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G = C8.D18order 288 = 25·32

1st non-split extension by C8 of D18 acting via D18/C9=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.2D6, C8.1D18, C12.8D12, C36.13D4, C4.15D36, Dic362C2, M4(2)⋊2D9, C72.1C22, C22.6D36, C36.33C23, D36.8C22, Dic18.8C22, C72⋊C22C2, (C2×C6).7D12, (C2×C18).6D4, (C2×C12).54D6, C2.16(C2×D36), (C2×C4).13D18, C18.14(C2×D4), C6.43(C2×D12), C91(C8.C22), C3.(C8.D6), (C2×Dic18)⋊8C2, (C9×M4(2))⋊2C2, C4.31(C22×D9), D365C2.4C2, (C2×C36).32C22, (C3×M4(2)).2S3, C12.184(C22×S3), SmallGroup(288,119)

Series: Derived Chief Lower central Upper central

C1C36 — C8.D18
C1C3C9C18C36D36D365C2 — C8.D18
C9C18C36 — C8.D18
C1C2C2×C4M4(2)

Generators and relations for C8.D18
 G = < a,b,c | a8=1, b18=c2=a4, bab-1=a5, cac-1=a-1, cbc-1=b17 >

Subgroups: 436 in 90 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C9, Dic3, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, D9, C18, C18, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C8.C22, Dic9, C36, D18, C2×C18, C24⋊C2, Dic12, C3×M4(2), C2×Dic6, C4○D12, C72, Dic18, Dic18, Dic18, C4×D9, D36, C2×Dic9, C9⋊D4, C2×C36, C8.D6, Dic36, C72⋊C2, C9×M4(2), C2×Dic18, D365C2, C8.D18
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, D12, C22×S3, C8.C22, D18, C2×D12, D36, C22×D9, C8.D6, C2×D36, C8.D18

Smallest permutation representation of C8.D18
On 144 points
Generators in S144
(1 57 84 127 19 39 102 109)(2 40 85 110 20 58 103 128)(3 59 86 129 21 41 104 111)(4 42 87 112 22 60 105 130)(5 61 88 131 23 43 106 113)(6 44 89 114 24 62 107 132)(7 63 90 133 25 45 108 115)(8 46 91 116 26 64 73 134)(9 65 92 135 27 47 74 117)(10 48 93 118 28 66 75 136)(11 67 94 137 29 49 76 119)(12 50 95 120 30 68 77 138)(13 69 96 139 31 51 78 121)(14 52 97 122 32 70 79 140)(15 71 98 141 33 53 80 123)(16 54 99 124 34 72 81 142)(17 37 100 143 35 55 82 125)(18 56 101 126 36 38 83 144)
(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 18 19 36)(2 35 20 17)(3 16 21 34)(4 33 22 15)(5 14 23 32)(6 31 24 13)(7 12 25 30)(8 29 26 11)(9 10 27 28)(37 128 55 110)(38 109 56 127)(39 126 57 144)(40 143 58 125)(41 124 59 142)(42 141 60 123)(43 122 61 140)(44 139 62 121)(45 120 63 138)(46 137 64 119)(47 118 65 136)(48 135 66 117)(49 116 67 134)(50 133 68 115)(51 114 69 132)(52 131 70 113)(53 112 71 130)(54 129 72 111)(73 76 91 94)(74 93 92 75)(77 108 95 90)(78 89 96 107)(79 106 97 88)(80 87 98 105)(81 104 99 86)(82 85 100 103)(83 102 101 84)

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

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

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

51 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E6A6B8A8B9A9B9C12A12B12C18A18B18C18D18E18F24A24B24C24D36A···36F36G36H36I72A···72L
order122234444466889991212121818181818182424242436···3636363672···72
size11236222363636244422222422244444442···24444···4

51 irreducible representations

dim111111222222222222444
type++++++++++++++++++---
imageC1C2C2C2C2C2S3D4D4D6D6D9D12D12D18D18D36D36C8.C22C8.D6C8.D18
kernelC8.D18Dic36C72⋊C2C9×M4(2)C2×Dic18D365C2C3×M4(2)C36C2×C18C24C2×C12M4(2)C12C2×C6C8C2×C4C4C22C9C3C1
# reps122111111213226366126

Matrix representation of C8.D18 in GL6(𝔽73)

7200000
0720000
00605470
003456013
0013214739
0034344734
,
70450000
28420000
005506868
003460680
0052345239
0013393952
,
3420000
45700000
005506868
002113068
0013393952
0052345239

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,60,34,13,34,0,0,5,5,21,34,0,0,47,60,47,47,0,0,0,13,39,34],[70,28,0,0,0,0,45,42,0,0,0,0,0,0,55,34,52,13,0,0,0,60,34,39,0,0,68,68,52,39,0,0,68,0,39,52],[3,45,0,0,0,0,42,70,0,0,0,0,0,0,55,21,13,52,0,0,0,13,39,34,0,0,68,0,39,52,0,0,68,68,52,39] >;

C8.D18 in GAP, Magma, Sage, TeX

C_8.D_{18}
% in TeX

G:=Group("C8.D18");
// GroupNames label

G:=SmallGroup(288,119);
// by ID

G=gap.SmallGroup(288,119);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,58,675,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^18=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^17>;
// generators/relations

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