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## G = C4×D36order 288 = 25·32

### Direct product of C4 and D36

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C4×D36
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — C2×D36 — C4×D36
 Lower central C9 — C18 — C4×D36
 Upper central C1 — C2×C4 — C42

Generators and relations for C4×D36
G = < a,b,c | a4=b36=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 656 in 141 conjugacy classes, 58 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×3], C22, C22 [×8], S3 [×4], C6 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], C9, Dic3 [×2], C12 [×4], C12, D6 [×8], C2×C6, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, D9 [×4], C18 [×3], C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×C12 [×3], C22×S3 [×2], C4×D4, Dic9 [×2], C36 [×4], C36, D18 [×4], D18 [×4], C2×C18, C4⋊Dic3, D6⋊C4 [×2], C4×C12, S3×C2×C4 [×2], C2×D12, C4×D9 [×4], D36 [×4], C2×Dic9 [×2], C2×C36 [×3], C22×D9 [×2], C4×D12, C4⋊Dic9, D18⋊C4 [×2], C4×C36, C2×C4×D9 [×2], C2×D36, C4×D36
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, D9, C4×S3 [×2], D12 [×2], C22×S3, C4×D4, D18 [×3], S3×C2×C4, C2×D12, C4○D12, C4×D9 [×2], D36 [×2], C22×D9, C4×D12, C2×C4×D9, C2×D36, D365C2, C4×D36

Smallest permutation representation of C4×D36
On 144 points
Generators in S144
(1 129 87 67)(2 130 88 68)(3 131 89 69)(4 132 90 70)(5 133 91 71)(6 134 92 72)(7 135 93 37)(8 136 94 38)(9 137 95 39)(10 138 96 40)(11 139 97 41)(12 140 98 42)(13 141 99 43)(14 142 100 44)(15 143 101 45)(16 144 102 46)(17 109 103 47)(18 110 104 48)(19 111 105 49)(20 112 106 50)(21 113 107 51)(22 114 108 52)(23 115 73 53)(24 116 74 54)(25 117 75 55)(26 118 76 56)(27 119 77 57)(28 120 78 58)(29 121 79 59)(30 122 80 60)(31 123 81 61)(32 124 82 62)(33 125 83 63)(34 126 84 64)(35 127 85 65)(36 128 86 66)
(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 77)(2 76)(3 75)(4 74)(5 73)(6 108)(7 107)(8 106)(9 105)(10 104)(11 103)(12 102)(13 101)(14 100)(15 99)(16 98)(17 97)(18 96)(19 95)(20 94)(21 93)(22 92)(23 91)(24 90)(25 89)(26 88)(27 87)(28 86)(29 85)(30 84)(31 83)(32 82)(33 81)(34 80)(35 79)(36 78)(37 113)(38 112)(39 111)(40 110)(41 109)(42 144)(43 143)(44 142)(45 141)(46 140)(47 139)(48 138)(49 137)(50 136)(51 135)(52 134)(53 133)(54 132)(55 131)(56 130)(57 129)(58 128)(59 127)(60 126)(61 125)(62 124)(63 123)(64 122)(65 121)(66 120)(67 119)(68 118)(69 117)(70 116)(71 115)(72 114)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 9A 9B 9C 12A ··· 12L 18A ··· 18I 36A ··· 36AJ order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 18 18 18 18 2 1 1 1 1 2 2 2 2 18 18 18 18 2 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 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D6 C4○D4 D9 C4×S3 D12 D18 C4○D12 C4×D9 D36 D36⋊5C2 kernel C4×D36 C4⋊Dic9 D18⋊C4 C4×C36 C2×C4×D9 C2×D36 D36 C4×C12 C36 C2×C12 C18 C42 C12 C12 C2×C4 C6 C4 C4 C2 # reps 1 1 2 1 2 1 8 1 2 3 2 3 4 4 9 4 12 12 12

Matrix representation of C4×D36 in GL3(𝔽37) generated by

 6 0 0 0 6 0 0 0 6
,
 36 0 0 0 4 8 0 29 12
,
 1 0 0 0 6 20 0 26 31
G:=sub<GL(3,GF(37))| [6,0,0,0,6,0,0,0,6],[36,0,0,0,4,29,0,8,12],[1,0,0,0,6,26,0,20,31] >;

C4×D36 in GAP, Magma, Sage, TeX

C_4\times D_{36}
% in TeX

G:=Group("C4xD36");
// GroupNames label

G:=SmallGroup(288,83);
// by ID

G=gap.SmallGroup(288,83);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,58,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^4=b^36=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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