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G = C36.53D4order 288 = 25·32

9th non-split extension by C36 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C36.53D4, M4(2).1D9, C22.1Dic18, C9⋊C8.1C4, (C2×C18).Q8, C12.5(C4×S3), C36.5(C2×C4), C4.13(C4×D9), C18.8(C4⋊C4), C92(C8.C4), (C2×C4).39D18, (C2×C12).41D6, (C2×C6).2Dic6, C4.28(C9⋊D4), C4.Dic9.2C2, C2.5(Dic9⋊C4), (C2×C36).19C22, C3.(C12.53D4), (C3×M4(2)).5S3, (C9×M4(2)).1C2, C12.123(C3⋊D4), C6.13(Dic3⋊C4), (C2×C9⋊C8).4C2, SmallGroup(288,29)

Series: Derived Chief Lower central Upper central

C1C36 — C36.53D4
C1C3C9C18C36C2×C36C2×C9⋊C8 — C36.53D4
C9C18C36 — C36.53D4
C1C4C2×C4M4(2)

Generators and relations for C36.53D4
 G = < a,b,c | a36=1, b4=a18, c2=a9, bab-1=cac-1=a17, cbc-1=a18b3 >

2C2
2C6
2C8
9C8
9C8
18C8
2C18
9M4(2)
9C2×C8
2C24
3C3⋊C8
3C3⋊C8
6C3⋊C8
9C8.C4
3C4.Dic3
3C2×C3⋊C8
2C9⋊C8
2C72
3C12.53D4

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

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

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

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

54 conjugacy classes

class 1 2A2B 3 4A4B4C6A6B8A8B8C8D8E8F8G8H9A9B9C12A12B12C18A18B18C18D18E18F24A24B24C24D36A···36F36G36H36I72A···72L
order122344466888888889991212121818181818182424242436···3636363672···72
size1122112244418181818363622222422244444442···24444···4

54 irreducible representations

dim11111222222222222244
type++++++-++-+-
imageC1C2C2C2C4S3D4Q8D6D9C4×S3C3⋊D4Dic6C8.C4D18C4×D9C9⋊D4Dic18C12.53D4C36.53D4
kernelC36.53D4C2×C9⋊C8C4.Dic9C9×M4(2)C9⋊C8C3×M4(2)C36C2×C18C2×C12M4(2)C12C12C2×C6C9C2×C4C4C4C22C3C1
# reps11114111132224366626

Matrix representation of C36.53D4 in GL4(𝔽73) generated by

32800
453100
00460
00046
,
591900
51400
00630
001251
,
511000
612200
002251
00051
G:=sub<GL(4,GF(73))| [3,45,0,0,28,31,0,0,0,0,46,0,0,0,0,46],[59,5,0,0,19,14,0,0,0,0,63,12,0,0,0,51],[51,61,0,0,10,22,0,0,0,0,22,0,0,0,51,51] >;

C36.53D4 in GAP, Magma, Sage, TeX

C_{36}._{53}D_4
% in TeX

G:=Group("C36.53D4");
// GroupNames label

G:=SmallGroup(288,29);
// by ID

G=gap.SmallGroup(288,29);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,346,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^36=1,b^4=a^18,c^2=a^9,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^18*b^3>;
// generators/relations

Export

Subgroup lattice of C36.53D4 in TeX

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