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G = C4×Dic13order 208 = 24·13

Direct product of C4 and Dic13

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×Dic13, C524C4, C132C42, C22.3D26, (C2×C52).7C2, C2.2(C4×D13), (C2×C4).6D13, C26.10(C2×C4), (C2×C26).3C22, C2.2(C2×Dic13), (C2×Dic13).6C2, SmallGroup(208,11)

Series: Derived Chief Lower central Upper central

C1C13 — C4×Dic13
C1C13C26C2×C26C2×Dic13 — C4×Dic13
C13 — C4×Dic13
C1C2×C4

Generators and relations for C4×Dic13
 G = < a,b,c | a4=b26=1, c2=b13, ab=ba, ac=ca, cbc-1=b-1 >

13C4
13C4
13C4
13C4
13C2×C4
13C2×C4
13C42

Smallest permutation representation of C4×Dic13
Regular action on 208 points
Generators in S208
(1 81 156 29)(2 82 131 30)(3 83 132 31)(4 84 133 32)(5 85 134 33)(6 86 135 34)(7 87 136 35)(8 88 137 36)(9 89 138 37)(10 90 139 38)(11 91 140 39)(12 92 141 40)(13 93 142 41)(14 94 143 42)(15 95 144 43)(16 96 145 44)(17 97 146 45)(18 98 147 46)(19 99 148 47)(20 100 149 48)(21 101 150 49)(22 102 151 50)(23 103 152 51)(24 104 153 52)(25 79 154 27)(26 80 155 28)(53 160 129 202)(54 161 130 203)(55 162 105 204)(56 163 106 205)(57 164 107 206)(58 165 108 207)(59 166 109 208)(60 167 110 183)(61 168 111 184)(62 169 112 185)(63 170 113 186)(64 171 114 187)(65 172 115 188)(66 173 116 189)(67 174 117 190)(68 175 118 191)(69 176 119 192)(70 177 120 193)(71 178 121 194)(72 179 122 195)(73 180 123 196)(74 181 124 197)(75 182 125 198)(76 157 126 199)(77 158 127 200)(78 159 128 201)
(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 60 14 73)(2 59 15 72)(3 58 16 71)(4 57 17 70)(5 56 18 69)(6 55 19 68)(7 54 20 67)(8 53 21 66)(9 78 22 65)(10 77 23 64)(11 76 24 63)(12 75 25 62)(13 74 26 61)(27 185 40 198)(28 184 41 197)(29 183 42 196)(30 208 43 195)(31 207 44 194)(32 206 45 193)(33 205 46 192)(34 204 47 191)(35 203 48 190)(36 202 49 189)(37 201 50 188)(38 200 51 187)(39 199 52 186)(79 169 92 182)(80 168 93 181)(81 167 94 180)(82 166 95 179)(83 165 96 178)(84 164 97 177)(85 163 98 176)(86 162 99 175)(87 161 100 174)(88 160 101 173)(89 159 102 172)(90 158 103 171)(91 157 104 170)(105 148 118 135)(106 147 119 134)(107 146 120 133)(108 145 121 132)(109 144 122 131)(110 143 123 156)(111 142 124 155)(112 141 125 154)(113 140 126 153)(114 139 127 152)(115 138 128 151)(116 137 129 150)(117 136 130 149)

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

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

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

C4×Dic13 is a maximal subgroup of
C52.8Q8  C1048C4  D527C4  C52.56D4  C52⋊C8  C26.C42  Dic13⋊C8  C42×D13  C42⋊D13  C23.11D26  C23.D26  Dic134D4  C23.6D26  Dic133Q8  C52⋊Q8  Dic13.Q8  C4.Dic26  C4⋊C47D13  D528C4  C4⋊C4⋊D13  C23.21D26  C52.17D4  C52⋊D4  Dic13⋊Q8  C52.23D4
C4×Dic13 is a maximal quotient of
C26.7C42  C1048C4  C26.10C42

64 conjugacy classes

class 1 2A2B2C4A4B4C4D4E···4L13A···13F26A···26R52A···52X
order122244444···413···1326···2652···52
size1111111113···132···22···22···2

64 irreducible representations

dim111112222
type++++-+
imageC1C2C2C4C4D13Dic13D26C4×D13
kernelC4×Dic13C2×Dic13C2×C52Dic13C52C2×C4C4C22C2
# reps12184612624

Matrix representation of C4×Dic13 in GL3(𝔽53) generated by

3000
0230
0023
,
100
0052
0127
,
5200
01530
04938
G:=sub<GL(3,GF(53))| [30,0,0,0,23,0,0,0,23],[1,0,0,0,0,1,0,52,27],[52,0,0,0,15,49,0,30,38] >;

C4×Dic13 in GAP, Magma, Sage, TeX

C_4\times {\rm Dic}_{13}
% in TeX

G:=Group("C4xDic13");
// GroupNames label

G:=SmallGroup(208,11);
// by ID

G=gap.SmallGroup(208,11);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,20,46,4804]);
// Polycyclic

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

Export

Subgroup lattice of C4×Dic13 in TeX

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