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G = C26.M4(2)  order 416 = 25·13

2nd non-split extension by C26 of M4(2) acting via M4(2)/C22=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C26.5M4(2), Dic13.23D4, (C2×C26)⋊1C8, C22⋊(C13⋊C8), C26.9(C2×C8), C132(C22⋊C8), C23.2(C13⋊C4), (C22×C26).3C4, C2.3(D13.D4), C26.10(C22⋊C4), (C2×Dic13).12C4, C2.3(C13⋊M4(2)), (C22×Dic13).7C2, (C2×Dic13).53C22, (C2×C13⋊C8)⋊2C2, C2.5(C2×C13⋊C8), (C2×C26).11(C2×C4), C22.14(C2×C13⋊C4), SmallGroup(416,87)

Series: Derived Chief Lower central Upper central

C1C26 — C26.M4(2)
C1C13C26Dic13C2×Dic13C2×C13⋊C8 — C26.M4(2)
C13C26 — C26.M4(2)
C1C22C23

Generators and relations for C26.M4(2)
 G = < a,b,c | a26=b8=c2=1, bab-1=a5, ac=ca, cbc=a13b5 >

2C2
2C2
2C22
2C22
13C4
13C4
26C4
2C26
2C26
13C2×C4
13C2×C4
26C8
26C8
26C2×C4
26C2×C4
2C2×C26
2Dic13
2C2×C26
13C2×C8
13C22×C4
13C2×C8
2C13⋊C8
2C2×Dic13
2C13⋊C8
2C2×Dic13
13C22⋊C8

Smallest permutation representation of C26.M4(2)
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)
(1 138 173 75 101 27 112 184)(2 133 172 54 102 48 111 189)(3 154 171 59 103 43 110 194)(4 149 170 64 104 38 109 199)(5 144 169 69 79 33 108 204)(6 139 168 74 80 28 107 183)(7 134 167 53 81 49 106 188)(8 155 166 58 82 44 105 193)(9 150 165 63 83 39 130 198)(10 145 164 68 84 34 129 203)(11 140 163 73 85 29 128 208)(12 135 162 78 86 50 127 187)(13 156 161 57 87 45 126 192)(14 151 160 62 88 40 125 197)(15 146 159 67 89 35 124 202)(16 141 158 72 90 30 123 207)(17 136 157 77 91 51 122 186)(18 131 182 56 92 46 121 191)(19 152 181 61 93 41 120 196)(20 147 180 66 94 36 119 201)(21 142 179 71 95 31 118 206)(22 137 178 76 96 52 117 185)(23 132 177 55 97 47 116 190)(24 153 176 60 98 42 115 195)(25 148 175 65 99 37 114 200)(26 143 174 70 100 32 113 205)
(27 151)(28 152)(29 153)(30 154)(31 155)(32 156)(33 131)(34 132)(35 133)(36 134)(37 135)(38 136)(39 137)(40 138)(41 139)(42 140)(43 141)(44 142)(45 143)(46 144)(47 145)(48 146)(49 147)(50 148)(51 149)(52 150)(53 201)(54 202)(55 203)(56 204)(57 205)(58 206)(59 207)(60 208)(61 183)(62 184)(63 185)(64 186)(65 187)(66 188)(67 189)(68 190)(69 191)(70 192)(71 193)(72 194)(73 195)(74 196)(75 197)(76 198)(77 199)(78 200)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F8A···8H13A13B13C26A···26U
order1222224444448···813131326···26
size11112213131313262626···264444···4

44 irreducible representations

dim1111112244444
type+++++-+-+
imageC1C2C2C4C4C8D4M4(2)C13⋊C4C13⋊C8C2×C13⋊C4C13⋊M4(2)D13.D4
kernelC26.M4(2)C2×C13⋊C8C22×Dic13C2×Dic13C22×C26C2×C26Dic13C26C23C22C22C2C2
# reps1212282236366

Matrix representation of C26.M4(2) in GL6(𝔽313)

31200000
03120000
0026626600
00472700
0000288162
0000151286
,
28840000
157250000
000010
000001
00777700
004923600
,
100000
1693120000
001000
000100
00003120
00000312

G:=sub<GL(6,GF(313))| [312,0,0,0,0,0,0,312,0,0,0,0,0,0,266,47,0,0,0,0,266,27,0,0,0,0,0,0,288,151,0,0,0,0,162,286],[288,157,0,0,0,0,4,25,0,0,0,0,0,0,0,0,77,49,0,0,0,0,77,236,0,0,1,0,0,0,0,0,0,1,0,0],[1,169,0,0,0,0,0,312,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,312,0,0,0,0,0,0,312] >;

C26.M4(2) in GAP, Magma, Sage, TeX

C_{26}.M_4(2)
% in TeX

G:=Group("C26.M4(2)");
// GroupNames label

G:=SmallGroup(416,87);
// by ID

G=gap.SmallGroup(416,87);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,86,9221,3473]);
// Polycyclic

G:=Group<a,b,c|a^26=b^8=c^2=1,b*a*b^-1=a^5,a*c=c*a,c*b*c=a^13*b^5>;
// generators/relations

Export

Subgroup lattice of C26.M4(2) in TeX

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