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G = C52⋊7M4(2)  order 400 = 24·52

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

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

 Derived series C1 — C5×C10 — C52⋊7M4(2)
 Chief series C1 — C5 — C52 — C5×C10 — C52⋊6C4 — C52⋊4C8 — C52⋊7M4(2)
 Lower central C52 — C5×C10 — C52⋊7M4(2)
 Upper central C1 — C2 — C4

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, C8, C2×C4, D5, C10, M4(2), Dic5, C20, D10, C52, C5⋊C8, C4×D5, C5⋊D5, C5×C10, C4.F5, C526C4, C5×C20, C2×C5⋊D5, C524C8, C4×C5⋊D5, C527M4(2)
Quotients: C1, C2, C4, C22, C2×C4, M4(2), F5, C2×F5, C4.F5, C5⋊F5, C2×C5⋊F5, C527M4(2)

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

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

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

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

34 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A ··· 5F 8A 8B 8C 8D 10A ··· 10F 20A ··· 20L order 1 2 2 4 4 4 5 ··· 5 8 8 8 8 10 ··· 10 20 ··· 20 size 1 1 50 2 25 25 4 ··· 4 50 50 50 50 4 ··· 4 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 2 4 4 4 type + + + + + image C1 C2 C2 C4 C4 M4(2) F5 C2×F5 C4.F5 kernel C52⋊7M4(2) C52⋊4C8 C4×C5⋊D5 C5×C20 C2×C5⋊D5 C52 C20 C10 C5 # reps 1 2 1 2 2 2 6 6 12

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

 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 5 0 0 0 0 0 0 0 40 1 0 0 0 0 0 0 40 0 1 0 0 0 0 40 39 40 40
,
 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 40 40 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 2 26 17 0 0 0 0 32 15 16 17 0 0 0 0 1 25 16 40 0 0 0 0 24 25 26 9 0 0 0 0 0 0 0 0 19 38 4 34 0 0 0 0 32 8 35 0 0 0 0 0 29 14 35 14 0 0 0 0 26 14 0 20
,
 1 0 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

`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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