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## G = D4×C36order 288 = 25·32

### Direct product of C36 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C36
 Chief series C1 — C3 — C6 — C2×C6 — C2×C18 — C2×C36 — C9×C22⋊C4 — D4×C36
 Lower central C1 — C2 — D4×C36
 Upper central C1 — C2×C36 — D4×C36

Generators and relations for D4×C36
G = < a,b,c | a36=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 186 in 141 conjugacy classes, 96 normal (36 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×3], C22, C22 [×4], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×4], C23 [×2], C9, C12 [×4], C12 [×3], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C18 [×3], C18 [×4], C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×4], C22×C6 [×2], C4×D4, C36 [×4], C36 [×3], C2×C18, C2×C18 [×4], C2×C18 [×4], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C22×C12 [×2], C6×D4, C2×C36 [×3], C2×C36 [×2], C2×C36 [×4], D4×C9 [×4], C22×C18 [×2], D4×C12, C4×C36, C9×C22⋊C4 [×2], C9×C4⋊C4, C22×C36 [×2], D4×C18, D4×C36
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×2], C23, C9, C12 [×4], C2×C6 [×7], C22×C4, C2×D4, C4○D4, C18 [×7], C2×C12 [×6], C3×D4 [×2], C22×C6, C4×D4, C36 [×4], C2×C18 [×7], C22×C12, C6×D4, C3×C4○D4, C2×C36 [×6], D4×C9 [×2], C22×C18, D4×C12, C22×C36, D4×C18, C9×C4○D4, D4×C36

Smallest permutation representation of D4×C36
On 144 points
Generators in S144
(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 40 93 111)(2 41 94 112)(3 42 95 113)(4 43 96 114)(5 44 97 115)(6 45 98 116)(7 46 99 117)(8 47 100 118)(9 48 101 119)(10 49 102 120)(11 50 103 121)(12 51 104 122)(13 52 105 123)(14 53 106 124)(15 54 107 125)(16 55 108 126)(17 56 73 127)(18 57 74 128)(19 58 75 129)(20 59 76 130)(21 60 77 131)(22 61 78 132)(23 62 79 133)(24 63 80 134)(25 64 81 135)(26 65 82 136)(27 66 83 137)(28 67 84 138)(29 68 85 139)(30 69 86 140)(31 70 87 141)(32 71 88 142)(33 72 89 143)(34 37 90 144)(35 38 91 109)(36 39 92 110)
(1 129)(2 130)(3 131)(4 132)(5 133)(6 134)(7 135)(8 136)(9 137)(10 138)(11 139)(12 140)(13 141)(14 142)(15 143)(16 144)(17 109)(18 110)(19 111)(20 112)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 108)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)(71 106)(72 107)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E ··· 4L 6A ··· 6F 6G ··· 6N 9A ··· 9F 12A ··· 12H 12I ··· 12X 18A ··· 18R 18S ··· 18AP 36A ··· 36X 36Y ··· 36BT order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 1 1 2 2 2 2 1 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C9 C12 C18 C18 C18 C18 C18 C36 D4 C4○D4 C3×D4 C3×C4○D4 D4×C9 C9×C4○D4 kernel D4×C36 C4×C36 C9×C22⋊C4 C9×C4⋊C4 C22×C36 D4×C18 D4×C12 D4×C9 C4×C12 C3×C22⋊C4 C3×C4⋊C4 C22×C12 C6×D4 C4×D4 C3×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C36 C18 C12 C6 C4 C2 # reps 1 1 2 1 2 1 2 8 2 4 2 4 2 6 16 6 12 6 12 6 48 2 2 4 4 12 12

Matrix representation of D4×C36 in GL4(𝔽37) generated by

 30 0 0 0 0 8 0 0 0 0 27 0 0 0 0 27
,
 36 0 0 0 0 1 0 0 0 0 16 31 0 0 12 21
,
 1 0 0 0 0 1 0 0 0 0 16 31 0 0 24 21
G:=sub<GL(4,GF(37))| [30,0,0,0,0,8,0,0,0,0,27,0,0,0,0,27],[36,0,0,0,0,1,0,0,0,0,16,12,0,0,31,21],[1,0,0,0,0,1,0,0,0,0,16,24,0,0,31,21] >;

D4×C36 in GAP, Magma, Sage, TeX

D_4\times C_{36}
% in TeX

G:=Group("D4xC36");
// GroupNames label

G:=SmallGroup(288,168);
// by ID

G=gap.SmallGroup(288,168);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,365,268,360]);
// Polycyclic

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

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