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G = C527M4(2)  order 400 = 24·52

3rd semidirect product of C52 and M4(2) acting via M4(2)/C4=C4

metabelian, supersoluble, monomial

Aliases: C20.3F5, C527M4(2), (C5×C20).3C4, C4.(C5⋊F5), C51(C4.F5), C524C83C2, C10.15(C2×F5), C526C4.19C22, (C4×C5⋊D5).6C2, (C2×C5⋊D5).8C4, C2.4(C2×C5⋊F5), (C5×C10).28(C2×C4), SmallGroup(400,150)

Series: Derived Chief Lower central Upper central

C1C5×C10 — C527M4(2)
C1C5C52C5×C10C526C4C524C8 — C527M4(2)
C52C5×C10 — C527M4(2)
C1C2C4

Generators and relations for C527M4(2)
 G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=a2, dad=a-1, cbc-1=b2, dbd=b-1, dcd=c5 >

Subgroups: 472 in 80 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5 [×6], C8 [×2], C2×C4, D5 [×6], C10 [×6], M4(2), Dic5 [×6], C20 [×6], D10 [×6], C52, C5⋊C8 [×12], C4×D5 [×6], C5⋊D5, C5×C10, C4.F5 [×6], C526C4, C5×C20, C2×C5⋊D5, C524C8 [×2], C4×C5⋊D5, C527M4(2)
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, M4(2), F5 [×6], C2×F5 [×6], C4.F5 [×6], C5⋊F5, C2×C5⋊F5, C527M4(2)

Smallest permutation representation of C527M4(2)
On 200 points
Generators in S200
(1 9 87 191 182)(2 88 183 10 192)(3 184 185 81 11)(4 186 12 177 82)(5 13 83 187 178)(6 84 179 14 188)(7 180 189 85 15)(8 190 16 181 86)(17 151 80 26 111)(18 73 112 152 27)(19 105 28 74 145)(20 29 146 106 75)(21 147 76 30 107)(22 77 108 148 31)(23 109 32 78 149)(24 25 150 110 79)(33 56 93 134 102)(34 94 103 49 135)(35 104 136 95 50)(36 129 51 97 96)(37 52 89 130 98)(38 90 99 53 131)(39 100 132 91 54)(40 133 55 101 92)(41 162 60 154 173)(42 61 174 163 155)(43 175 156 62 164)(44 157 165 176 63)(45 166 64 158 169)(46 57 170 167 159)(47 171 160 58 168)(48 153 161 172 59)(65 113 200 124 140)(66 193 141 114 125)(67 142 126 194 115)(68 127 116 143 195)(69 117 196 128 144)(70 197 137 118 121)(71 138 122 198 119)(72 123 120 139 199)
(1 100 62 118 106)(2 63 107 101 119)(3 108 120 64 102)(4 113 103 109 57)(5 104 58 114 110)(6 59 111 97 115)(7 112 116 60 98)(8 117 99 105 61)(9 132 164 121 75)(10 165 76 133 122)(11 77 123 166 134)(12 124 135 78 167)(13 136 168 125 79)(14 161 80 129 126)(15 73 127 162 130)(16 128 131 74 163)(17 96 67 84 48)(18 68 41 89 85)(19 42 86 69 90)(20 87 91 43 70)(21 92 71 88 44)(22 72 45 93 81)(23 46 82 65 94)(24 83 95 47 66)(25 187 50 171 193)(26 51 194 188 172)(27 195 173 52 189)(28 174 190 196 53)(29 191 54 175 197)(30 55 198 192 176)(31 199 169 56 185)(32 170 186 200 49)(33 184 148 139 158)(34 149 159 177 140)(35 160 141 150 178)(36 142 179 153 151)(37 180 152 143 154)(38 145 155 181 144)(39 156 137 146 182)(40 138 183 157 147)
(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)
(2 6)(4 8)(9 182)(10 179)(11 184)(12 181)(13 178)(14 183)(15 180)(16 177)(17 55)(18 52)(19 49)(20 54)(21 51)(22 56)(23 53)(24 50)(25 95)(26 92)(27 89)(28 94)(29 91)(30 96)(31 93)(32 90)(33 77)(34 74)(35 79)(36 76)(37 73)(38 78)(39 75)(40 80)(41 195)(42 200)(43 197)(44 194)(45 199)(46 196)(47 193)(48 198)(57 117)(58 114)(59 119)(60 116)(61 113)(62 118)(63 115)(64 120)(65 174)(66 171)(67 176)(68 173)(69 170)(70 175)(71 172)(72 169)(81 185)(82 190)(83 187)(84 192)(85 189)(86 186)(87 191)(88 188)(97 107)(98 112)(99 109)(100 106)(101 111)(102 108)(103 105)(104 110)(121 156)(122 153)(123 158)(124 155)(125 160)(126 157)(127 154)(128 159)(129 147)(130 152)(131 149)(132 146)(133 151)(134 148)(135 145)(136 150)(137 164)(138 161)(139 166)(140 163)(141 168)(142 165)(143 162)(144 167)

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

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

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

34 conjugacy classes

class 1 2A2B4A4B4C5A···5F8A8B8C8D10A···10F20A···20L
order1224445···5888810···1020···20
size1150225254···4505050504···44···4

34 irreducible representations

dim111112444
type+++++
imageC1C2C2C4C4M4(2)F5C2×F5C4.F5
kernelC527M4(2)C524C8C4×C5⋊D5C5×C20C2×C5⋊D5C52C20C10C5
# reps1212226612

Matrix representation of C527M4(2) in GL8(𝔽41)

10000000
01000000
00100000
00010000
00001500
000004010
000004001
000040394040
,
01000000
00100000
00010000
404040400000
00001000
00000100
00000010
00000001
,
1226170000
321516170000
12516400000
24252690000
00001938434
0000328350
000029143514
00002614020
,
10000000
404040400000
00010000
00100000
00001000
000040404040
00000001
00000010

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,5,40,40,39,0,0,0,0,0,1,0,40,0,0,0,0,0,0,1,40],[0,0,0,40,0,0,0,0,1,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,32,1,24,0,0,0,0,2,15,25,25,0,0,0,0,26,16,16,26,0,0,0,0,17,17,40,9,0,0,0,0,0,0,0,0,19,32,29,26,0,0,0,0,38,8,14,14,0,0,0,0,4,35,35,0,0,0,0,0,34,0,14,20],[1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,1,0,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,1,0] >;

C527M4(2) in GAP, Magma, Sage, TeX

C_5^2\rtimes_7M_4(2)
% in TeX

G:=Group("C5^2:7M4(2)");
// GroupNames label

G:=SmallGroup(400,150);
// by ID

G=gap.SmallGroup(400,150);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,50,964,496,5765,2897]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^2,d*a*d=a^-1,c*b*c^-1=b^2,d*b*d=b^-1,d*c*d=c^5>;
// generators/relations

׿
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