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## G = C2×D4⋊2D9order 288 = 25·32

### Direct product of C2 and D4⋊2D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C2×D4⋊2D9
 Chief series C1 — C3 — C9 — C18 — D18 — C22×D9 — C2×C4×D9 — C2×D4⋊2D9
 Lower central C9 — C18 — C2×D4⋊2D9
 Upper central C1 — C22 — C2×D4

Generators and relations for C2×D42D9
G = < a,b,c,d,e | a2=b4=c2=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 872 in 246 conjugacy classes, 108 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×4], C22 [×8], S3 [×2], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×15], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, C9, Dic3 [×6], C12 [×2], D6 [×4], C2×C6, C2×C6 [×4], C2×C6 [×4], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], D9 [×2], C18, C18 [×2], C18 [×4], Dic6 [×4], C4×S3 [×4], C2×Dic3 [×11], C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×C6 [×2], C2×C4○D4, Dic9 [×6], C36 [×2], D18 [×2], D18 [×2], C2×C18, C2×C18 [×4], C2×C18 [×4], C2×Dic6, S3×C2×C4, D42S3 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, Dic18 [×4], C4×D9 [×4], C2×Dic9, C2×Dic9 [×10], C9⋊D4 [×8], C2×C36, D4×C9 [×4], C22×D9, C22×C18 [×2], C2×D42S3, C2×Dic18, C2×C4×D9, D42D9 [×8], C22×Dic9 [×2], C2×C9⋊D4 [×2], D4×C18, C2×D42D9
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, D9, C22×S3 [×7], C2×C4○D4, D18 [×7], D42S3 [×2], S3×C23, C22×D9 [×7], C2×D42S3, D42D9 [×2], C23×D9, C2×D42D9

Smallest permutation representation of C2×D42D9
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)
(1 109 10 118)(2 110 11 119)(3 111 12 120)(4 112 13 121)(5 113 14 122)(6 114 15 123)(7 115 16 124)(8 116 17 125)(9 117 18 126)(19 127 28 136)(20 128 29 137)(21 129 30 138)(22 130 31 139)(23 131 32 140)(24 132 33 141)(25 133 34 142)(26 134 35 143)(27 135 36 144)(37 82 46 73)(38 83 47 74)(39 84 48 75)(40 85 49 76)(41 86 50 77)(42 87 51 78)(43 88 52 79)(44 89 53 80)(45 90 54 81)(55 100 64 91)(56 101 65 92)(57 102 66 93)(58 103 67 94)(59 104 68 95)(60 105 69 96)(61 106 70 97)(62 107 71 98)(63 108 72 99)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 136)(38 137)(39 138)(40 139)(41 140)(42 141)(43 142)(44 143)(45 144)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 133)(53 134)(54 135)(55 118)(56 119)(57 120)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)
(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 9)(2 8)(3 7)(4 6)(10 18)(11 17)(12 16)(13 15)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(37 45)(38 44)(39 43)(40 42)(46 54)(47 53)(48 52)(49 51)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(73 81)(74 80)(75 79)(76 78)(82 90)(83 89)(84 88)(85 87)(91 108)(92 107)(93 106)(94 105)(95 104)(96 103)(97 102)(98 101)(99 100)(109 117)(110 116)(111 115)(112 114)(118 126)(119 125)(120 124)(121 123)(127 144)(128 143)(129 142)(130 141)(131 140)(132 139)(133 138)(134 137)(135 136)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 12A 12B 18A ··· 18I 18J ··· 18U 36A ··· 36F order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 9 9 9 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 2 18 18 2 2 2 9 9 9 9 18 18 18 18 2 2 2 4 4 4 4 2 2 2 4 4 2 ··· 2 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 C4○D4 D9 D18 D18 D18 D4⋊2S3 D4⋊2D9 kernel C2×D4⋊2D9 C2×Dic18 C2×C4×D9 D4⋊2D9 C22×Dic9 C2×C9⋊D4 D4×C18 C6×D4 C2×C12 C3×D4 C22×C6 C18 C2×D4 C2×C4 D4 C23 C6 C2 # reps 1 1 1 8 2 2 1 1 1 4 2 4 3 3 12 6 2 6

Matrix representation of C2×D42D9 in GL5(𝔽37)

 36 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 36 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 31 0 0 0 0 18 6
,
 36 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36 24 0 0 0 0 1
,
 1 0 0 0 0 0 11 17 0 0 0 20 31 0 0 0 0 0 1 0 0 0 0 0 1
,
 36 0 0 0 0 0 11 17 0 0 0 6 26 0 0 0 0 0 1 0 0 0 0 34 36

G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[36,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,31,18,0,0,0,0,6],[36,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,24,1],[1,0,0,0,0,0,11,20,0,0,0,17,31,0,0,0,0,0,1,0,0,0,0,0,1],[36,0,0,0,0,0,11,6,0,0,0,17,26,0,0,0,0,0,1,34,0,0,0,0,36] >;

C2×D42D9 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_2D_9
% in TeX

G:=Group("C2xD4:2D9");
// GroupNames label

G:=SmallGroup(288,357);
// by ID

G=gap.SmallGroup(288,357);
# by ID

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

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

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