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G = C72.C4order 288 = 25·32

1st non-split extension by C72 of C4 acting via C4/C2=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, C91(C8.C4), (C2×C24).14S3, C36.35(C2×C4), (C2×C4).72D18, C4.6(C2×Dic9), C3.(C24.C4), (C2×C12).392D6, C2.3(C4⋊Dic9), C6.7(C4⋊Dic3), (C2×C6).12Dic6, C4.Dic9.1C2, (C2×C36).95C22, C12.41(C2×Dic3), SmallGroup(288,20)

Series: Derived Chief Lower central Upper central

C1C36 — C72.C4
C1C3C9C18C36C2×C36C4.Dic9 — C72.C4
C9C18C36 — C72.C4
C1C4C2×C4C2×C8

Generators and relations for C72.C4
 G = < a,b,c | a8=1, b18=a4, c2=b9, ab=ba, cac-1=a3, cbc-1=b17 >

2C2
2C6
18C8
18C8
2C18
9M4(2)
9M4(2)
6C3⋊C8
6C3⋊C8
9C8.C4
3C4.Dic3
3C4.Dic3
2C9⋊C8
2C9⋊C8
3C24.C4

Smallest permutation representation of C72.C4
On 144 points
Generators in S144
(1 59 28 50 19 41 10 68)(2 60 29 51 20 42 11 69)(3 61 30 52 21 43 12 70)(4 62 31 53 22 44 13 71)(5 63 32 54 23 45 14 72)(6 64 33 55 24 46 15 37)(7 65 34 56 25 47 16 38)(8 66 35 57 26 48 17 39)(9 67 36 58 27 49 18 40)(73 113 82 122 91 131 100 140)(74 114 83 123 92 132 101 141)(75 115 84 124 93 133 102 142)(76 116 85 125 94 134 103 143)(77 117 86 126 95 135 104 144)(78 118 87 127 96 136 105 109)(79 119 88 128 97 137 106 110)(80 120 89 129 98 138 107 111)(81 121 90 130 99 139 108 112)
(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 81 10 90 19 99 28 108)(2 98 11 107 20 80 29 89)(3 79 12 88 21 97 30 106)(4 96 13 105 22 78 31 87)(5 77 14 86 23 95 32 104)(6 94 15 103 24 76 33 85)(7 75 16 84 25 93 34 102)(8 92 17 101 26 74 35 83)(9 73 18 82 27 91 36 100)(37 116 46 125 55 134 64 143)(38 133 47 142 56 115 65 124)(39 114 48 123 57 132 66 141)(40 131 49 140 58 113 67 122)(41 112 50 121 59 130 68 139)(42 129 51 138 60 111 69 120)(43 110 52 119 61 128 70 137)(44 127 53 136 62 109 71 118)(45 144 54 117 63 126 72 135)

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

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

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

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.C4Dic9D18D36Dic18C24.C4C72.C4
kernelC72.C4C4.Dic9C2×C72C72C2×C24C36C2×C18C24C2×C12C2×C8C12C2×C6C9C8C2×C4C4C22C3C1
# reps12141112132246366824

Matrix representation of C72.C4 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] >;

C72.C4 in GAP, Magma, Sage, TeX

C_{72}.C_4
% in TeX

G:=Group("C72.C4");
// 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

Export

Subgroup lattice of C72.C4 in TeX

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