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G = C52.53D4order 416 = 25·13

10th non-split extension by C52 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52.53D4, M4(2).1D13, C22.1Dic26, (C2×C26).Q8, C132C8.1C4, C52.26(C2×C4), (C2×C4).37D26, C4.13(C4×D13), C26.15(C4⋊C4), C134(C8.C4), C52.4C4.2C2, C4.28(C13⋊D4), (C2×C52).12C22, C2.5(C26.D4), (C13×M4(2)).1C2, (C2×C132C8).4C2, SmallGroup(416,29)

Series: Derived Chief Lower central Upper central

C1C52 — C52.53D4
C1C13C26C52C2×C52C2×C132C8 — C52.53D4
C13C26C52 — C52.53D4
C1C4C2×C4M4(2)

Generators and relations for C52.53D4
 G = < a,b,c | a52=1, b4=a26, c2=a13, bab-1=cac-1=a25, cbc-1=a26b3 >

2C2
2C26
2C8
13C8
13C8
26C8
13M4(2)
13C2×C8
2C104
2C132C8
13C8.C4

Smallest permutation representation of C52.53D4
On 208 points
Generators in S208
(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)(157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)
(1 150 40 137 27 124 14 111)(2 123 41 110 28 149 15 136)(3 148 42 135 29 122 16 109)(4 121 43 108 30 147 17 134)(5 146 44 133 31 120 18 107)(6 119 45 106 32 145 19 132)(7 144 46 131 33 118 20 105)(8 117 47 156 34 143 21 130)(9 142 48 129 35 116 22 155)(10 115 49 154 36 141 23 128)(11 140 50 127 37 114 24 153)(12 113 51 152 38 139 25 126)(13 138 52 125 39 112 26 151)(53 163 66 176 79 189 92 202)(54 188 67 201 80 162 93 175)(55 161 68 174 81 187 94 200)(56 186 69 199 82 160 95 173)(57 159 70 172 83 185 96 198)(58 184 71 197 84 158 97 171)(59 157 72 170 85 183 98 196)(60 182 73 195 86 208 99 169)(61 207 74 168 87 181 100 194)(62 180 75 193 88 206 101 167)(63 205 76 166 89 179 102 192)(64 178 77 191 90 204 103 165)(65 203 78 164 91 177 104 190)
(1 168 14 181 27 194 40 207)(2 193 15 206 28 167 41 180)(3 166 16 179 29 192 42 205)(4 191 17 204 30 165 43 178)(5 164 18 177 31 190 44 203)(6 189 19 202 32 163 45 176)(7 162 20 175 33 188 46 201)(8 187 21 200 34 161 47 174)(9 160 22 173 35 186 48 199)(10 185 23 198 36 159 49 172)(11 158 24 171 37 184 50 197)(12 183 25 196 38 157 51 170)(13 208 26 169 39 182 52 195)(53 106 66 119 79 132 92 145)(54 131 67 144 80 105 93 118)(55 156 68 117 81 130 94 143)(56 129 69 142 82 155 95 116)(57 154 70 115 83 128 96 141)(58 127 71 140 84 153 97 114)(59 152 72 113 85 126 98 139)(60 125 73 138 86 151 99 112)(61 150 74 111 87 124 100 137)(62 123 75 136 88 149 101 110)(63 148 76 109 89 122 102 135)(64 121 77 134 90 147 103 108)(65 146 78 107 91 120 104 133)

G:=sub<Sym(208)| (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)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,150,40,137,27,124,14,111)(2,123,41,110,28,149,15,136)(3,148,42,135,29,122,16,109)(4,121,43,108,30,147,17,134)(5,146,44,133,31,120,18,107)(6,119,45,106,32,145,19,132)(7,144,46,131,33,118,20,105)(8,117,47,156,34,143,21,130)(9,142,48,129,35,116,22,155)(10,115,49,154,36,141,23,128)(11,140,50,127,37,114,24,153)(12,113,51,152,38,139,25,126)(13,138,52,125,39,112,26,151)(53,163,66,176,79,189,92,202)(54,188,67,201,80,162,93,175)(55,161,68,174,81,187,94,200)(56,186,69,199,82,160,95,173)(57,159,70,172,83,185,96,198)(58,184,71,197,84,158,97,171)(59,157,72,170,85,183,98,196)(60,182,73,195,86,208,99,169)(61,207,74,168,87,181,100,194)(62,180,75,193,88,206,101,167)(63,205,76,166,89,179,102,192)(64,178,77,191,90,204,103,165)(65,203,78,164,91,177,104,190), (1,168,14,181,27,194,40,207)(2,193,15,206,28,167,41,180)(3,166,16,179,29,192,42,205)(4,191,17,204,30,165,43,178)(5,164,18,177,31,190,44,203)(6,189,19,202,32,163,45,176)(7,162,20,175,33,188,46,201)(8,187,21,200,34,161,47,174)(9,160,22,173,35,186,48,199)(10,185,23,198,36,159,49,172)(11,158,24,171,37,184,50,197)(12,183,25,196,38,157,51,170)(13,208,26,169,39,182,52,195)(53,106,66,119,79,132,92,145)(54,131,67,144,80,105,93,118)(55,156,68,117,81,130,94,143)(56,129,69,142,82,155,95,116)(57,154,70,115,83,128,96,141)(58,127,71,140,84,153,97,114)(59,152,72,113,85,126,98,139)(60,125,73,138,86,151,99,112)(61,150,74,111,87,124,100,137)(62,123,75,136,88,149,101,110)(63,148,76,109,89,122,102,135)(64,121,77,134,90,147,103,108)(65,146,78,107,91,120,104,133)>;

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)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,150,40,137,27,124,14,111)(2,123,41,110,28,149,15,136)(3,148,42,135,29,122,16,109)(4,121,43,108,30,147,17,134)(5,146,44,133,31,120,18,107)(6,119,45,106,32,145,19,132)(7,144,46,131,33,118,20,105)(8,117,47,156,34,143,21,130)(9,142,48,129,35,116,22,155)(10,115,49,154,36,141,23,128)(11,140,50,127,37,114,24,153)(12,113,51,152,38,139,25,126)(13,138,52,125,39,112,26,151)(53,163,66,176,79,189,92,202)(54,188,67,201,80,162,93,175)(55,161,68,174,81,187,94,200)(56,186,69,199,82,160,95,173)(57,159,70,172,83,185,96,198)(58,184,71,197,84,158,97,171)(59,157,72,170,85,183,98,196)(60,182,73,195,86,208,99,169)(61,207,74,168,87,181,100,194)(62,180,75,193,88,206,101,167)(63,205,76,166,89,179,102,192)(64,178,77,191,90,204,103,165)(65,203,78,164,91,177,104,190), (1,168,14,181,27,194,40,207)(2,193,15,206,28,167,41,180)(3,166,16,179,29,192,42,205)(4,191,17,204,30,165,43,178)(5,164,18,177,31,190,44,203)(6,189,19,202,32,163,45,176)(7,162,20,175,33,188,46,201)(8,187,21,200,34,161,47,174)(9,160,22,173,35,186,48,199)(10,185,23,198,36,159,49,172)(11,158,24,171,37,184,50,197)(12,183,25,196,38,157,51,170)(13,208,26,169,39,182,52,195)(53,106,66,119,79,132,92,145)(54,131,67,144,80,105,93,118)(55,156,68,117,81,130,94,143)(56,129,69,142,82,155,95,116)(57,154,70,115,83,128,96,141)(58,127,71,140,84,153,97,114)(59,152,72,113,85,126,98,139)(60,125,73,138,86,151,99,112)(61,150,74,111,87,124,100,137)(62,123,75,136,88,149,101,110)(63,148,76,109,89,122,102,135)(64,121,77,134,90,147,103,108)(65,146,78,107,91,120,104,133) );

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),(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)], [(1,150,40,137,27,124,14,111),(2,123,41,110,28,149,15,136),(3,148,42,135,29,122,16,109),(4,121,43,108,30,147,17,134),(5,146,44,133,31,120,18,107),(6,119,45,106,32,145,19,132),(7,144,46,131,33,118,20,105),(8,117,47,156,34,143,21,130),(9,142,48,129,35,116,22,155),(10,115,49,154,36,141,23,128),(11,140,50,127,37,114,24,153),(12,113,51,152,38,139,25,126),(13,138,52,125,39,112,26,151),(53,163,66,176,79,189,92,202),(54,188,67,201,80,162,93,175),(55,161,68,174,81,187,94,200),(56,186,69,199,82,160,95,173),(57,159,70,172,83,185,96,198),(58,184,71,197,84,158,97,171),(59,157,72,170,85,183,98,196),(60,182,73,195,86,208,99,169),(61,207,74,168,87,181,100,194),(62,180,75,193,88,206,101,167),(63,205,76,166,89,179,102,192),(64,178,77,191,90,204,103,165),(65,203,78,164,91,177,104,190)], [(1,168,14,181,27,194,40,207),(2,193,15,206,28,167,41,180),(3,166,16,179,29,192,42,205),(4,191,17,204,30,165,43,178),(5,164,18,177,31,190,44,203),(6,189,19,202,32,163,45,176),(7,162,20,175,33,188,46,201),(8,187,21,200,34,161,47,174),(9,160,22,173,35,186,48,199),(10,185,23,198,36,159,49,172),(11,158,24,171,37,184,50,197),(12,183,25,196,38,157,51,170),(13,208,26,169,39,182,52,195),(53,106,66,119,79,132,92,145),(54,131,67,144,80,105,93,118),(55,156,68,117,81,130,94,143),(56,129,69,142,82,155,95,116),(57,154,70,115,83,128,96,141),(58,127,71,140,84,153,97,114),(59,152,72,113,85,126,98,139),(60,125,73,138,86,151,99,112),(61,150,74,111,87,124,100,137),(62,123,75,136,88,149,101,110),(63,148,76,109,89,122,102,135),(64,121,77,134,90,147,103,108),(65,146,78,107,91,120,104,133)])

74 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D8E8F8G8H13A···13F26A···26F26G···26L52A···52L52M···52R104A···104X
order1224448888888813···1326···2626···2652···5252···52104···104
size112112442626262652522···22···24···42···24···44···4

74 irreducible representations

dim11111222222224
type+++++-++-
imageC1C2C2C2C4D4Q8D13C8.C4D26C4×D13C13⋊D4Dic26C52.53D4
kernelC52.53D4C2×C132C8C52.4C4C13×M4(2)C132C8C52C2×C26M4(2)C13C2×C4C4C4C22C1
# reps111141164612121212

Matrix representation of C52.53D4 in GL4(𝔽313) generated by

25000
02500
0012625
0016110
,
125000
030800
00211130
00245102
,
030800
308000
0023341
005080
G:=sub<GL(4,GF(313))| [25,0,0,0,0,25,0,0,0,0,126,16,0,0,25,110],[125,0,0,0,0,308,0,0,0,0,211,245,0,0,130,102],[0,308,0,0,308,0,0,0,0,0,233,50,0,0,41,80] >;

C52.53D4 in GAP, Magma, Sage, TeX

C_{52}._{53}D_4
% in TeX

G:=Group("C52.53D4");
// GroupNames label

G:=SmallGroup(416,29);
// by ID

G=gap.SmallGroup(416,29);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,121,31,86,297,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^52=1,b^4=a^26,c^2=a^13,b*a*b^-1=c*a*c^-1=a^25,c*b*c^-1=a^26*b^3>;
// generators/relations

Export

Subgroup lattice of C52.53D4 in TeX

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