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## G = C52⋊7C8order 200 = 23·52

### 2nd semidirect product of C52 and C8 acting via C8/C4=C2

Aliases: C527C8, C20.6D5, C10.4Dic5, C52(C52C8), (C5×C10).6C4, (C5×C20).4C2, C4.2(C5⋊D5), C2.(C526C4), SmallGroup(200,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊7C8
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — C52⋊7C8
 Lower central C52 — C52⋊7C8
 Upper central C1 — C4

Generators and relations for C527C8
G = < a,b,c | a5=b5=c8=1, ab=ba, cac-1=a-1, cbc-1=b-1 >

Smallest permutation representation of C527C8
Regular action on 200 points
Generators in S200
(1 48 62 189 11)(2 12 190 63 41)(3 42 64 191 13)(4 14 192 57 43)(5 44 58 185 15)(6 16 186 59 45)(7 46 60 187 9)(8 10 188 61 47)(17 174 164 145 112)(18 105 146 165 175)(19 176 166 147 106)(20 107 148 167 169)(21 170 168 149 108)(22 109 150 161 171)(23 172 162 151 110)(24 111 152 163 173)(25 182 104 139 34)(26 35 140 97 183)(27 184 98 141 36)(28 37 142 99 177)(29 178 100 143 38)(30 39 144 101 179)(31 180 102 137 40)(32 33 138 103 181)(49 117 74 85 196)(50 197 86 75 118)(51 119 76 87 198)(52 199 88 77 120)(53 113 78 81 200)(54 193 82 79 114)(55 115 80 83 194)(56 195 84 73 116)(65 158 89 130 121)(66 122 131 90 159)(67 160 91 132 123)(68 124 133 92 153)(69 154 93 134 125)(70 126 135 94 155)(71 156 95 136 127)(72 128 129 96 157)
(1 165 118 89 29)(2 30 90 119 166)(3 167 120 91 31)(4 32 92 113 168)(5 161 114 93 25)(6 26 94 115 162)(7 163 116 95 27)(8 28 96 117 164)(9 152 73 156 36)(10 37 157 74 145)(11 146 75 158 38)(12 39 159 76 147)(13 148 77 160 40)(14 33 153 78 149)(15 150 79 154 34)(16 35 155 80 151)(17 61 99 128 196)(18 197 121 100 62)(19 63 101 122 198)(20 199 123 102 64)(21 57 103 124 200)(22 193 125 104 58)(23 59 97 126 194)(24 195 127 98 60)(41 179 131 51 176)(42 169 52 132 180)(43 181 133 53 170)(44 171 54 134 182)(45 183 135 55 172)(46 173 56 136 184)(47 177 129 49 174)(48 175 50 130 178)(65 143 189 105 86)(66 87 106 190 144)(67 137 191 107 88)(68 81 108 192 138)(69 139 185 109 82)(70 83 110 186 140)(71 141 187 111 84)(72 85 112 188 142)
(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)(177 178 179 180 181 182 183 184)(185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200)

G:=sub<Sym(200)| (1,48,62,189,11)(2,12,190,63,41)(3,42,64,191,13)(4,14,192,57,43)(5,44,58,185,15)(6,16,186,59,45)(7,46,60,187,9)(8,10,188,61,47)(17,174,164,145,112)(18,105,146,165,175)(19,176,166,147,106)(20,107,148,167,169)(21,170,168,149,108)(22,109,150,161,171)(23,172,162,151,110)(24,111,152,163,173)(25,182,104,139,34)(26,35,140,97,183)(27,184,98,141,36)(28,37,142,99,177)(29,178,100,143,38)(30,39,144,101,179)(31,180,102,137,40)(32,33,138,103,181)(49,117,74,85,196)(50,197,86,75,118)(51,119,76,87,198)(52,199,88,77,120)(53,113,78,81,200)(54,193,82,79,114)(55,115,80,83,194)(56,195,84,73,116)(65,158,89,130,121)(66,122,131,90,159)(67,160,91,132,123)(68,124,133,92,153)(69,154,93,134,125)(70,126,135,94,155)(71,156,95,136,127)(72,128,129,96,157), (1,165,118,89,29)(2,30,90,119,166)(3,167,120,91,31)(4,32,92,113,168)(5,161,114,93,25)(6,26,94,115,162)(7,163,116,95,27)(8,28,96,117,164)(9,152,73,156,36)(10,37,157,74,145)(11,146,75,158,38)(12,39,159,76,147)(13,148,77,160,40)(14,33,153,78,149)(15,150,79,154,34)(16,35,155,80,151)(17,61,99,128,196)(18,197,121,100,62)(19,63,101,122,198)(20,199,123,102,64)(21,57,103,124,200)(22,193,125,104,58)(23,59,97,126,194)(24,195,127,98,60)(41,179,131,51,176)(42,169,52,132,180)(43,181,133,53,170)(44,171,54,134,182)(45,183,135,55,172)(46,173,56,136,184)(47,177,129,49,174)(48,175,50,130,178)(65,143,189,105,86)(66,87,106,190,144)(67,137,191,107,88)(68,81,108,192,138)(69,139,185,109,82)(70,83,110,186,140)(71,141,187,111,84)(72,85,112,188,142), (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)(177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200)>;

G:=Group( (1,48,62,189,11)(2,12,190,63,41)(3,42,64,191,13)(4,14,192,57,43)(5,44,58,185,15)(6,16,186,59,45)(7,46,60,187,9)(8,10,188,61,47)(17,174,164,145,112)(18,105,146,165,175)(19,176,166,147,106)(20,107,148,167,169)(21,170,168,149,108)(22,109,150,161,171)(23,172,162,151,110)(24,111,152,163,173)(25,182,104,139,34)(26,35,140,97,183)(27,184,98,141,36)(28,37,142,99,177)(29,178,100,143,38)(30,39,144,101,179)(31,180,102,137,40)(32,33,138,103,181)(49,117,74,85,196)(50,197,86,75,118)(51,119,76,87,198)(52,199,88,77,120)(53,113,78,81,200)(54,193,82,79,114)(55,115,80,83,194)(56,195,84,73,116)(65,158,89,130,121)(66,122,131,90,159)(67,160,91,132,123)(68,124,133,92,153)(69,154,93,134,125)(70,126,135,94,155)(71,156,95,136,127)(72,128,129,96,157), (1,165,118,89,29)(2,30,90,119,166)(3,167,120,91,31)(4,32,92,113,168)(5,161,114,93,25)(6,26,94,115,162)(7,163,116,95,27)(8,28,96,117,164)(9,152,73,156,36)(10,37,157,74,145)(11,146,75,158,38)(12,39,159,76,147)(13,148,77,160,40)(14,33,153,78,149)(15,150,79,154,34)(16,35,155,80,151)(17,61,99,128,196)(18,197,121,100,62)(19,63,101,122,198)(20,199,123,102,64)(21,57,103,124,200)(22,193,125,104,58)(23,59,97,126,194)(24,195,127,98,60)(41,179,131,51,176)(42,169,52,132,180)(43,181,133,53,170)(44,171,54,134,182)(45,183,135,55,172)(46,173,56,136,184)(47,177,129,49,174)(48,175,50,130,178)(65,143,189,105,86)(66,87,106,190,144)(67,137,191,107,88)(68,81,108,192,138)(69,139,185,109,82)(70,83,110,186,140)(71,141,187,111,84)(72,85,112,188,142), (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)(177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200) );

G=PermutationGroup([[(1,48,62,189,11),(2,12,190,63,41),(3,42,64,191,13),(4,14,192,57,43),(5,44,58,185,15),(6,16,186,59,45),(7,46,60,187,9),(8,10,188,61,47),(17,174,164,145,112),(18,105,146,165,175),(19,176,166,147,106),(20,107,148,167,169),(21,170,168,149,108),(22,109,150,161,171),(23,172,162,151,110),(24,111,152,163,173),(25,182,104,139,34),(26,35,140,97,183),(27,184,98,141,36),(28,37,142,99,177),(29,178,100,143,38),(30,39,144,101,179),(31,180,102,137,40),(32,33,138,103,181),(49,117,74,85,196),(50,197,86,75,118),(51,119,76,87,198),(52,199,88,77,120),(53,113,78,81,200),(54,193,82,79,114),(55,115,80,83,194),(56,195,84,73,116),(65,158,89,130,121),(66,122,131,90,159),(67,160,91,132,123),(68,124,133,92,153),(69,154,93,134,125),(70,126,135,94,155),(71,156,95,136,127),(72,128,129,96,157)], [(1,165,118,89,29),(2,30,90,119,166),(3,167,120,91,31),(4,32,92,113,168),(5,161,114,93,25),(6,26,94,115,162),(7,163,116,95,27),(8,28,96,117,164),(9,152,73,156,36),(10,37,157,74,145),(11,146,75,158,38),(12,39,159,76,147),(13,148,77,160,40),(14,33,153,78,149),(15,150,79,154,34),(16,35,155,80,151),(17,61,99,128,196),(18,197,121,100,62),(19,63,101,122,198),(20,199,123,102,64),(21,57,103,124,200),(22,193,125,104,58),(23,59,97,126,194),(24,195,127,98,60),(41,179,131,51,176),(42,169,52,132,180),(43,181,133,53,170),(44,171,54,134,182),(45,183,135,55,172),(46,173,56,136,184),(47,177,129,49,174),(48,175,50,130,178),(65,143,189,105,86),(66,87,106,190,144),(67,137,191,107,88),(68,81,108,192,138),(69,139,185,109,82),(70,83,110,186,140),(71,141,187,111,84),(72,85,112,188,142)], [(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),(177,178,179,180,181,182,183,184),(185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200)]])

C527C8 is a maximal subgroup of
C524C16  C525C16  D5×C52C8  C20.30D10  C522D8  D20.D5  C522Q16  C8×C5⋊D5  C40⋊D5  C20.59D10  C527D8  C528SD16  C5210SD16  C527Q16
C527C8 is a maximal quotient of
C527C16

56 conjugacy classes

 class 1 2 4A 4B 5A ··· 5L 8A 8B 8C 8D 10A ··· 10L 20A ··· 20X order 1 2 4 4 5 ··· 5 8 8 8 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 25 25 25 25 2 ··· 2 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + - image C1 C2 C4 C8 D5 Dic5 C5⋊2C8 kernel C52⋊7C8 C5×C20 C5×C10 C52 C20 C10 C5 # reps 1 1 2 4 12 12 24

Matrix representation of C527C8 in GL5(𝔽41)

 1 0 0 0 0 0 40 6 0 0 0 35 35 0 0 0 0 0 6 40 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 40 6 0 0 0 0 0 0 1 0 0 0 40 6
,
 3 0 0 0 0 0 39 26 0 0 0 14 2 0 0 0 0 0 15 2 0 0 0 10 26

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,35,0,0,0,6,35,0,0,0,0,0,6,1,0,0,0,40,0],[1,0,0,0,0,0,0,40,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,1,6],[3,0,0,0,0,0,39,14,0,0,0,26,2,0,0,0,0,0,15,10,0,0,0,2,26] >;

C527C8 in GAP, Magma, Sage, TeX

C_5^2\rtimes_7C_8
% in TeX

G:=Group("C5^2:7C8");
// GroupNames label

G:=SmallGroup(200,16);
// by ID

G=gap.SmallGroup(200,16);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,10,26,643,4004]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^8=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

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