Copied to
clipboard

G = C68.4C4order 272 = 24·17

1st non-split extension by C68 of C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C68.4C4, C4.Dic17, C4.15D34, C174M4(2), C22.Dic17, C68.15C22, C173C85C2, (C2×C34).5C4, (C2×C68).5C2, (C2×C4).2D17, C34.14(C2×C4), C2.3(C2×Dic17), SmallGroup(272,10)

Series: Derived Chief Lower central Upper central

C1C34 — C68.4C4
C1C17C34C68C173C8 — C68.4C4
C17C34 — C68.4C4
C1C4C2×C4

Generators and relations for C68.4C4
 G = < a,b | a68=1, b4=a34, bab-1=a-1 >

2C2
2C34
17C8
17C8
17M4(2)

Smallest permutation representation of C68.4C4
On 136 points
Generators in S136
(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)
(1 111 18 94 35 77 52 128)(2 110 19 93 36 76 53 127)(3 109 20 92 37 75 54 126)(4 108 21 91 38 74 55 125)(5 107 22 90 39 73 56 124)(6 106 23 89 40 72 57 123)(7 105 24 88 41 71 58 122)(8 104 25 87 42 70 59 121)(9 103 26 86 43 69 60 120)(10 102 27 85 44 136 61 119)(11 101 28 84 45 135 62 118)(12 100 29 83 46 134 63 117)(13 99 30 82 47 133 64 116)(14 98 31 81 48 132 65 115)(15 97 32 80 49 131 66 114)(16 96 33 79 50 130 67 113)(17 95 34 78 51 129 68 112)

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

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

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

74 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D17A···17H34A···34X68A···68AF
order122444888817···1734···3468···68
size112112343434342···22···22···2

74 irreducible representations

dim11111222222
type++++-+-
imageC1C2C2C4C4M4(2)D17Dic17D34Dic17C68.4C4
kernelC68.4C4C173C8C2×C68C68C2×C34C17C2×C4C4C4C22C1
# reps121222888832

Matrix representation of C68.4C4 in GL2(𝔽137) generated by

440
0109
,
01
370
G:=sub<GL(2,GF(137))| [44,0,0,109],[0,37,1,0] >;

C68.4C4 in GAP, Magma, Sage, TeX

C_{68}._4C_4
% in TeX

G:=Group("C68.4C4");
// GroupNames label

G:=SmallGroup(272,10);
// by ID

G=gap.SmallGroup(272,10);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17,20,101,42,6404]);
// Polycyclic

G:=Group<a,b|a^68=1,b^4=a^34,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C68.4C4 in TeX

׿
×
𝔽