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## G = C52⋊13M4(2)  order 400 = 24·52

### 3rd semidirect product of C52 and M4(2) acting via M4(2)/C22=C4

Aliases: C102.6C4, C5213M4(2), (C2×C10).6F5, C524C84C2, C10.19(C2×F5), C526C4.8C4, C22.(C5⋊F5), C52(C22.F5), C526C4.21C22, C2.6(C2×C5⋊F5), (C5×C10).32(C2×C4), (C2×C526C4).9C2, SmallGroup(400,154)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C52⋊13M4(2)
 Chief series C1 — C5 — C52 — C5×C10 — C52⋊6C4 — C52⋊4C8 — C52⋊13M4(2)
 Lower central C52 — C5×C10 — C52⋊13M4(2)
 Upper central C1 — C2 — C22

Generators and relations for C5213M4(2)
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=a3, ad=da, cbc-1=b3, bd=db, dcd=c5 >

Subgroups: 376 in 80 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C2, C4, C22, C5, C8, C2×C4, C10, C10, M4(2), Dic5, C2×C10, C52, C5⋊C8, C2×Dic5, C5×C10, C5×C10, C22.F5, C526C4, C102, C524C8, C2×C526C4, C5213M4(2)
Quotients: C1, C2, C4, C22, C2×C4, M4(2), F5, C2×F5, C22.F5, C5⋊F5, C2×C5⋊F5, C5213M4(2)

Smallest permutation representation of C5213M4(2)
On 200 points
Generators in S200
```(1 144 82 189 15)(2 190 137 16 83)(3 9 191 84 138)(4 85 10 139 192)(5 140 86 185 11)(6 186 141 12 87)(7 13 187 88 142)(8 81 14 143 188)(17 174 145 154 71)(18 155 175 72 146)(19 65 156 147 176)(20 148 66 169 157)(21 170 149 158 67)(22 159 171 68 150)(23 69 160 151 172)(24 152 70 173 153)(25 41 50 164 63)(26 165 42 64 51)(27 57 166 52 43)(28 53 58 44 167)(29 45 54 168 59)(30 161 46 60 55)(31 61 162 56 47)(32 49 62 48 163)(33 80 94 196 110)(34 197 73 111 95)(35 112 198 96 74)(36 89 105 75 199)(37 76 90 200 106)(38 193 77 107 91)(39 108 194 92 78)(40 93 109 79 195)(97 115 179 135 126)(98 136 116 127 180)(99 128 129 181 117)(100 182 121 118 130)(101 119 183 131 122)(102 132 120 123 184)(103 124 133 177 113)(104 178 125 114 134)
(1 72 168 89 100)(2 90 65 101 161)(3 102 91 162 66)(4 163 103 67 92)(5 68 164 93 104)(6 94 69 97 165)(7 98 95 166 70)(8 167 99 71 96)(9 132 38 56 169)(10 49 133 170 39)(11 171 50 40 134)(12 33 172 135 51)(13 136 34 52 173)(14 53 129 174 35)(15 175 54 36 130)(16 37 176 131 55)(17 74 81 28 128)(18 29 75 121 82)(19 122 30 83 76)(20 84 123 77 31)(21 78 85 32 124)(22 25 79 125 86)(23 126 26 87 80)(24 88 127 73 27)(41 195 114 185 159)(42 186 196 160 115)(43 153 187 116 197)(44 117 154 198 188)(45 199 118 189 155)(46 190 200 156 119)(47 157 191 120 193)(48 113 158 194 192)(57 152 142 180 111)(58 181 145 112 143)(59 105 182 144 146)(60 137 106 147 183)(61 148 138 184 107)(62 177 149 108 139)(63 109 178 140 150)(64 141 110 151 179)
(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)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)(33 37)(35 39)(42 46)(44 48)(49 53)(51 55)(58 62)(60 64)(65 69)(67 71)(74 78)(76 80)(81 85)(83 87)(90 94)(92 96)(97 101)(99 103)(106 110)(108 112)(113 117)(115 119)(122 126)(124 128)(129 133)(131 135)(137 141)(139 143)(145 149)(147 151)(154 158)(156 160)(161 165)(163 167)(170 174)(172 176)(177 181)(179 183)(186 190)(188 192)(194 198)(196 200)```

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

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

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

34 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A ··· 5F 8A 8B 8C 8D 10A ··· 10R order 1 2 2 4 4 4 5 ··· 5 8 8 8 8 10 ··· 10 size 1 1 2 25 25 50 4 ··· 4 50 50 50 50 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 C22.F5 kernel C52⋊13M4(2) C52⋊4C8 C2×C52⋊6C4 C52⋊6C4 C102 C52 C2×C10 C10 C5 # reps 1 2 1 2 2 2 6 6 12

Matrix representation of C5213M4(2) in GL8(𝔽41)

 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 37 0 0 0 0 0 0 0 10 10 0 0 0 0 0 0 32 0 16 0 0 0 0 0 38 0 0 18
,
 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 37 0 0 0 0 0 0 0 10 10 0 0 0 0 0 0 32 0 16 0 0 0 0 0 38 0 0 18
,
 8 23 28 17 0 0 0 0 5 35 18 26 0 0 0 0 24 32 6 11 0 0 0 0 15 20 9 33 0 0 0 0 0 0 0 0 14 0 19 0 0 0 0 0 29 0 4 1 0 0 0 0 5 40 27 0 0 0 0 0 0 0 38 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 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 40 0 0 0 0 0 4 0 0 40

`G:=sub<GL(8,GF(41))| [0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,0,37,10,32,38,0,0,0,0,0,10,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,18],[0,0,0,40,0,0,0,0,1,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,37,10,32,38,0,0,0,0,0,10,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,18],[8,5,24,15,0,0,0,0,23,35,32,20,0,0,0,0,28,18,6,9,0,0,0,0,17,26,11,33,0,0,0,0,0,0,0,0,14,29,5,0,0,0,0,0,0,0,40,0,0,0,0,0,19,4,27,38,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,5,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;`

C5213M4(2) in GAP, Magma, Sage, TeX

`C_5^2\rtimes_{13}M_4(2)`
`% in TeX`

`G:=Group("C5^2:13M4(2)");`
`// GroupNames label`

`G:=SmallGroup(400,154);`
`// by ID`

`G=gap.SmallGroup(400,154);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,50,964,496,5765,2897]);`
`// 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^3,a*d=d*a,c*b*c^-1=b^3,b*d=d*b,d*c*d=c^5>;`
`// generators/relations`

׿
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