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## G = C68.4C4order 272 = 24·17

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

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

 Derived series C1 — C34 — C68.4C4
 Chief series C1 — C17 — C34 — C68 — C17⋊3C8 — C68.4C4
 Lower central C17 — C34 — C68.4C4
 Upper central C1 — C4 — C2×C4

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

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 2A 2B 4A 4B 4C 8A 8B 8C 8D 17A ··· 17H 34A ··· 34X 68A ··· 68AF order 1 2 2 4 4 4 8 8 8 8 17 ··· 17 34 ··· 34 68 ··· 68 size 1 1 2 1 1 2 34 34 34 34 2 ··· 2 2 ··· 2 2 ··· 2

74 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 M4(2) D17 Dic17 D34 Dic17 C68.4C4 kernel C68.4C4 C17⋊3C8 C2×C68 C68 C2×C34 C17 C2×C4 C4 C4 C22 C1 # reps 1 2 1 2 2 2 8 8 8 8 32

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

 44 0 0 109
,
 0 1 37 0
`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`

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