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G = C4.Dic25order 400 = 24·52

The non-split extension by C4 of Dic25 acting via Dic25/C50=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C4.Dic25, C100.4C4, C4.15D50, C254M4(2), C20.54D10, C20.4Dic5, C22.Dic25, C100.15C22, C252C85C2, (C2×C50).5C4, (C2×C4).2D25, (C2×C100).4C2, C50.14(C2×C4), (C2×C20).10D5, C5.(C4.Dic5), (C2×C10).5Dic5, C2.3(C2×Dic25), C10.14(C2×Dic5), SmallGroup(400,10)

Series: Derived Chief Lower central Upper central

C1C50 — C4.Dic25
C1C5C25C50C100C252C8 — C4.Dic25
C25C50 — C4.Dic25
C1C4C2×C4

Generators and relations for C4.Dic25
 G = < a,b,c | a4=1, b50=a2, c2=b25, ab=ba, cac-1=a-1, cbc-1=b49 >

2C2
2C10
25C8
25C8
2C50
25M4(2)
5C52C8
5C52C8
5C4.Dic5

Smallest permutation representation of C4.Dic25
On 200 points
Generators in S200
(1 26 51 76)(2 27 52 77)(3 28 53 78)(4 29 54 79)(5 30 55 80)(6 31 56 81)(7 32 57 82)(8 33 58 83)(9 34 59 84)(10 35 60 85)(11 36 61 86)(12 37 62 87)(13 38 63 88)(14 39 64 89)(15 40 65 90)(16 41 66 91)(17 42 67 92)(18 43 68 93)(19 44 69 94)(20 45 70 95)(21 46 71 96)(22 47 72 97)(23 48 73 98)(24 49 74 99)(25 50 75 100)(101 176 151 126)(102 177 152 127)(103 178 153 128)(104 179 154 129)(105 180 155 130)(106 181 156 131)(107 182 157 132)(108 183 158 133)(109 184 159 134)(110 185 160 135)(111 186 161 136)(112 187 162 137)(113 188 163 138)(114 189 164 139)(115 190 165 140)(116 191 166 141)(117 192 167 142)(118 193 168 143)(119 194 169 144)(120 195 170 145)(121 196 171 146)(122 197 172 147)(123 198 173 148)(124 199 174 149)(125 200 175 150)
(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)
(1 182 26 107 51 132 76 157)(2 131 27 156 52 181 77 106)(3 180 28 105 53 130 78 155)(4 129 29 154 54 179 79 104)(5 178 30 103 55 128 80 153)(6 127 31 152 56 177 81 102)(7 176 32 101 57 126 82 151)(8 125 33 150 58 175 83 200)(9 174 34 199 59 124 84 149)(10 123 35 148 60 173 85 198)(11 172 36 197 61 122 86 147)(12 121 37 146 62 171 87 196)(13 170 38 195 63 120 88 145)(14 119 39 144 64 169 89 194)(15 168 40 193 65 118 90 143)(16 117 41 142 66 167 91 192)(17 166 42 191 67 116 92 141)(18 115 43 140 68 165 93 190)(19 164 44 189 69 114 94 139)(20 113 45 138 70 163 95 188)(21 162 46 187 71 112 96 137)(22 111 47 136 72 161 97 186)(23 160 48 185 73 110 98 135)(24 109 49 134 74 159 99 184)(25 158 50 183 75 108 100 133)

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

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

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

106 conjugacy classes

class 1 2A2B4A4B4C5A5B8A8B8C8D10A···10F20A···20H25A···25J50A···50AD100A···100AN
order12244455888810···1020···2025···2550···50100···100
size11211222505050502···22···22···22···22···2

106 irreducible representations

dim1111122222222222
type++++-+-+-+-
imageC1C2C2C4C4D5M4(2)Dic5D10Dic5D25C4.Dic5Dic25D50Dic25C4.Dic25
kernelC4.Dic25C252C8C2×C100C100C2×C50C2×C20C25C20C20C2×C10C2×C4C5C4C4C22C1
# reps121222222210810101040

Matrix representation of C4.Dic25 in GL2(𝔽401) generated by

200
120381
,
2850
91121
,
18395
318383
G:=sub<GL(2,GF(401))| [20,120,0,381],[285,91,0,121],[18,318,395,383] >;

C4.Dic25 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_{25}
% in TeX

G:=Group("C4.Dic25");
// GroupNames label

G:=SmallGroup(400,10);
// by ID

G=gap.SmallGroup(400,10);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,50,4324,628,11525]);
// Polycyclic

G:=Group<a,b,c|a^4=1,b^50=a^2,c^2=b^25,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^49>;
// generators/relations

Export

Subgroup lattice of C4.Dic25 in TeX

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