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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 [×3], C4, C4 [×3], C22 [×3], C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], C13, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], D13 [×3], C26, C4○D8, Dic13, C52, C52 [×2], D26, D26 [×2], C132C8, C104, C4×D13, C4×D13 [×2], D52 [×2], D52 [×2], Q8×C13 [×2], C8×D13, D104, Q8⋊D13 [×2], C13×Q16, D52⋊C2 [×2], D104⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C4○D8, D26 [×3], 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)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 208)(8 207)(9 206)(10 205)(11 204)(12 203)(13 202)(14 201)(15 200)(16 199)(17 198)(18 197)(19 196)(20 195)(21 194)(22 193)(23 192)(24 191)(25 190)(26 189)(27 188)(28 187)(29 186)(30 185)(31 184)(32 183)(33 182)(34 181)(35 180)(36 179)(37 178)(38 177)(39 176)(40 175)(41 174)(42 173)(43 172)(44 171)(45 170)(46 169)(47 168)(48 167)(49 166)(50 165)(51 164)(52 163)(53 162)(54 161)(55 160)(56 159)(57 158)(58 157)(59 156)(60 155)(61 154)(62 153)(63 152)(64 151)(65 150)(66 149)(67 148)(68 147)(69 146)(70 145)(71 144)(72 143)(73 142)(74 141)(75 140)(76 139)(77 138)(78 137)(79 136)(80 135)(81 134)(82 133)(83 132)(84 131)(85 130)(86 129)(87 128)(88 127)(89 126)(90 125)(91 124)(92 123)(93 122)(94 121)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)
(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)(105 117)(106 142)(107 167)(108 192)(109 113)(110 138)(111 163)(112 188)(114 134)(115 159)(116 184)(118 130)(119 155)(120 180)(121 205)(122 126)(123 151)(124 176)(125 201)(127 147)(128 172)(129 197)(131 143)(132 168)(133 193)(135 139)(136 164)(137 189)(140 160)(141 185)(144 156)(145 181)(146 206)(148 152)(149 177)(150 202)(153 173)(154 198)(157 169)(158 194)(161 165)(162 190)(166 186)(170 182)(171 207)(174 178)(175 203)(179 199)(183 195)(187 191)(196 208)(200 204)

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

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

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

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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