82D9
G = < a,b,c,d,e | a2=b4=d9=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

Subgroups: 512 in 102 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×2], Q8, C23, C9, C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C8, SD16 [×4], C2×D4, C2×Q8, D9 [×2], C18, C18 [×2], C3⋊C8 [×2], D12 [×3], C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, C22×S3, C2×SD16, C36 [×2], C36 [×2], D18 [×4], C2×C18, C2×C3⋊C8, Q82S3 [×4], C2×D12, C6×Q8, C9⋊C8 [×2], D36 [×2], D36, C2×C36, C2×C36, Q8×C9 [×2], Q8×C9, C22×D9, C2×Q82S3, C2×C9⋊C8, Q82D9 [×4], C2×D36, Q8×C18, C2×Q82D9
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], C2×D4, D9, C3⋊D4 [×2], C22×S3, C2×SD16, D18 [×3], Q82S3 [×2], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C2×Q82S3, Q82D9 [×2], C2×C9⋊D4, C2×Q82D9

Smallest permutation representation of C2×Q82D9
On 144 points
Generators in S144
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(1 28 10 19)(2 29 11 20)(3 30 12 21)(4 31 13 22)(5 32 14 23)(6 33 15 24)(7 34 16 25)(8 35 17 26)(9 36 18 27)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)(73 100 82 91)(74 101 83 92)(75 102 84 93)(76 103 85 94)(77 104 86 95)(78 105 87 96)(79 106 88 97)(80 107 89 98)(81 108 90 99)(109 127 118 136)(110 128 119 137)(111 129 120 138)(112 130 121 139)(113 131 122 140)(114 132 123 141)(115 133 124 142)(116 134 125 143)(117 135 126 144)
(1 118 10 109)(2 119 11 110)(3 120 12 111)(4 121 13 112)(5 122 14 113)(6 123 15 114)(7 124 16 115)(8 125 17 116)(9 126 18 117)(19 136 28 127)(20 137 29 128)(21 138 30 129)(22 139 31 130)(23 140 32 131)(24 141 33 132)(25 142 34 133)(26 143 35 134)(27 144 36 135)(37 73 46 82)(38 74 47 83)(39 75 48 84)(40 76 49 85)(41 77 50 86)(42 78 51 87)(43 79 52 88)(44 80 53 89)(45 81 54 90)(55 91 64 100)(56 92 65 101)(57 93 66 102)(58 94 67 103)(59 95 68 104)(60 96 69 105)(61 97 70 106)(62 98 71 107)(63 99 72 108)
(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 81)(2 80)(3 79)(4 78)(5 77)(6 76)(7 75)(8 74)(9 73)(10 90)(11 89)(12 88)(13 87)(14 86)(15 85)(16 84)(17 83)(18 82)(19 108)(20 107)(21 106)(22 105)(23 104)(24 103)(25 102)(26 101)(27 100)(28 99)(29 98)(30 97)(31 96)(32 95)(33 94)(34 93)(35 92)(36 91)(37 135)(38 134)(39 133)(40 132)(41 131)(42 130)(43 129)(44 128)(45 127)(46 144)(47 143)(48 142)(49 141)(50 140)(51 139)(52 138)(53 137)(54 136)(55 117)(56 116)(57 115)(58 114)(59 113)(60 112)(61 111)(62 110)(63 109)(64 126)(65 125)(66 124)(67 123)(68 122)(69 121)(70 120)(71 119)(72 118)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 9A 9B 9C 12A ··· 12F 18A ··· 18I 36A ··· 36R order 1 2 2 2 2 2 3 4 4 4 4 6 6 6 8 8 8 8 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 36 36 2 2 2 4 4 2 2 2 18 18 18 18 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 SD16 D9 C3⋊D4 C3⋊D4 D18 D18 C9⋊D4 C9⋊D4 Q8⋊2S3 Q8⋊2D9 kernel C2×Q8⋊2D9 C2×C9⋊C8 Q8⋊2D9 C2×D36 Q8×C18 C6×Q8 C36 C2×C18 C2×C12 C3×Q8 C18 C2×Q8 C12 C2×C6 C2×C4 Q8 C4 C22 C6 C2 # reps 1 1 4 1 1 1 1 1 1 2 4 3 2 2 3 6 6 6 2 6

Matrix representation of C2×Q82D9 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 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 30 60 0 0 0 0 13 43 0 0 0 0 0 0 0 71 0 0 0 0 36 0 0 0 0 0 0 0 6 6 0 0 0 0 6 67
,
 45 42 0 0 0 0 31 3 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
,
 45 42 0 0 0 0 70 28 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 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],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[30,13,0,0,0,0,60,43,0,0,0,0,0,0,0,36,0,0,0,0,71,0,0,0,0,0,0,0,6,6,0,0,0,0,6,67],[45,31,0,0,0,0,42,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[45,70,0,0,0,0,42,28,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;

C2×Q82D9 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_2D_9
% in TeX

G:=Group("C2xQ8:2D9");
// GroupNames label

G:=SmallGroup(288,152);
// by ID

G=gap.SmallGroup(288,152);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,100,675,185,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^9=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

׿
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