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G = D36.2C4order 288 = 25·32

The non-split extension by D36 of C4 acting through Inn(D36)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D36.2C4, C24.90D6, C8.22D18, C72.23C22, C36.37C23, Dic18.2C4, (C2×C8)⋊7D9, (C8×D9)⋊6C2, C91(C8○D4), (C2×C72)⋊10C2, C8⋊D97C2, C3.(C8○D12), C4.10(C4×D9), C9⋊D4.2C4, C12.59(C4×S3), (C2×C24).29S3, C36.20(C2×C4), D18.1(C2×C4), (C2×C4).80D18, C9⋊C8.11C22, C22.2(C4×D9), (C2×C12).393D6, C4.Dic911C2, Dic9.3(C2×C4), C4.37(C22×D9), D365C2.6C2, C18.14(C22×C4), (C2×C36).96C22, (C4×D9).15C22, C12.198(C22×S3), C6.53(S3×C2×C4), C2.15(C2×C4×D9), (C2×C6).41(C4×S3), (C2×C18).15(C2×C4), SmallGroup(288,112)

Series: Derived Chief Lower central Upper central

C1C18 — D36.2C4
C1C3C9C18C36C4×D9D365C2 — D36.2C4
C9C18 — D36.2C4
C1C8C2×C8

Generators and relations for D36.2C4
 G = < a,b,c | a36=b2=1, c4=a18, bab=a-1, ac=ca, bc=cb >

Subgroups: 332 in 93 conjugacy classes, 46 normal (32 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6, C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, C9, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, D9 [×2], C18, C18, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, C8○D4, Dic9 [×2], C36 [×2], D18 [×2], C2×C18, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C2×C24, C4○D12, C9⋊C8 [×2], C72 [×2], Dic18, C4×D9 [×2], D36, C9⋊D4 [×2], C2×C36, C8○D12, C8×D9 [×2], C8⋊D9 [×2], C4.Dic9, C2×C72, D365C2, D36.2C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, D9, C4×S3 [×2], C22×S3, C8○D4, D18 [×3], S3×C2×C4, C4×D9 [×2], C22×D9, C8○D12, C2×C4×D9, D36.2C4

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

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

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

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

84 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E6A6B6C8A8B8C8D8E8F8G8H8I8J9A9B9C12A12B12C12D18A···18I24A···24H36A···36L72A···72X
order1222234444466688888888889991212121218···1824···2436···3672···72
size1121818211218182221111221818181822222222···22···22···22···2

84 irreducible representations

dim1111111112222222222222
type++++++++++++
imageC1C2C2C2C2C2C4C4C4S3D6D6D9C4×S3C4×S3C8○D4D18D18C4×D9C4×D9C8○D12D36.2C4
kernelD36.2C4C8×D9C8⋊D9C4.Dic9C2×C72D365C2Dic18D36C9⋊D4C2×C24C24C2×C12C2×C8C12C2×C6C9C8C2×C4C4C22C3C1
# reps12211122412132246366824

Matrix representation of D36.2C4 in GL4(𝔽73) generated by

66700
665900
003145
00283
,
1100
07200
00721
0001
,
51000
05100
00720
00072
G:=sub<GL(4,GF(73))| [66,66,0,0,7,59,0,0,0,0,31,28,0,0,45,3],[1,0,0,0,1,72,0,0,0,0,72,0,0,0,1,1],[51,0,0,0,0,51,0,0,0,0,72,0,0,0,0,72] >;

D36.2C4 in GAP, Magma, Sage, TeX

D_{36}._2C_4
% in TeX

G:=Group("D36.2C4");
// GroupNames label

G:=SmallGroup(288,112);
// by ID

G=gap.SmallGroup(288,112);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,58,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^36=b^2=1,c^4=a^18,b*a*b=a^-1,a*c=c*a,b*c=c*b>;
// generators/relations

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