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G = C44.C8order 352 = 25·11

1st non-split extension by C44 of C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C88.4C4, C44.1C8, C8.22D22, C112M5(2), C8.2Dic11, C88.22C22, C4.(C11⋊C8), C11⋊C165C2, (C2×C22).3C8, C22.9(C2×C8), (C2×C44).8C4, C22.(C11⋊C8), (C2×C8).7D11, (C2×C88).10C2, C44.39(C2×C4), (C2×C4).5Dic11, C4.11(C2×Dic11), C2.4(C2×C11⋊C8), SmallGroup(352,18)

Series: Derived Chief Lower central Upper central

C1C22 — C44.C8
C1C11C22C44C88C11⋊C16 — C44.C8
C11C22 — C44.C8
C1C8C2×C8

Generators and relations for C44.C8
 G = < a,b | a44=1, b8=a22, bab-1=a-1 >

2C2
2C22
11C16
11C16
11M5(2)

Smallest permutation representation of C44.C8
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)
(1 174 58 126 34 141 47 93 23 152 80 104 12 163 69 115)(2 173 59 125 35 140 48 92 24 151 81 103 13 162 70 114)(3 172 60 124 36 139 49 91 25 150 82 102 14 161 71 113)(4 171 61 123 37 138 50 90 26 149 83 101 15 160 72 112)(5 170 62 122 38 137 51 89 27 148 84 100 16 159 73 111)(6 169 63 121 39 136 52 132 28 147 85 99 17 158 74 110)(7 168 64 120 40 135 53 131 29 146 86 98 18 157 75 109)(8 167 65 119 41 134 54 130 30 145 87 97 19 156 76 108)(9 166 66 118 42 133 55 129 31 144 88 96 20 155 77 107)(10 165 67 117 43 176 56 128 32 143 45 95 21 154 78 106)(11 164 68 116 44 175 57 127 33 142 46 94 22 153 79 105)

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

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

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

100 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D8E8F11A···11E16A···16H22A···22O44A···44T88A···88AN
order12244488888811···1116···1622···2244···4488···88
size1121121111222···222···222···22···22···2

100 irreducible representations

dim111111122222222
type++++-+-
imageC1C2C2C4C4C8C8D11M5(2)Dic11D22Dic11C11⋊C8C11⋊C8C44.C8
kernelC44.C8C11⋊C16C2×C88C88C2×C44C44C2×C22C2×C8C11C8C8C2×C4C4C22C1
# reps121224454555101040

Matrix representation of C44.C8 in GL2(𝔽353) generated by

319305
0218
,
245186
214108
G:=sub<GL(2,GF(353))| [319,0,305,218],[245,214,186,108] >;

C44.C8 in GAP, Magma, Sage, TeX

C_{44}.C_8
% in TeX

G:=Group("C44.C8");
// GroupNames label

G:=SmallGroup(352,18);
// by ID

G=gap.SmallGroup(352,18);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,24,217,50,69,11525]);
// Polycyclic

G:=Group<a,b|a^44=1,b^8=a^22,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C44.C8 in TeX

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