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G = C8.Dic9  order 288 = 25·32

1st non-split extension by C8 of Dic9 acting via Dic9/C18=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C72.1C4, C36.34D4, C4.18D36, C8.1Dic9, C12.53D12, C24.1Dic3, C22.2Dic18, (C2×C8).5D9, (C2×C72).7C2, C18.4(C4⋊C4), (C2×C18).3Q8, C9⋊1(C8.C4), C36.35(C2×C4), (C2×C24).14S3, (C2×C4).72D18, C4.6(C2×Dic9), C3.(C8.Dic3), (C2×C12).392D6, C6.7(C4⋊Dic3), C2.3(C4⋊Dic9), (C2×C6).12Dic6, C4.Dic9.1C2, (C2×C36).95C22, C12.41(C2×Dic3), SmallGroup(288,20)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C36 — C8.Dic9
Chief series
C1 — C3 — C9 — C18 — C36 — C2×C36 — C4.Dic9 — C8.Dic9
Lower central
C9 — C18 — C36 — C8.Dic9
Upper central
C1 — C4 — C2×C4 — C2×C8

Generators and relations for C8.Dic9
 G = < a,b,c | a8=1, b18=a4, c2=b9, ab=ba, cac-1=a3, cbc-1=b17 >

C1
C2
2C2
C3
C4
C22
C4
C6
2C6
C9
C2×C4
C8
C8
18C8
18C8
C12
C12
C2×C6
C18
2C18
C2×C8
9M4(2)
9M4(2)
C24
C24
C2×C12
6C3⋊C8
6C3⋊C8
C2×C18
C36
C36
9C8.C4
C2×C24
3C4.Dic3
3C4.Dic3
C2×C36
C72
C72
2C9⋊C8
2C9⋊C8
3C8.Dic3
C4.Dic9
C2×C72
C4.Dic9
C8.Dic9

Smallest permutation representation of C8.Dic9
►On 144 points
Generators in S144
(1 64 28 55 19 46 10 37)(2 65 29 56 20 47 11 38)(3 66 30 57 21 48 12 39)(4 67 31 58 22 49 13 40)(5 68 32 59 23 50 14 41)(6 69 33 60 24 51 15 42)(7 70 34 61 25 52 16 43)(8 71 35 62 26 53 17 44)(9 72 36 63 27 54 18 45)(73 114 82 123 91 132 100 141)(74 115 83 124 92 133 101 142)(75 116 84 125 93 134 102 143)(76 117 85 126 94 135 103 144)(77 118 86 127 95 136 104 109)(78 119 87 128 96 137 105 110)(79 120 88 129 97 138 106 111)(80 121 89 130 98 139 107 112)(81 122 90 131 99 140 108 113)

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

78 conjugacy classes

class 1 2A2B 3 4A4B4C6A6B6C8A8B8C8D8E8F8G8H9A9B9C12A12B12C12D18A···18I24A···24H36A···36L72A···72X
order1223444666888888889991212121218···1824···2436···3672···72
size112211222222223636363622222222···22···22···22···2

78 irreducible representations

dim1111222222222222222
type+++++--+++--++-
imageC1C2C2C4S3D4Q8Dic3D6D9D12Dic6C8.C4Dic9D18D36Dic18C8.Dic3C8.Dic9
kernelC8.Dic9C4.Dic9C2×C72C72C2×C24C36C2×C18C24C2×C12C2×C8C12C2×C6C9C8C2×C4C4C22C3C1
# reps12141112132246366824

Matrix representation of C8.Dic9 ►in GL2(𝔽73) generated by

510
010
,
540
050
,
01
270
G:=sub<GL(2,GF(73))| [51,0,0,10],[54,0,0,50],[0,27,1,0] >;
 

C8.Dic9 in GAP, Magma, Sage, TeX 

C_8.{\rm Dic}_9
 
% in TeX
 
G:=Group("C8.Dic9");
 
// GroupNames label
 
G:=SmallGroup(288,20);
 
// by ID
 
G=gap.SmallGroup(288,20);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,64,100,675,80,6725,292,9414]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=1,b^18=a^4,c^2=b^9,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^17>;
 
// generators/relations
 

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