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## G = C2×D36⋊5C2order 288 = 25·32

### Direct product of C2 and D36⋊5C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C2×D36⋊5C2
 Chief series C1 — C3 — C9 — C18 — D18 — C22×D9 — C2×C4×D9 — C2×D36⋊5C2
 Lower central C9 — C18 — C2×D36⋊5C2
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×D365C2
G = < a,b,c,d | a2=b36=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd=b18c >

Subgroups: 952 in 246 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×10], S3 [×4], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], C9, Dic3 [×4], C12 [×4], D6 [×8], C2×C6, C2×C6 [×2], C2×C6 [×2], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], D9 [×4], C18, C18 [×2], C18 [×2], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×4], C22×S3 [×2], C22×C6, C2×C4○D4, Dic9 [×4], C36 [×4], D18 [×4], D18 [×4], C2×C18, C2×C18 [×2], C2×C18 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, Dic18 [×4], C4×D9 [×8], D36 [×4], C2×Dic9 [×2], C9⋊D4 [×8], C2×C36 [×2], C2×C36 [×4], C22×D9 [×2], C22×C18, C2×C4○D12, C2×Dic18, C2×C4×D9 [×2], C2×D36, D365C2 [×8], C2×C9⋊D4 [×2], C22×C36, C2×D365C2
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, D9, C22×S3 [×7], C2×C4○D4, D18 [×7], C4○D12 [×2], S3×C23, C22×D9 [×7], C2×C4○D12, D365C2 [×2], C23×D9, C2×D365C2

Smallest permutation representation of C2×D365C2
On 144 points
Generators in S144
(1 124)(2 125)(3 126)(4 127)(5 128)(6 129)(7 130)(8 131)(9 132)(10 133)(11 134)(12 135)(13 136)(14 137)(15 138)(16 139)(17 140)(18 141)(19 142)(20 143)(21 144)(22 109)(23 110)(24 111)(25 112)(26 113)(27 114)(28 115)(29 116)(30 117)(31 118)(32 119)(33 120)(34 121)(35 122)(36 123)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 97)(45 98)(46 99)(47 100)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 73)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)
(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)
(1 123)(2 122)(3 121)(4 120)(5 119)(6 118)(7 117)(8 116)(9 115)(10 114)(11 113)(12 112)(13 111)(14 110)(15 109)(16 144)(17 143)(18 142)(19 141)(20 140)(21 139)(22 138)(23 137)(24 136)(25 135)(26 134)(27 133)(28 132)(29 131)(30 130)(31 129)(32 128)(33 127)(34 126)(35 125)(36 124)(37 75)(38 74)(39 73)(40 108)(41 107)(42 106)(43 105)(44 104)(45 103)(46 102)(47 101)(48 100)(49 99)(50 98)(51 97)(52 96)(53 95)(54 94)(55 93)(56 92)(57 91)(58 90)(59 89)(60 88)(61 87)(62 86)(63 85)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 85)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 97)(25 98)(26 99)(27 100)(28 101)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 73)(37 140)(38 141)(39 142)(40 143)(41 144)(42 109)(43 110)(44 111)(45 112)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 127)(61 128)(62 129)(63 130)(64 131)(65 132)(66 133)(67 134)(68 135)(69 136)(70 137)(71 138)(72 139)

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

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

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

84 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 ··· 6G 9A 9B 9C 12A ··· 12H 18A ··· 18U 36A ··· 36X order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 18 18 18 18 2 1 1 1 1 2 2 18 18 18 18 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 C4○D4 D9 D18 D18 C4○D12 D36⋊5C2 kernel C2×D36⋊5C2 C2×Dic18 C2×C4×D9 C2×D36 D36⋊5C2 C2×C9⋊D4 C22×C36 C22×C12 C2×C12 C22×C6 C18 C22×C4 C2×C4 C23 C6 C2 # reps 1 1 2 1 8 2 1 1 6 1 4 3 18 3 8 24

Matrix representation of C2×D365C2 in GL3(𝔽37) generated by

 36 0 0 0 36 0 0 0 36
,
 36 0 0 0 25 33 0 4 29
,
 36 0 0 0 12 4 0 29 25
,
 36 0 0 0 30 23 0 14 7
G:=sub<GL(3,GF(37))| [36,0,0,0,36,0,0,0,36],[36,0,0,0,25,4,0,33,29],[36,0,0,0,12,29,0,4,25],[36,0,0,0,30,14,0,23,7] >;

C2×D365C2 in GAP, Magma, Sage, TeX

C_2\times D_{36}\rtimes_5C_2
% in TeX

G:=Group("C2xD36:5C2");
// GroupNames label

G:=SmallGroup(288,355);
// by ID

G=gap.SmallGroup(288,355);
# by ID

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

G:=Group<a,b,c,d|a^2=b^36=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d=b^18*c>;
// generators/relations

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