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G = C524Q8order 200 = 23·52

2nd semidirect product of C52 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial

Aliases: C524Q8, C20.3D5, C52Dic10, C10.12D10, C4.(C5⋊D5), (C5×C20).1C2, C526C4.2C2, (C5×C10).11C22, C2.3(C2×C5⋊D5), SmallGroup(200,32)

Series: Derived Chief Lower central Upper central

C1C5×C10 — C524Q8
C1C5C52C5×C10C526C4 — C524Q8
C52C5×C10 — C524Q8
C1C2C4

Generators and relations for C524Q8
 G = < a,b,c,d | a5=b5=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

25C4
25C4
25Q8
5Dic5
5Dic5
5Dic5
5Dic5
5Dic5
5Dic5
5Dic5
5Dic5
5Dic5
5Dic5
5Dic5
5Dic5
5Dic10
5Dic10
5Dic10
5Dic10
5Dic10
5Dic10

Smallest permutation representation of C524Q8
Regular action on 200 points
Generators in S200
(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)
(1 44 28 31 38)(2 45 29 32 39)(3 41 30 33 40)(4 42 26 34 36)(5 43 27 35 37)(6 200 25 16 11)(7 196 21 17 12)(8 197 22 18 13)(9 198 23 19 14)(10 199 24 20 15)(46 61 56 67 51)(47 62 57 68 52)(48 63 58 69 53)(49 64 59 70 54)(50 65 60 66 55)(71 86 81 92 76)(72 87 82 93 77)(73 88 83 94 78)(74 89 84 95 79)(75 90 85 91 80)(96 106 111 102 116)(97 107 112 103 117)(98 108 113 104 118)(99 109 114 105 119)(100 110 115 101 120)(121 130 131 136 141)(122 126 132 137 142)(123 127 133 138 143)(124 128 134 139 144)(125 129 135 140 145)(146 155 156 161 166)(147 151 157 162 167)(148 152 158 163 168)(149 153 159 164 169)(150 154 160 165 170)(171 180 181 186 191)(172 176 182 187 192)(173 177 183 188 193)(174 178 184 189 194)(175 179 185 190 195)
(1 113 58 83)(2 114 59 84)(3 115 60 85)(4 111 56 81)(5 112 57 82)(6 145 195 170)(7 141 191 166)(8 142 192 167)(9 143 193 168)(10 144 194 169)(11 140 190 165)(12 136 186 161)(13 137 187 162)(14 138 188 163)(15 139 189 164)(16 135 185 160)(17 131 181 156)(18 132 182 157)(19 133 183 158)(20 134 184 159)(21 130 180 155)(22 126 176 151)(23 127 177 152)(24 128 178 153)(25 129 179 154)(26 116 51 76)(27 117 52 77)(28 118 53 78)(29 119 54 79)(30 120 55 80)(31 98 48 73)(32 99 49 74)(33 100 50 75)(34 96 46 71)(35 97 47 72)(36 106 61 86)(37 107 62 87)(38 108 63 88)(39 109 64 89)(40 110 65 90)(41 101 66 91)(42 102 67 92)(43 103 68 93)(44 104 69 94)(45 105 70 95)(121 171 146 196)(122 172 147 197)(123 173 148 198)(124 174 149 199)(125 175 150 200)
(1 157 58 132)(2 156 59 131)(3 160 60 135)(4 159 56 134)(5 158 57 133)(6 100 195 75)(7 99 191 74)(8 98 192 73)(9 97 193 72)(10 96 194 71)(11 110 190 90)(12 109 186 89)(13 108 187 88)(14 107 188 87)(15 106 189 86)(16 115 185 85)(17 114 181 84)(18 113 182 83)(19 112 183 82)(20 111 184 81)(21 105 180 95)(22 104 176 94)(23 103 177 93)(24 102 178 92)(25 101 179 91)(26 149 51 124)(27 148 52 123)(28 147 53 122)(29 146 54 121)(30 150 55 125)(31 167 48 142)(32 166 49 141)(33 170 50 145)(34 169 46 144)(35 168 47 143)(36 164 61 139)(37 163 62 138)(38 162 63 137)(39 161 64 136)(40 165 65 140)(41 154 66 129)(42 153 67 128)(43 152 68 127)(44 151 69 126)(45 155 70 130)(76 199 116 174)(77 198 117 173)(78 197 118 172)(79 196 119 171)(80 200 120 175)

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

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

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

C524Q8 is a maximal subgroup of
C523SD16  C523Q16  C402D5  C40.D5  C528SD16  C527Q16  D5×Dic10  D205D5  C20.50D10  C20.D10  Q8×C5⋊D5
C524Q8 is a maximal quotient of
C102.22C22  C203Dic5

53 conjugacy classes

class 1  2 4A4B4C5A···5L10A···10L20A···20X
order124445···510···1020···20
size11250502···22···22···2

53 irreducible representations

dim1112222
type+++-++-
imageC1C2C2Q8D5D10Dic10
kernelC524Q8C526C4C5×C20C52C20C10C5
# reps1211121224

Matrix representation of C524Q8 in GL4(𝔽41) generated by

344000
1000
0010
0001
,
0100
403400
0071
003340
,
40000
04000
003028
002211
,
1700
04000
001410
00927
G:=sub<GL(4,GF(41))| [34,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,1,34,0,0,0,0,7,33,0,0,1,40],[40,0,0,0,0,40,0,0,0,0,30,22,0,0,28,11],[1,0,0,0,7,40,0,0,0,0,14,9,0,0,10,27] >;

C524Q8 in GAP, Magma, Sage, TeX

C_5^2\rtimes_4Q_8
% in TeX

G:=Group("C5^2:4Q8");
// GroupNames label

G:=SmallGroup(200,32);
// by ID

G=gap.SmallGroup(200,32);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,20,61,26,643,4004]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of C524Q8 in TeX

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