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G = C36⋊D4order 288 = 25·32

3rd semidirect product of C36 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C363D4, Dic91D4, C23.15D18, (C2×D4)⋊6D9, (D4×C18)⋊4C2, (C2×D36)⋊9C2, C41(C9⋊D4), C92(C41D4), (C6×D4).9S3, C2.28(D4×D9), C3.(C123D4), (C4×Dic9)⋊6C2, (C2×C4).53D18, (C2×C12).61D6, C6.103(S3×D4), C18.52(C2×D4), (C22×C6).53D6, C12.14(C3⋊D4), (C2×C18).55C23, (C2×C36).39C22, C22.62(C22×D9), (C22×C18).22C22, (C2×Dic9).41C22, (C22×D9).12C22, (C2×C9⋊D4)⋊7C2, C2.16(C2×C9⋊D4), C6.100(C2×C3⋊D4), (C2×C6).212(C22×S3), SmallGroup(288,150)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C36⋊D4
C1C3C9C18C2×C18C22×D9C2×D36 — C36⋊D4
C9C2×C18 — C36⋊D4
C1C22C2×D4

Generators and relations for C36⋊D4
 G = < a,b,c | a36=b4=c2=1, bab-1=a17, cac=a-1, cbc=b-1 >

Subgroups: 820 in 162 conjugacy classes, 48 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×4], C22, C22 [×12], S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×2], D4 [×12], C23 [×2], C23 [×2], C9, Dic3 [×4], C12 [×2], D6 [×6], C2×C6, C2×C6 [×6], C42, C2×D4, C2×D4 [×5], D9 [×2], C18, C18 [×2], C18 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12, C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C41D4, Dic9 [×4], C36 [×2], D18 [×6], C2×C18, C2×C18 [×6], C4×Dic3, C2×D12, C2×C3⋊D4 [×4], C6×D4, D36 [×2], C2×Dic9 [×2], C9⋊D4 [×8], C2×C36, D4×C9 [×2], C22×D9 [×2], C22×C18 [×2], C123D4, C4×Dic9, C2×D36, C2×C9⋊D4 [×4], D4×C18, C36⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D9, C3⋊D4 [×2], C22×S3, C41D4, D18 [×3], S3×D4 [×2], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C123D4, D4×D9 [×2], C2×C9⋊D4, C36⋊D4

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

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F6A6B6C6D6E6F6G9A9B9C12A12B18A···18I18J···18U36A···36F
order1222222234444446666666999121218···1818···1836···36
size1111443636222181818182224444222442···24···44···4

54 irreducible representations

dim11111222222222244
type+++++++++++++++
imageC1C2C2C2C2S3D4D4D6D6D9C3⋊D4D18D18C9⋊D4S3×D4D4×D9
kernelC36⋊D4C4×Dic9C2×D36C2×C9⋊D4D4×C18C6×D4Dic9C36C2×C12C22×C6C2×D4C12C2×C4C23C4C6C2
# reps111411421234361226

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

171100
26600
00201
00617
,
21300
113500
00360
00036
,
202600
61700
0010
003436
G:=sub<GL(4,GF(37))| [17,26,0,0,11,6,0,0,0,0,20,6,0,0,1,17],[2,11,0,0,13,35,0,0,0,0,36,0,0,0,0,36],[20,6,0,0,26,17,0,0,0,0,1,34,0,0,0,36] >;

C36⋊D4 in GAP, Magma, Sage, TeX

C_{36}\rtimes D_4
% in TeX

G:=Group("C36:D4");
// GroupNames label

G:=SmallGroup(288,150);
// by ID

G=gap.SmallGroup(288,150);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,219,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^36=b^4=c^2=1,b*a*b^-1=a^17,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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