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## G = M4(2)×C26order 416 = 25·13

### Direct product of C26 and M4(2)

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — M4(2)×C26
 Chief series C1 — C2 — C4 — C52 — C104 — C13×M4(2) — M4(2)×C26
 Lower central C1 — C2 — M4(2)×C26
 Upper central C1 — C2×C52 — M4(2)×C26

Generators and relations for M4(2)×C26
G = < a,b,c | a26=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Subgroups: 76 in 68 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C13, C2×C8 [×2], M4(2) [×4], C22×C4, C26, C26 [×2], C26 [×2], C2×M4(2), C52 [×2], C52 [×2], C2×C26, C2×C26 [×2], C2×C26 [×2], C104 [×4], C2×C52 [×2], C2×C52 [×4], C22×C26, C2×C104 [×2], C13×M4(2) [×4], C22×C52, M4(2)×C26
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C13, M4(2) [×2], C22×C4, C26 [×7], C2×M4(2), C52 [×4], C2×C26 [×7], C2×C52 [×6], C22×C26, C13×M4(2) [×2], C22×C52, M4(2)×C26

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

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

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

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

260 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A ··· 8H 13A ··· 13L 26A ··· 26AJ 26AK ··· 26BH 52A ··· 52AV 52AW ··· 52BT 104A ··· 104CR order 1 2 2 2 2 2 4 4 4 4 4 4 8 ··· 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 1 1 2 2 1 1 1 1 2 2 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

260 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C4 C4 C13 C26 C26 C26 C52 C52 M4(2) C13×M4(2) kernel M4(2)×C26 C2×C104 C13×M4(2) C22×C52 C2×C52 C22×C26 C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C23 C26 C2 # reps 1 2 4 1 6 2 12 24 48 12 72 24 4 48

Matrix representation of M4(2)×C26 in GL3(𝔽313) generated by

 312 0 0 0 48 0 0 0 48
,
 25 0 0 0 139 311 0 283 174
,
 1 0 0 0 312 0 0 174 1
G:=sub<GL(3,GF(313))| [312,0,0,0,48,0,0,0,48],[25,0,0,0,139,283,0,311,174],[1,0,0,0,312,174,0,0,1] >;

M4(2)×C26 in GAP, Magma, Sage, TeX

M_4(2)\times C_{26}
% in TeX

G:=Group("M4(2)xC26");
// GroupNames label

G:=SmallGroup(416,191);
// by ID

G=gap.SmallGroup(416,191);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,624,2521,88]);
// Polycyclic

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

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