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G = D104⋊C2order 416 = 25·13

5th semidirect product of D104 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D1045C2, Q163D13, D26.7D4, C8.10D26, Q8.5D26, C104.8C22, C52.10C23, D52.5C22, Dic13.26D4, (C8×D13)⋊3C2, C134(C4○D8), Q8⋊D134C2, (C13×Q16)⋊3C2, C2.24(D4×D13), C26.36(C2×D4), D52⋊C23C2, C132C8.8C22, C4.10(C22×D13), (Q8×C13).5C22, (C4×D13).20C22, SmallGroup(416,140)

Series: Derived Chief Lower central Upper central

C1C52 — D104⋊C2
C1C13C26C52C4×D13D52⋊C2 — D104⋊C2
C13C26C52 — D104⋊C2
C1C2C4Q16

Generators and relations for D104⋊C2
 G = < a,b,c | a104=b2=c2=1, bab=a-1, cac=a25, cbc=a76b >

Subgroups: 512 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, Q8, C13, C2×C8, D8, SD16, Q16, C4○D4, D13, C26, C4○D8, Dic13, C52, C52, D26, D26, C132C8, C104, C4×D13, C4×D13, D52, D52, Q8×C13, C8×D13, D104, Q8⋊D13, C13×Q16, D52⋊C2, D104⋊C2
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C4○D8, D26, C22×D13, D4×D13, D104⋊C2

Smallest permutation representation of D104⋊C2
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)
(1 118)(2 117)(3 116)(4 115)(5 114)(6 113)(7 112)(8 111)(9 110)(10 109)(11 108)(12 107)(13 106)(14 105)(15 208)(16 207)(17 206)(18 205)(19 204)(20 203)(21 202)(22 201)(23 200)(24 199)(25 198)(26 197)(27 196)(28 195)(29 194)(30 193)(31 192)(32 191)(33 190)(34 189)(35 188)(36 187)(37 186)(38 185)(39 184)(40 183)(41 182)(42 181)(43 180)(44 179)(45 178)(46 177)(47 176)(48 175)(49 174)(50 173)(51 172)(52 171)(53 170)(54 169)(55 168)(56 167)(57 166)(58 165)(59 164)(60 163)(61 162)(62 161)(63 160)(64 159)(65 158)(66 157)(67 156)(68 155)(69 154)(70 153)(71 152)(72 151)(73 150)(74 149)(75 148)(76 147)(77 146)(78 145)(79 144)(80 143)(81 142)(82 141)(83 140)(84 139)(85 138)(86 137)(87 136)(88 135)(89 134)(90 133)(91 132)(92 131)(93 130)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)(101 122)(102 121)(103 120)(104 119)
(2 26)(3 51)(4 76)(5 101)(6 22)(7 47)(8 72)(9 97)(10 18)(11 43)(12 68)(13 93)(15 39)(16 64)(17 89)(19 35)(20 60)(21 85)(23 31)(24 56)(25 81)(28 52)(29 77)(30 102)(32 48)(33 73)(34 98)(36 44)(37 69)(38 94)(41 65)(42 90)(45 61)(46 86)(49 57)(50 82)(54 78)(55 103)(58 74)(59 99)(62 70)(63 95)(67 91)(71 87)(75 83)(80 104)(84 100)(88 96)(105 133)(106 158)(107 183)(108 208)(109 129)(110 154)(111 179)(112 204)(113 125)(114 150)(115 175)(116 200)(117 121)(118 146)(119 171)(120 196)(122 142)(123 167)(124 192)(126 138)(127 163)(128 188)(130 134)(131 159)(132 184)(135 155)(136 180)(137 205)(139 151)(140 176)(141 201)(143 147)(144 172)(145 197)(148 168)(149 193)(152 164)(153 189)(156 160)(157 185)(161 181)(162 206)(165 177)(166 202)(169 173)(170 198)(174 194)(178 190)(182 186)(187 207)(191 203)(195 199)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E8A8B8C8D13A···13F26A···26F52A···52F52G···52R104A···104L
order1222244444888813···1326···2652···5252···52104···104
size1126525224413132226262···22···24···48···84···4

56 irreducible representations

dim11111122222244
type+++++++++++++
imageC1C2C2C2C2C2D4D4D13C4○D8D26D26D4×D13D104⋊C2
kernelD104⋊C2C8×D13D104Q8⋊D13C13×Q16D52⋊C2Dic13D26Q16C13C8Q8C2C1
# reps1112121164612612

Matrix representation of D104⋊C2 in GL4(𝔽313) generated by

133700
27628000
001250
00182308
,
30027600
131300
00262260
0030351
,
1000
26131200
0010
00128312
G:=sub<GL(4,GF(313))| [13,276,0,0,37,280,0,0,0,0,125,182,0,0,0,308],[300,13,0,0,276,13,0,0,0,0,262,303,0,0,260,51],[1,261,0,0,0,312,0,0,0,0,1,128,0,0,0,312] >;

D104⋊C2 in GAP, Magma, Sage, TeX

D_{104}\rtimes C_2
% in TeX

G:=Group("D104:C2");
// GroupNames label

G:=SmallGroup(416,140);
// by ID

G=gap.SmallGroup(416,140);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,103,362,116,86,297,159,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^104=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^25,c*b*c=a^76*b>;
// generators/relations

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