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G = C524C8order 200 = 23·52

3rd semidirect product of C52 and C8 acting via C8/C2=C4

metabelian, supersoluble, monomial, A-group

Aliases: C524C8, C10.3F5, C51(C5⋊C8), (C5×C10).3C4, C2.(C5⋊F5), C526C4.3C2, SmallGroup(200,20)

Series: Derived Chief Lower central Upper central

C1C52 — C524C8
C1C5C52C5×C10C526C4 — C524C8
C52 — C524C8
C1C2

Generators and relations for C524C8
 G = < a,b,c | a5=b5=c8=1, ab=ba, cac-1=a3, cbc-1=b3 >

25C4
25C8
5Dic5
5Dic5
5Dic5
5Dic5
5Dic5
5Dic5
5C5⋊C8
5C5⋊C8
5C5⋊C8
5C5⋊C8
5C5⋊C8
5C5⋊C8

Character table of C524C8

 class 124A4B5A5B5C5D5E5F8A8B8C8D10A10B10C10D10E10F
 size 11252544444425252525444444
ρ111111111111111111111    trivial
ρ21111111111-1-1-1-1111111    linear of order 2
ρ311-1-1111111i-ii-i111111    linear of order 4
ρ411-1-1111111-ii-ii111111    linear of order 4
ρ51-1-ii111111ζ87ζ85ζ83ζ8-1-1-1-1-1-1    linear of order 8
ρ61-1-ii111111ζ83ζ8ζ87ζ85-1-1-1-1-1-1    linear of order 8
ρ71-1i-i111111ζ85ζ87ζ8ζ83-1-1-1-1-1-1    linear of order 8
ρ81-1i-i111111ζ8ζ83ζ85ζ87-1-1-1-1-1-1    linear of order 8
ρ94400-1-1-14-1-100004-1-1-1-1-1    orthogonal lifted from F5
ρ104400-14-1-1-1-10000-1-1-1-14-1    orthogonal lifted from F5
ρ114400-1-1-1-1-140000-1-14-1-1-1    orthogonal lifted from F5
ρ124400-1-1-1-14-10000-14-1-1-1-1    orthogonal lifted from F5
ρ134400-1-14-1-1-10000-1-1-1-1-14    orthogonal lifted from F5
ρ1444004-1-1-1-1-10000-1-1-14-1-1    orthogonal lifted from F5
ρ154-400-1-1-14-1-10000-411111    symplectic lifted from C5⋊C8, Schur index 2
ρ164-400-14-1-1-1-100001111-41    symplectic lifted from C5⋊C8, Schur index 2
ρ174-400-1-14-1-1-1000011111-4    symplectic lifted from C5⋊C8, Schur index 2
ρ184-4004-1-1-1-1-10000111-411    symplectic lifted from C5⋊C8, Schur index 2
ρ194-400-1-1-1-14-100001-41111    symplectic lifted from C5⋊C8, Schur index 2
ρ204-400-1-1-1-1-14000011-4111    symplectic lifted from C5⋊C8, Schur index 2

Smallest permutation representation of C524C8
Regular action on 200 points
Generators in S200
(1 48 9 189 153)(2 190 41 154 10)(3 155 191 11 42)(4 12 156 43 192)(5 44 13 185 157)(6 186 45 158 14)(7 159 187 15 46)(8 16 160 47 188)(17 56 57 162 169)(18 163 49 170 58)(19 171 164 59 50)(20 60 172 51 165)(21 52 61 166 173)(22 167 53 174 62)(23 175 168 63 54)(24 64 176 55 161)(25 143 38 104 178)(26 97 144 179 39)(27 180 98 40 137)(28 33 181 138 99)(29 139 34 100 182)(30 101 140 183 35)(31 184 102 36 141)(32 37 177 142 103)(65 83 147 80 194)(66 73 84 195 148)(67 196 74 149 85)(68 150 197 86 75)(69 87 151 76 198)(70 77 88 199 152)(71 200 78 145 81)(72 146 193 82 79)(89 130 121 115 108)(90 116 131 109 122)(91 110 117 123 132)(92 124 111 133 118)(93 134 125 119 112)(94 120 135 105 126)(95 106 113 127 136)(96 128 107 129 114)
(1 23 68 89 104)(2 90 24 97 69)(3 98 91 70 17)(4 71 99 18 92)(5 19 72 93 100)(6 94 20 101 65)(7 102 95 66 21)(8 67 103 22 96)(9 168 197 121 25)(10 122 161 26 198)(11 27 123 199 162)(12 200 28 163 124)(13 164 193 125 29)(14 126 165 30 194)(15 31 127 195 166)(16 196 32 167 128)(33 49 111 156 78)(34 157 50 79 112)(35 80 158 105 51)(36 106 73 52 159)(37 53 107 160 74)(38 153 54 75 108)(39 76 154 109 55)(40 110 77 56 155)(41 131 176 179 151)(42 180 132 152 169)(43 145 181 170 133)(44 171 146 134 182)(45 135 172 183 147)(46 184 136 148 173)(47 149 177 174 129)(48 175 150 130 178)(57 191 137 117 88)(58 118 192 81 138)(59 82 119 139 185)(60 140 83 186 120)(61 187 141 113 84)(62 114 188 85 142)(63 86 115 143 189)(64 144 87 190 116)
(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)| (1,48,9,189,153)(2,190,41,154,10)(3,155,191,11,42)(4,12,156,43,192)(5,44,13,185,157)(6,186,45,158,14)(7,159,187,15,46)(8,16,160,47,188)(17,56,57,162,169)(18,163,49,170,58)(19,171,164,59,50)(20,60,172,51,165)(21,52,61,166,173)(22,167,53,174,62)(23,175,168,63,54)(24,64,176,55,161)(25,143,38,104,178)(26,97,144,179,39)(27,180,98,40,137)(28,33,181,138,99)(29,139,34,100,182)(30,101,140,183,35)(31,184,102,36,141)(32,37,177,142,103)(65,83,147,80,194)(66,73,84,195,148)(67,196,74,149,85)(68,150,197,86,75)(69,87,151,76,198)(70,77,88,199,152)(71,200,78,145,81)(72,146,193,82,79)(89,130,121,115,108)(90,116,131,109,122)(91,110,117,123,132)(92,124,111,133,118)(93,134,125,119,112)(94,120,135,105,126)(95,106,113,127,136)(96,128,107,129,114), (1,23,68,89,104)(2,90,24,97,69)(3,98,91,70,17)(4,71,99,18,92)(5,19,72,93,100)(6,94,20,101,65)(7,102,95,66,21)(8,67,103,22,96)(9,168,197,121,25)(10,122,161,26,198)(11,27,123,199,162)(12,200,28,163,124)(13,164,193,125,29)(14,126,165,30,194)(15,31,127,195,166)(16,196,32,167,128)(33,49,111,156,78)(34,157,50,79,112)(35,80,158,105,51)(36,106,73,52,159)(37,53,107,160,74)(38,153,54,75,108)(39,76,154,109,55)(40,110,77,56,155)(41,131,176,179,151)(42,180,132,152,169)(43,145,181,170,133)(44,171,146,134,182)(45,135,172,183,147)(46,184,136,148,173)(47,149,177,174,129)(48,175,150,130,178)(57,191,137,117,88)(58,118,192,81,138)(59,82,119,139,185)(60,140,83,186,120)(61,187,141,113,84)(62,114,188,85,142)(63,86,115,143,189)(64,144,87,190,116), (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( (1,48,9,189,153)(2,190,41,154,10)(3,155,191,11,42)(4,12,156,43,192)(5,44,13,185,157)(6,186,45,158,14)(7,159,187,15,46)(8,16,160,47,188)(17,56,57,162,169)(18,163,49,170,58)(19,171,164,59,50)(20,60,172,51,165)(21,52,61,166,173)(22,167,53,174,62)(23,175,168,63,54)(24,64,176,55,161)(25,143,38,104,178)(26,97,144,179,39)(27,180,98,40,137)(28,33,181,138,99)(29,139,34,100,182)(30,101,140,183,35)(31,184,102,36,141)(32,37,177,142,103)(65,83,147,80,194)(66,73,84,195,148)(67,196,74,149,85)(68,150,197,86,75)(69,87,151,76,198)(70,77,88,199,152)(71,200,78,145,81)(72,146,193,82,79)(89,130,121,115,108)(90,116,131,109,122)(91,110,117,123,132)(92,124,111,133,118)(93,134,125,119,112)(94,120,135,105,126)(95,106,113,127,136)(96,128,107,129,114), (1,23,68,89,104)(2,90,24,97,69)(3,98,91,70,17)(4,71,99,18,92)(5,19,72,93,100)(6,94,20,101,65)(7,102,95,66,21)(8,67,103,22,96)(9,168,197,121,25)(10,122,161,26,198)(11,27,123,199,162)(12,200,28,163,124)(13,164,193,125,29)(14,126,165,30,194)(15,31,127,195,166)(16,196,32,167,128)(33,49,111,156,78)(34,157,50,79,112)(35,80,158,105,51)(36,106,73,52,159)(37,53,107,160,74)(38,153,54,75,108)(39,76,154,109,55)(40,110,77,56,155)(41,131,176,179,151)(42,180,132,152,169)(43,145,181,170,133)(44,171,146,134,182)(45,135,172,183,147)(46,184,136,148,173)(47,149,177,174,129)(48,175,150,130,178)(57,191,137,117,88)(58,118,192,81,138)(59,82,119,139,185)(60,140,83,186,120)(61,187,141,113,84)(62,114,188,85,142)(63,86,115,143,189)(64,144,87,190,116), (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([(1,48,9,189,153),(2,190,41,154,10),(3,155,191,11,42),(4,12,156,43,192),(5,44,13,185,157),(6,186,45,158,14),(7,159,187,15,46),(8,16,160,47,188),(17,56,57,162,169),(18,163,49,170,58),(19,171,164,59,50),(20,60,172,51,165),(21,52,61,166,173),(22,167,53,174,62),(23,175,168,63,54),(24,64,176,55,161),(25,143,38,104,178),(26,97,144,179,39),(27,180,98,40,137),(28,33,181,138,99),(29,139,34,100,182),(30,101,140,183,35),(31,184,102,36,141),(32,37,177,142,103),(65,83,147,80,194),(66,73,84,195,148),(67,196,74,149,85),(68,150,197,86,75),(69,87,151,76,198),(70,77,88,199,152),(71,200,78,145,81),(72,146,193,82,79),(89,130,121,115,108),(90,116,131,109,122),(91,110,117,123,132),(92,124,111,133,118),(93,134,125,119,112),(94,120,135,105,126),(95,106,113,127,136),(96,128,107,129,114)], [(1,23,68,89,104),(2,90,24,97,69),(3,98,91,70,17),(4,71,99,18,92),(5,19,72,93,100),(6,94,20,101,65),(7,102,95,66,21),(8,67,103,22,96),(9,168,197,121,25),(10,122,161,26,198),(11,27,123,199,162),(12,200,28,163,124),(13,164,193,125,29),(14,126,165,30,194),(15,31,127,195,166),(16,196,32,167,128),(33,49,111,156,78),(34,157,50,79,112),(35,80,158,105,51),(36,106,73,52,159),(37,53,107,160,74),(38,153,54,75,108),(39,76,154,109,55),(40,110,77,56,155),(41,131,176,179,151),(42,180,132,152,169),(43,145,181,170,133),(44,171,146,134,182),(45,135,172,183,147),(46,184,136,148,173),(47,149,177,174,129),(48,175,150,130,178),(57,191,137,117,88),(58,118,192,81,138),(59,82,119,139,185),(60,140,83,186,120),(61,187,141,113,84),(62,114,188,85,142),(63,86,115,143,189),(64,144,87,190,116)], [(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)])

C524C8 is a maximal subgroup of   C52⋊C16  D10.2F5  C524M4(2)  C20.F5  C527M4(2)  C5213M4(2)
C524C8 is a maximal quotient of   C524C16

Matrix representation of C524C8 in GL8(𝔽41)

040100000
040010000
040000000
140000000
00001000
00000100
00000010
00000001
,
401000000
400100000
400010000
400000000
00000100
00000010
00000001
000040404040
,
6152340000
8827400000
3314110000
71635260000
000023222722
000050191
000019105
00004044018

G:=sub<GL(8,GF(41))| [0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,1,40],[6,8,33,7,0,0,0,0,15,8,14,16,0,0,0,0,2,27,1,35,0,0,0,0,34,40,1,26,0,0,0,0,0,0,0,0,23,5,19,40,0,0,0,0,22,0,1,4,0,0,0,0,27,19,0,40,0,0,0,0,22,1,5,18] >;

C524C8 in GAP, Magma, Sage, TeX

C_5^2\rtimes_4C_8
% in TeX

G:=Group("C5^2:4C8");
// GroupNames label

G:=SmallGroup(200,20);
// by ID

G=gap.SmallGroup(200,20);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,10,26,323,328,2004,2009]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^8=1,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C524C8 in TeX
Character table of C524C8 in TeX

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