metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C25⋊2M4(2), Dic25.3C4, Dic25.8C22, C25⋊C8⋊2C2, C50.6(C2×C4), (C2×C50).2C4, C22.(C25⋊C4), (C2×C10).2F5, C10.11(C2×F5), C5.(C22.F5), (C2×Dic25).5C2, C2.6(C2×C25⋊C4), SmallGroup(400,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C25⋊M4(2)
G = < a,b,c | a25=b8=c2=1, bab-1=a18, ac=ca, cbc=b5 >
Smallest permutation representation of C25⋊M4(2)
►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)
(1 170 51 131 33 176 90 117)(2 152 75 149 34 183 89 110)(3 159 74 142 35 190 88 103)(4 166 73 135 36 197 87 121)(5 173 72 128 37 179 86 114)(6 155 71 146 38 186 85 107)(7 162 70 139 39 193 84 125)(8 169 69 132 40 200 83 118)(9 151 68 150 41 182 82 111)(10 158 67 143 42 189 81 104)(11 165 66 136 43 196 80 122)(12 172 65 129 44 178 79 115)(13 154 64 147 45 185 78 108)(14 161 63 140 46 192 77 101)(15 168 62 133 47 199 76 119)(16 175 61 126 48 181 100 112)(17 157 60 144 49 188 99 105)(18 164 59 137 50 195 98 123)(19 171 58 130 26 177 97 116)(20 153 57 148 27 184 96 109)(21 160 56 141 28 191 95 102)(22 167 55 134 29 198 94 120)(23 174 54 127 30 180 93 113)(24 156 53 145 31 187 92 106)(25 163 52 138 32 194 91 124)
(101 140)(102 141)(103 142)(104 143)(105 144)(106 145)(107 146)(108 147)(109 148)(110 149)(111 150)(112 126)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(151 182)(152 183)(153 184)(154 185)(155 186)(156 187)(157 188)(158 189)(159 190)(160 191)(161 192)(162 193)(163 194)(164 195)(165 196)(166 197)(167 198)(168 199)(169 200)(170 176)(171 177)(172 178)(173 179)(174 180)(175 181)
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), (1,170,51,131,33,176,90,117)(2,152,75,149,34,183,89,110)(3,159,74,142,35,190,88,103)(4,166,73,135,36,197,87,121)(5,173,72,128,37,179,86,114)(6,155,71,146,38,186,85,107)(7,162,70,139,39,193,84,125)(8,169,69,132,40,200,83,118)(9,151,68,150,41,182,82,111)(10,158,67,143,42,189,81,104)(11,165,66,136,43,196,80,122)(12,172,65,129,44,178,79,115)(13,154,64,147,45,185,78,108)(14,161,63,140,46,192,77,101)(15,168,62,133,47,199,76,119)(16,175,61,126,48,181,100,112)(17,157,60,144,49,188,99,105)(18,164,59,137,50,195,98,123)(19,171,58,130,26,177,97,116)(20,153,57,148,27,184,96,109)(21,160,56,141,28,191,95,102)(22,167,55,134,29,198,94,120)(23,174,54,127,30,180,93,113)(24,156,53,145,31,187,92,106)(25,163,52,138,32,194,91,124), (101,140)(102,141)(103,142)(104,143)(105,144)(106,145)(107,146)(108,147)(109,148)(110,149)(111,150)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(151,182)(152,183)(153,184)(154,185)(155,186)(156,187)(157,188)(158,189)(159,190)(160,191)(161,192)(162,193)(163,194)(164,195)(165,196)(166,197)(167,198)(168,199)(169,200)(170,176)(171,177)(172,178)(173,179)(174,180)(175,181)>;
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), (1,170,51,131,33,176,90,117)(2,152,75,149,34,183,89,110)(3,159,74,142,35,190,88,103)(4,166,73,135,36,197,87,121)(5,173,72,128,37,179,86,114)(6,155,71,146,38,186,85,107)(7,162,70,139,39,193,84,125)(8,169,69,132,40,200,83,118)(9,151,68,150,41,182,82,111)(10,158,67,143,42,189,81,104)(11,165,66,136,43,196,80,122)(12,172,65,129,44,178,79,115)(13,154,64,147,45,185,78,108)(14,161,63,140,46,192,77,101)(15,168,62,133,47,199,76,119)(16,175,61,126,48,181,100,112)(17,157,60,144,49,188,99,105)(18,164,59,137,50,195,98,123)(19,171,58,130,26,177,97,116)(20,153,57,148,27,184,96,109)(21,160,56,141,28,191,95,102)(22,167,55,134,29,198,94,120)(23,174,54,127,30,180,93,113)(24,156,53,145,31,187,92,106)(25,163,52,138,32,194,91,124), (101,140)(102,141)(103,142)(104,143)(105,144)(106,145)(107,146)(108,147)(109,148)(110,149)(111,150)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(151,182)(152,183)(153,184)(154,185)(155,186)(156,187)(157,188)(158,189)(159,190)(160,191)(161,192)(162,193)(163,194)(164,195)(165,196)(166,197)(167,198)(168,199)(169,200)(170,176)(171,177)(172,178)(173,179)(174,180)(175,181) );
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)], [(1,170,51,131,33,176,90,117),(2,152,75,149,34,183,89,110),(3,159,74,142,35,190,88,103),(4,166,73,135,36,197,87,121),(5,173,72,128,37,179,86,114),(6,155,71,146,38,186,85,107),(7,162,70,139,39,193,84,125),(8,169,69,132,40,200,83,118),(9,151,68,150,41,182,82,111),(10,158,67,143,42,189,81,104),(11,165,66,136,43,196,80,122),(12,172,65,129,44,178,79,115),(13,154,64,147,45,185,78,108),(14,161,63,140,46,192,77,101),(15,168,62,133,47,199,76,119),(16,175,61,126,48,181,100,112),(17,157,60,144,49,188,99,105),(18,164,59,137,50,195,98,123),(19,171,58,130,26,177,97,116),(20,153,57,148,27,184,96,109),(21,160,56,141,28,191,95,102),(22,167,55,134,29,198,94,120),(23,174,54,127,30,180,93,113),(24,156,53,145,31,187,92,106),(25,163,52,138,32,194,91,124)], [(101,140),(102,141),(103,142),(104,143),(105,144),(106,145),(107,146),(108,147),(109,148),(110,149),(111,150),(112,126),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(151,182),(152,183),(153,184),(154,185),(155,186),(156,187),(157,188),(158,189),(159,190),(160,191),(161,192),(162,193),(163,194),(164,195),(165,196),(166,197),(167,198),(168,199),(169,200),(170,176),(171,177),(172,178),(173,179),(174,180),(175,181)]])
34 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 25A | ··· | 25E | 50A | ··· | 50O |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 25 | ··· | 25 | 50 | ··· | 50 |
| size | 1 | 1 | 2 | 25 | 25 | 50 | 4 | 50 | 50 | 50 | 50 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C4 | C4 | M4(2) | F5 | C2×F5 | C22.F5 | C25⋊C4 | C2×C25⋊C4 | C25⋊M4(2) |
| kernel | C25⋊M4(2) | C25⋊C8 | C2×Dic25 | Dic25 | C2×C50 | C25 | C2×C10 | C10 | C5 | C22 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 5 | 5 | 10 |
Matrix representation of C25⋊M4(2) ►in GL4(𝔽401) generated by
| 9 | 266 | 0 | 0 |
| 135 | 292 | 0 | 0 |
| 0 | 0 | 134 | 72 |
| 0 | 0 | 329 | 90 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 361 | 277 | 0 | 0 |
| 346 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 400 | 0 |
| 0 | 0 | 0 | 400 |
G:=sub<GL(4,GF(401))| [9,135,0,0,266,292,0,0,0,0,134,329,0,0,72,90],[0,0,361,346,0,0,277,40,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,400,0,0,0,0,400] >;
C25⋊M4(2) in GAP, Magma, Sage, TeX
C_{25}\rtimes M_4(2) % in TeX
G:=Group("C25:M4(2)"); // GroupNames label
G:=SmallGroup(400,33);
// by ID
G=gap.SmallGroup(400,33);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,50,3364,2896,178,5765,2897]);
// Polycyclic
G:=Group<a,b,c|a^25=b^8=c^2=1,b*a*b^-1=a^18,a*c=c*a,c*b*c=b^5>;
// generators/relations
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