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G = C88.C4order 352 = 25·11

1st non-split extension by C88 of C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C88.1C4, C4.18D44, C44.34D4, C8.1Dic11, C22.2Dic22, (C2×C88).7C2, (C2×C8).5D11, C22.7(C4⋊C4), (C2×C22).3Q8, C44.35(C2×C4), (C2×C4).70D22, C111(C8.C4), C4.8(C2×Dic11), C2.5(C44⋊C4), C44.C4.1C2, (C2×C44).97C22, SmallGroup(352,25)

Series: Derived Chief Lower central Upper central

C1C44 — C88.C4
C1C11C22C44C2×C44C44.C4 — C88.C4
C11C22C44 — C88.C4
C1C4C2×C4C2×C8

Generators and relations for C88.C4
 G = < a,b | a88=1, b4=a44, bab-1=a43 >

2C2
2C22
22C8
22C8
11M4(2)
11M4(2)
2C11⋊C8
2C11⋊C8
11C8.C4

Smallest permutation representation of C88.C4
On 176 points
Generators in S176
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176)
(1 91 23 157 45 135 67 113)(2 134 24 112 46 90 68 156)(3 89 25 155 47 133 69 111)(4 132 26 110 48 176 70 154)(5 175 27 153 49 131 71 109)(6 130 28 108 50 174 72 152)(7 173 29 151 51 129 73 107)(8 128 30 106 52 172 74 150)(9 171 31 149 53 127 75 105)(10 126 32 104 54 170 76 148)(11 169 33 147 55 125 77 103)(12 124 34 102 56 168 78 146)(13 167 35 145 57 123 79 101)(14 122 36 100 58 166 80 144)(15 165 37 143 59 121 81 99)(16 120 38 98 60 164 82 142)(17 163 39 141 61 119 83 97)(18 118 40 96 62 162 84 140)(19 161 41 139 63 117 85 95)(20 116 42 94 64 160 86 138)(21 159 43 137 65 115 87 93)(22 114 44 92 66 158 88 136)

G:=sub<Sym(176)| (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)(89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,91,23,157,45,135,67,113)(2,134,24,112,46,90,68,156)(3,89,25,155,47,133,69,111)(4,132,26,110,48,176,70,154)(5,175,27,153,49,131,71,109)(6,130,28,108,50,174,72,152)(7,173,29,151,51,129,73,107)(8,128,30,106,52,172,74,150)(9,171,31,149,53,127,75,105)(10,126,32,104,54,170,76,148)(11,169,33,147,55,125,77,103)(12,124,34,102,56,168,78,146)(13,167,35,145,57,123,79,101)(14,122,36,100,58,166,80,144)(15,165,37,143,59,121,81,99)(16,120,38,98,60,164,82,142)(17,163,39,141,61,119,83,97)(18,118,40,96,62,162,84,140)(19,161,41,139,63,117,85,95)(20,116,42,94,64,160,86,138)(21,159,43,137,65,115,87,93)(22,114,44,92,66,158,88,136)>;

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)(89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,91,23,157,45,135,67,113)(2,134,24,112,46,90,68,156)(3,89,25,155,47,133,69,111)(4,132,26,110,48,176,70,154)(5,175,27,153,49,131,71,109)(6,130,28,108,50,174,72,152)(7,173,29,151,51,129,73,107)(8,128,30,106,52,172,74,150)(9,171,31,149,53,127,75,105)(10,126,32,104,54,170,76,148)(11,169,33,147,55,125,77,103)(12,124,34,102,56,168,78,146)(13,167,35,145,57,123,79,101)(14,122,36,100,58,166,80,144)(15,165,37,143,59,121,81,99)(16,120,38,98,60,164,82,142)(17,163,39,141,61,119,83,97)(18,118,40,96,62,162,84,140)(19,161,41,139,63,117,85,95)(20,116,42,94,64,160,86,138)(21,159,43,137,65,115,87,93)(22,114,44,92,66,158,88,136) );

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),(89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)], [(1,91,23,157,45,135,67,113),(2,134,24,112,46,90,68,156),(3,89,25,155,47,133,69,111),(4,132,26,110,48,176,70,154),(5,175,27,153,49,131,71,109),(6,130,28,108,50,174,72,152),(7,173,29,151,51,129,73,107),(8,128,30,106,52,172,74,150),(9,171,31,149,53,127,75,105),(10,126,32,104,54,170,76,148),(11,169,33,147,55,125,77,103),(12,124,34,102,56,168,78,146),(13,167,35,145,57,123,79,101),(14,122,36,100,58,166,80,144),(15,165,37,143,59,121,81,99),(16,120,38,98,60,164,82,142),(17,163,39,141,61,119,83,97),(18,118,40,96,62,162,84,140),(19,161,41,139,63,117,85,95),(20,116,42,94,64,160,86,138),(21,159,43,137,65,115,87,93),(22,114,44,92,66,158,88,136)])

94 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D8E8F8G8H11A···11E22A···22O44A···44T88A···88AN
order1224448888888811···1122···2244···4488···88
size1121122222444444442···22···22···22···2

94 irreducible representations

dim1111222222222
type++++-+-++-
imageC1C2C2C4D4Q8D11C8.C4Dic11D22D44Dic22C88.C4
kernelC88.C4C44.C4C2×C88C88C44C2×C22C2×C8C11C8C2×C4C4C22C1
# reps12141154105101040

Matrix representation of C88.C4 in GL2(𝔽89) generated by

290
2546
,
3146
1958
G:=sub<GL(2,GF(89))| [29,25,0,46],[31,19,46,58] >;

C88.C4 in GAP, Magma, Sage, TeX

C_{88}.C_4
% in TeX

G:=Group("C88.C4");
// GroupNames label

G:=SmallGroup(352,25);
// by ID

G=gap.SmallGroup(352,25);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,24,121,55,86,579,69,11525]);
// Polycyclic

G:=Group<a,b|a^88=1,b^4=a^44,b*a*b^-1=a^43>;
// generators/relations

Export

Subgroup lattice of C88.C4 in TeX

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