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G = D26.C8order 416 = 25·13

3rd non-split extension by D26 of C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C104.5C4, D26.3C8, C131M5(2), Dic13.3C8, C13⋊C161C2, C8.3(C13⋊C4), C26.2(C2×C8), C52.14(C2×C4), (C4×D13).6C4, (C8×D13).8C2, C2.3(D13⋊C8), C132C8.15C22, C4.15(C2×C13⋊C4), SmallGroup(416,65)

Series: Derived Chief Lower central Upper central

C1C26 — D26.C8
C1C13C26C52C132C8C13⋊C16 — D26.C8
C13C26 — D26.C8
C1C4C8

Generators and relations for D26.C8
 G = < a,b,c | a26=b2=1, c8=a13, bab=a-1, cac-1=a5, cbc-1=a17b >

26C2
13C22
13C4
2D13
13C8
13C2×C4
13C16
13C2×C8
13C16
13M5(2)

Smallest permutation representation of D26.C8
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 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)(13 14)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(47 52)(48 51)(49 50)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(69 78)(70 77)(71 76)(72 75)(73 74)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(95 104)(96 103)(97 102)(98 101)(99 100)(105 109)(106 108)(110 130)(111 129)(112 128)(113 127)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)(131 143)(132 142)(133 141)(134 140)(135 139)(136 138)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)(158 182)(159 181)(160 180)(161 179)(162 178)(163 177)(164 176)(165 175)(166 174)(167 173)(168 172)(169 171)(183 197)(184 196)(185 195)(186 194)(187 193)(188 192)(189 191)(198 208)(199 207)(200 206)(201 205)(202 204)
(1 184 87 144 50 177 61 114 14 197 100 131 37 164 74 127)(2 205 86 149 51 172 60 119 15 192 99 136 38 159 73 106)(3 200 85 154 52 167 59 124 16 187 98 141 39 180 72 111)(4 195 84 133 27 162 58 129 17 208 97 146 40 175 71 116)(5 190 83 138 28 157 57 108 18 203 96 151 41 170 70 121)(6 185 82 143 29 178 56 113 19 198 95 156 42 165 69 126)(7 206 81 148 30 173 55 118 20 193 94 135 43 160 68 105)(8 201 80 153 31 168 54 123 21 188 93 140 44 181 67 110)(9 196 79 132 32 163 53 128 22 183 92 145 45 176 66 115)(10 191 104 137 33 158 78 107 23 204 91 150 46 171 65 120)(11 186 103 142 34 179 77 112 24 199 90 155 47 166 64 125)(12 207 102 147 35 174 76 117 25 194 89 134 48 161 63 130)(13 202 101 152 36 169 75 122 26 189 88 139 49 182 62 109)

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

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

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

44 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D8E8F13A13B13C16A···16H26A26B26C52A···52F104A···104L
order12244488888813131316···1626262652···52104···104
size11261126221313131344426···264444···44···4

44 irreducible representations

dim111111124444
type+++++
imageC1C2C2C4C4C8C8M5(2)C13⋊C4C2×C13⋊C4D13⋊C8D26.C8
kernelD26.C8C13⋊C16C8×D13C104C4×D13Dic13D26C13C8C4C2C1
# reps1212244433612

Matrix representation of D26.C8 in GL4(𝔽5) generated by

3441
3414
3342
2412
,
3240
3311
4242
3410
,
1212
3010
2023
2022
G:=sub<GL(4,GF(5))| [3,3,3,2,4,4,3,4,4,1,4,1,1,4,2,2],[3,3,4,3,2,3,2,4,4,1,4,1,0,1,2,0],[1,3,2,2,2,0,0,0,1,1,2,2,2,0,3,2] >;

D26.C8 in GAP, Magma, Sage, TeX

D_{26}.C_8
% in TeX

G:=Group("D26.C8");
// GroupNames label

G:=SmallGroup(416,65);
// by ID

G=gap.SmallGroup(416,65);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,55,50,69,9221,3473]);
// Polycyclic

G:=Group<a,b,c|a^26=b^2=1,c^8=a^13,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^17*b>;
// generators/relations

Export

Subgroup lattice of D26.C8 in TeX

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