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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

Aliases: D4×C36, C427C18, C4⋊C47C18, C3.(D4×C12), C41(C2×C36), C366(C2×C4), (D4×C12).C3, (C4×C36)⋊11C2, C2.3(D4×C18), C6.66(C6×D4), C22⋊C46C18, (C4×C12).16C6, C222(C2×C36), (C22×C4)⋊4C18, (C22×C36)⋊4C2, (C3×D4).4C12, (C2×D4).7C18, (C6×D4).26C6, C18.66(C2×D4), C12.86(C3×D4), C12.33(C2×C12), (D4×C18).14C2, C2.4(C22×C36), C18.39(C4○D4), C6.32(C22×C12), (C2×C18).73C23, (C22×C12).13C6, C23.12(C2×C18), C18.32(C22×C4), (C2×C36).120C22, C22.6(C22×C18), (C22×C18).26C22, (C9×C4⋊C4)⋊16C2, (C2×C18)⋊4(C2×C4), C2.2(C9×C4○D4), (C3×C4⋊C4).25C6, (C2×C6).9(C2×C12), C6.39(C3×C4○D4), (C9×C22⋊C4)⋊14C2, (C2×C4).18(C2×C18), (C2×C12).81(C2×C6), (C3×C22⋊C4).15C6, (C2×C6).78(C22×C6), (C22×C6).44(C2×C6), SmallGroup(288,168)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C36
C1C3C6C2×C6C2×C18C2×C36C9×C22⋊C4 — D4×C36
C1C2 — D4×C36
C1C2×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 2A2B2C2D2E2F2G3A3B4A4B4C4D4E···4L6A···6F6G···6N9A···9F12A···12H12I···12X18A···18R18S···18AP36A···36X36Y···36BT
order122222223344444···46···66···69···912···1212···1218···1818···1836···3636···36
size111122221111112···21···12···21···11···12···21···12···21···12···2

180 irreducible representations

dim111111111111111111111222222
type+++++++
imageC1C2C2C2C2C2C3C4C6C6C6C6C6C9C12C18C18C18C18C18C36D4C4○D4C3×D4C3×C4○D4D4×C9C9×C4○D4
kernelD4×C36C4×C36C9×C22⋊C4C9×C4⋊C4C22×C36D4×C18D4×C12D4×C9C4×C12C3×C22⋊C4C3×C4⋊C4C22×C12C6×D4C4×D4C3×D4C42C22⋊C4C4⋊C4C22×C4C2×D4D4C36C18C12C6C4C2
# reps112121282424261661261264822441212

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

30000
0800
00270
00027
,
36000
0100
001631
001221
,
1000
0100
001631
002421
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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