Copied to
clipboard

## G = C4○D4×C26order 416 = 25·13

### Direct product of C26 and C4○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C26
 Chief series C1 — C2 — C26 — C2×C26 — D4×C13 — C13×C4○D4 — C4○D4×C26
 Lower central C1 — C2 — C4○D4×C26
 Upper central C1 — C2×C52 — C4○D4×C26

Generators and relations for C4○D4×C26
G = < a,b,c,d | a26=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 188 in 164 conjugacy classes, 140 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C13, C22×C4, C2×D4, C2×Q8, C4○D4, C26, C26, C26, C2×C4○D4, C52, C2×C26, C2×C26, C2×C26, C2×C52, C2×C52, D4×C13, Q8×C13, C22×C26, C22×C52, D4×C26, Q8×C26, C13×C4○D4, C4○D4×C26
Quotients: C1, C2, C22, C23, C13, C4○D4, C24, C26, C2×C4○D4, C2×C26, C22×C26, C13×C4○D4, C23×C26, C4○D4×C26

Smallest permutation representation of C4○D4×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)
(1 199 124 154)(2 200 125 155)(3 201 126 156)(4 202 127 131)(5 203 128 132)(6 204 129 133)(7 205 130 134)(8 206 105 135)(9 207 106 136)(10 208 107 137)(11 183 108 138)(12 184 109 139)(13 185 110 140)(14 186 111 141)(15 187 112 142)(16 188 113 143)(17 189 114 144)(18 190 115 145)(19 191 116 146)(20 192 117 147)(21 193 118 148)(22 194 119 149)(23 195 120 150)(24 196 121 151)(25 197 122 152)(26 198 123 153)(27 62 181 83)(28 63 182 84)(29 64 157 85)(30 65 158 86)(31 66 159 87)(32 67 160 88)(33 68 161 89)(34 69 162 90)(35 70 163 91)(36 71 164 92)(37 72 165 93)(38 73 166 94)(39 74 167 95)(40 75 168 96)(41 76 169 97)(42 77 170 98)(43 78 171 99)(44 53 172 100)(45 54 173 101)(46 55 174 102)(47 56 175 103)(48 57 176 104)(49 58 177 79)(50 59 178 80)(51 60 179 81)(52 61 180 82)
(1 96 124 75)(2 97 125 76)(3 98 126 77)(4 99 127 78)(5 100 128 53)(6 101 129 54)(7 102 130 55)(8 103 105 56)(9 104 106 57)(10 79 107 58)(11 80 108 59)(12 81 109 60)(13 82 110 61)(14 83 111 62)(15 84 112 63)(16 85 113 64)(17 86 114 65)(18 87 115 66)(19 88 116 67)(20 89 117 68)(21 90 118 69)(22 91 119 70)(23 92 120 71)(24 93 121 72)(25 94 122 73)(26 95 123 74)(27 141 181 186)(28 142 182 187)(29 143 157 188)(30 144 158 189)(31 145 159 190)(32 146 160 191)(33 147 161 192)(34 148 162 193)(35 149 163 194)(36 150 164 195)(37 151 165 196)(38 152 166 197)(39 153 167 198)(40 154 168 199)(41 155 169 200)(42 156 170 201)(43 131 171 202)(44 132 172 203)(45 133 173 204)(46 134 174 205)(47 135 175 206)(48 136 176 207)(49 137 177 208)(50 138 178 183)(51 139 179 184)(52 140 180 185)
(1 111)(2 112)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 121)(12 122)(13 123)(14 124)(15 125)(16 126)(17 127)(18 128)(19 129)(20 130)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(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)(131 189)(132 190)(133 191)(134 192)(135 193)(136 194)(137 195)(138 196)(139 197)(140 198)(141 199)(142 200)(143 201)(144 202)(145 203)(146 204)(147 205)(148 206)(149 207)(150 208)(151 183)(152 184)(153 185)(154 186)(155 187)(156 188)(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), (1,199,124,154)(2,200,125,155)(3,201,126,156)(4,202,127,131)(5,203,128,132)(6,204,129,133)(7,205,130,134)(8,206,105,135)(9,207,106,136)(10,208,107,137)(11,183,108,138)(12,184,109,139)(13,185,110,140)(14,186,111,141)(15,187,112,142)(16,188,113,143)(17,189,114,144)(18,190,115,145)(19,191,116,146)(20,192,117,147)(21,193,118,148)(22,194,119,149)(23,195,120,150)(24,196,121,151)(25,197,122,152)(26,198,123,153)(27,62,181,83)(28,63,182,84)(29,64,157,85)(30,65,158,86)(31,66,159,87)(32,67,160,88)(33,68,161,89)(34,69,162,90)(35,70,163,91)(36,71,164,92)(37,72,165,93)(38,73,166,94)(39,74,167,95)(40,75,168,96)(41,76,169,97)(42,77,170,98)(43,78,171,99)(44,53,172,100)(45,54,173,101)(46,55,174,102)(47,56,175,103)(48,57,176,104)(49,58,177,79)(50,59,178,80)(51,60,179,81)(52,61,180,82), (1,96,124,75)(2,97,125,76)(3,98,126,77)(4,99,127,78)(5,100,128,53)(6,101,129,54)(7,102,130,55)(8,103,105,56)(9,104,106,57)(10,79,107,58)(11,80,108,59)(12,81,109,60)(13,82,110,61)(14,83,111,62)(15,84,112,63)(16,85,113,64)(17,86,114,65)(18,87,115,66)(19,88,116,67)(20,89,117,68)(21,90,118,69)(22,91,119,70)(23,92,120,71)(24,93,121,72)(25,94,122,73)(26,95,123,74)(27,141,181,186)(28,142,182,187)(29,143,157,188)(30,144,158,189)(31,145,159,190)(32,146,160,191)(33,147,161,192)(34,148,162,193)(35,149,163,194)(36,150,164,195)(37,151,165,196)(38,152,166,197)(39,153,167,198)(40,154,168,199)(41,155,169,200)(42,156,170,201)(43,131,171,202)(44,132,172,203)(45,133,173,204)(46,134,174,205)(47,135,175,206)(48,136,176,207)(49,137,177,208)(50,138,178,183)(51,139,179,184)(52,140,180,185), (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,121)(12,122)(13,123)(14,124)(15,125)(16,126)(17,127)(18,128)(19,129)(20,130)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(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)(131,189)(132,190)(133,191)(134,192)(135,193)(136,194)(137,195)(138,196)(139,197)(140,198)(141,199)(142,200)(143,201)(144,202)(145,203)(146,204)(147,205)(148,206)(149,207)(150,208)(151,183)(152,184)(153,185)(154,186)(155,187)(156,188)(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), (1,199,124,154)(2,200,125,155)(3,201,126,156)(4,202,127,131)(5,203,128,132)(6,204,129,133)(7,205,130,134)(8,206,105,135)(9,207,106,136)(10,208,107,137)(11,183,108,138)(12,184,109,139)(13,185,110,140)(14,186,111,141)(15,187,112,142)(16,188,113,143)(17,189,114,144)(18,190,115,145)(19,191,116,146)(20,192,117,147)(21,193,118,148)(22,194,119,149)(23,195,120,150)(24,196,121,151)(25,197,122,152)(26,198,123,153)(27,62,181,83)(28,63,182,84)(29,64,157,85)(30,65,158,86)(31,66,159,87)(32,67,160,88)(33,68,161,89)(34,69,162,90)(35,70,163,91)(36,71,164,92)(37,72,165,93)(38,73,166,94)(39,74,167,95)(40,75,168,96)(41,76,169,97)(42,77,170,98)(43,78,171,99)(44,53,172,100)(45,54,173,101)(46,55,174,102)(47,56,175,103)(48,57,176,104)(49,58,177,79)(50,59,178,80)(51,60,179,81)(52,61,180,82), (1,96,124,75)(2,97,125,76)(3,98,126,77)(4,99,127,78)(5,100,128,53)(6,101,129,54)(7,102,130,55)(8,103,105,56)(9,104,106,57)(10,79,107,58)(11,80,108,59)(12,81,109,60)(13,82,110,61)(14,83,111,62)(15,84,112,63)(16,85,113,64)(17,86,114,65)(18,87,115,66)(19,88,116,67)(20,89,117,68)(21,90,118,69)(22,91,119,70)(23,92,120,71)(24,93,121,72)(25,94,122,73)(26,95,123,74)(27,141,181,186)(28,142,182,187)(29,143,157,188)(30,144,158,189)(31,145,159,190)(32,146,160,191)(33,147,161,192)(34,148,162,193)(35,149,163,194)(36,150,164,195)(37,151,165,196)(38,152,166,197)(39,153,167,198)(40,154,168,199)(41,155,169,200)(42,156,170,201)(43,131,171,202)(44,132,172,203)(45,133,173,204)(46,134,174,205)(47,135,175,206)(48,136,176,207)(49,137,177,208)(50,138,178,183)(51,139,179,184)(52,140,180,185), (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,121)(12,122)(13,123)(14,124)(15,125)(16,126)(17,127)(18,128)(19,129)(20,130)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(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)(131,189)(132,190)(133,191)(134,192)(135,193)(136,194)(137,195)(138,196)(139,197)(140,198)(141,199)(142,200)(143,201)(144,202)(145,203)(146,204)(147,205)(148,206)(149,207)(150,208)(151,183)(152,184)(153,185)(154,186)(155,187)(156,188)(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)], [(1,199,124,154),(2,200,125,155),(3,201,126,156),(4,202,127,131),(5,203,128,132),(6,204,129,133),(7,205,130,134),(8,206,105,135),(9,207,106,136),(10,208,107,137),(11,183,108,138),(12,184,109,139),(13,185,110,140),(14,186,111,141),(15,187,112,142),(16,188,113,143),(17,189,114,144),(18,190,115,145),(19,191,116,146),(20,192,117,147),(21,193,118,148),(22,194,119,149),(23,195,120,150),(24,196,121,151),(25,197,122,152),(26,198,123,153),(27,62,181,83),(28,63,182,84),(29,64,157,85),(30,65,158,86),(31,66,159,87),(32,67,160,88),(33,68,161,89),(34,69,162,90),(35,70,163,91),(36,71,164,92),(37,72,165,93),(38,73,166,94),(39,74,167,95),(40,75,168,96),(41,76,169,97),(42,77,170,98),(43,78,171,99),(44,53,172,100),(45,54,173,101),(46,55,174,102),(47,56,175,103),(48,57,176,104),(49,58,177,79),(50,59,178,80),(51,60,179,81),(52,61,180,82)], [(1,96,124,75),(2,97,125,76),(3,98,126,77),(4,99,127,78),(5,100,128,53),(6,101,129,54),(7,102,130,55),(8,103,105,56),(9,104,106,57),(10,79,107,58),(11,80,108,59),(12,81,109,60),(13,82,110,61),(14,83,111,62),(15,84,112,63),(16,85,113,64),(17,86,114,65),(18,87,115,66),(19,88,116,67),(20,89,117,68),(21,90,118,69),(22,91,119,70),(23,92,120,71),(24,93,121,72),(25,94,122,73),(26,95,123,74),(27,141,181,186),(28,142,182,187),(29,143,157,188),(30,144,158,189),(31,145,159,190),(32,146,160,191),(33,147,161,192),(34,148,162,193),(35,149,163,194),(36,150,164,195),(37,151,165,196),(38,152,166,197),(39,153,167,198),(40,154,168,199),(41,155,169,200),(42,156,170,201),(43,131,171,202),(44,132,172,203),(45,133,173,204),(46,134,174,205),(47,135,175,206),(48,136,176,207),(49,137,177,208),(50,138,178,183),(51,139,179,184),(52,140,180,185)], [(1,111),(2,112),(3,113),(4,114),(5,115),(6,116),(7,117),(8,118),(9,119),(10,120),(11,121),(12,122),(13,123),(14,124),(15,125),(16,126),(17,127),(18,128),(19,129),(20,130),(21,105),(22,106),(23,107),(24,108),(25,109),(26,110),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52),(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),(131,189),(132,190),(133,191),(134,192),(135,193),(136,194),(137,195),(138,196),(139,197),(140,198),(141,199),(142,200),(143,201),(144,202),(145,203),(146,204),(147,205),(148,206),(149,207),(150,208),(151,183),(152,184),(153,185),(154,186),(155,187),(156,188),(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 ··· 2I 4A 4B 4C 4D 4E ··· 4J 13A ··· 13L 26A ··· 26AJ 26AK ··· 26DD 52A ··· 52AV 52AW ··· 52DP order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

260 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C2 C13 C26 C26 C26 C26 C4○D4 C13×C4○D4 kernel C4○D4×C26 C22×C52 D4×C26 Q8×C26 C13×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C26 C2 # reps 1 3 3 1 8 12 36 36 12 96 4 48

Matrix representation of C4○D4×C26 in GL3(𝔽53) generated by

 52 0 0 0 46 0 0 0 46
,
 1 0 0 0 30 0 0 0 30
,
 1 0 0 0 36 2 0 14 17
,
 1 0 0 0 52 0 0 36 1
G:=sub<GL(3,GF(53))| [52,0,0,0,46,0,0,0,46],[1,0,0,0,30,0,0,0,30],[1,0,0,0,36,14,0,2,17],[1,0,0,0,52,36,0,0,1] >;

C4○D4×C26 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{26}
% in TeX

G:=Group("C4oD4xC26");
// GroupNames label

G:=SmallGroup(416,230);
// by ID

G=gap.SmallGroup(416,230);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-13,-2,2521,950]);
// Polycyclic

G:=Group<a,b,c,d|a^26=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations

׿
×
𝔽