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G = C36.C8order 288 = 25·32

1st non-split extension by C36 of C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C36.1C8, C72.4C4, C92M5(2), C8.21D18, C24.87D6, C8.2Dic9, C24.7Dic3, C72.22C22, C4.(C9⋊C8), C9⋊C165C2, C22.(C9⋊C8), (C2×C8).7D9, C12.2(C3⋊C8), (C2×C18).3C8, C18.9(C2×C8), (C2×C24).24S3, C36.41(C2×C4), (C2×C36).10C4, (C2×C72).10C2, (C2×C4).5Dic9, C3.(C12.C8), C4.10(C2×Dic9), C12.50(C2×Dic3), (C2×C12).17Dic3, C6.9(C2×C3⋊C8), C2.4(C2×C9⋊C8), (C2×C6).4(C3⋊C8), SmallGroup(288,19)

Series: Derived Chief Lower central Upper central

C1C18 — C36.C8
C1C3C9C18C36C72C9⋊C16 — C36.C8
C9C18 — C36.C8
C1C8C2×C8

Generators and relations for C36.C8
 G = < a,b | a72=1, b4=a18, bab-1=a53 >

2C2
2C6
2C18
9C16
9C16
9M5(2)
3C3⋊C16
3C3⋊C16
3C12.C8

Smallest permutation representation of C36.C8
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)
(1 127 10 100 19 73 28 118 37 91 46 136 55 109 64 82)(2 108 11 81 20 126 29 99 38 144 47 117 56 90 65 135)(3 89 12 134 21 107 30 80 39 125 48 98 57 143 66 116)(4 142 13 115 22 88 31 133 40 106 49 79 58 124 67 97)(5 123 14 96 23 141 32 114 41 87 50 132 59 105 68 78)(6 104 15 77 24 122 33 95 42 140 51 113 60 86 69 131)(7 85 16 130 25 103 34 76 43 121 52 94 61 139 70 112)(8 138 17 111 26 84 35 129 44 102 53 75 62 120 71 93)(9 119 18 92 27 137 36 110 45 83 54 128 63 101 72 74)

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

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

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

84 conjugacy classes

class 1 2A2B 3 4A4B4C6A6B6C8A8B8C8D8E8F9A9B9C12A12B12C12D16A···16H18A···18I24A···24H36A···36L72A···72X
order12234446668888889991212121216···1618···1824···2436···3672···72
size1122112222111122222222218···182···22···22···22···2

84 irreducible representations

dim1111111222222222222222
type++++-+-+-+-
imageC1C2C2C4C4C8C8S3Dic3D6Dic3D9C3⋊C8C3⋊C8M5(2)Dic9D18Dic9C9⋊C8C9⋊C8C12.C8C36.C8
kernelC36.C8C9⋊C16C2×C72C72C2×C36C36C2×C18C2×C24C24C24C2×C12C2×C8C12C2×C6C9C8C8C2×C4C4C22C3C1
# reps12122441111322433366824

Matrix representation of C36.C8 in GL2(𝔽433) generated by

3580
0217
,
01
2850
G:=sub<GL(2,GF(433))| [358,0,0,217],[0,285,1,0] >;

C36.C8 in GAP, Magma, Sage, TeX

C_{36}.C_8
% in TeX

G:=Group("C36.C8");
// GroupNames label

G:=SmallGroup(288,19);
// by ID

G=gap.SmallGroup(288,19);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,58,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b|a^72=1,b^4=a^18,b*a*b^-1=a^53>;
// generators/relations

Export

Subgroup lattice of C36.C8 in TeX

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