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G = D1085C2order 432 = 24·33

The semidirect product of D108 and C2 acting through Inn(D108)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D1085C2, C4.12D54, C36.65D6, Dic545C2, C12.65D18, C54.4C23, C22.2D54, D54.1C22, C108.12C22, Dic27.2C22, (C2×C4)⋊3D27, (C2×C108)⋊4C2, (C4×D27)⋊4C2, C271(C4○D4), C27⋊D43C2, C9.(C4○D12), (C2×C36).11S3, (C2×C18).32D6, (C2×C12).11D9, (C2×C6).32D18, C3.(D365C2), C6.31(C22×D9), C2.5(C22×D27), (C2×C54).11C22, C18.31(C22×S3), SmallGroup(432,46)

Series: Derived Chief Lower central Upper central

C1C54 — D1085C2
C1C3C9C27C54D54C4×D27 — D1085C2
C27C54 — D1085C2
C1C4C2×C4

Generators and relations for D1085C2
 G = < a,b,c | a108=b2=c2=1, bab=a-1, ac=ca, cbc=a54b >

Subgroups: 632 in 80 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6, C6, C2×C4, C2×C4 [×2], D4 [×3], Q8, C9, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C4○D4, D9 [×2], C18, C18, Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, C27, Dic9 [×2], C36 [×2], D18 [×2], C2×C18, C4○D12, D27 [×2], C54, C54, Dic18, C4×D9 [×2], D36, C9⋊D4 [×2], C2×C36, Dic27 [×2], C108 [×2], D54 [×2], C2×C54, D365C2, Dic54, C4×D27 [×2], D108, C27⋊D4 [×2], C2×C108, D1085C2
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], C4○D4, D9, C22×S3, D18 [×3], C4○D12, D27, C22×D9, D54 [×3], D365C2, C22×D27, D1085C2

Smallest permutation representation of D1085C2
On 216 points
Generators in S216
(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)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216)
(1 108)(2 107)(3 106)(4 105)(5 104)(6 103)(7 102)(8 101)(9 100)(10 99)(11 98)(12 97)(13 96)(14 95)(15 94)(16 93)(17 92)(18 91)(19 90)(20 89)(21 88)(22 87)(23 86)(24 85)(25 84)(26 83)(27 82)(28 81)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 72)(38 71)(39 70)(40 69)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(109 204)(110 203)(111 202)(112 201)(113 200)(114 199)(115 198)(116 197)(117 196)(118 195)(119 194)(120 193)(121 192)(122 191)(123 190)(124 189)(125 188)(126 187)(127 186)(128 185)(129 184)(130 183)(131 182)(132 181)(133 180)(134 179)(135 178)(136 177)(137 176)(138 175)(139 174)(140 173)(141 172)(142 171)(143 170)(144 169)(145 168)(146 167)(147 166)(148 165)(149 164)(150 163)(151 162)(152 161)(153 160)(154 159)(155 158)(156 157)(205 216)(206 215)(207 214)(208 213)(209 212)(210 211)
(1 130)(2 131)(3 132)(4 133)(5 134)(6 135)(7 136)(8 137)(9 138)(10 139)(11 140)(12 141)(13 142)(14 143)(15 144)(16 145)(17 146)(18 147)(19 148)(20 149)(21 150)(22 151)(23 152)(24 153)(25 154)(26 155)(27 156)(28 157)(29 158)(30 159)(31 160)(32 161)(33 162)(34 163)(35 164)(36 165)(37 166)(38 167)(39 168)(40 169)(41 170)(42 171)(43 172)(44 173)(45 174)(46 175)(47 176)(48 177)(49 178)(50 179)(51 180)(52 181)(53 182)(54 183)(55 184)(56 185)(57 186)(58 187)(59 188)(60 189)(61 190)(62 191)(63 192)(64 193)(65 194)(66 195)(67 196)(68 197)(69 198)(70 199)(71 200)(72 201)(73 202)(74 203)(75 204)(76 205)(77 206)(78 207)(79 208)(80 209)(81 210)(82 211)(83 212)(84 213)(85 214)(86 215)(87 216)(88 109)(89 110)(90 111)(91 112)(92 113)(93 114)(94 115)(95 116)(96 117)(97 118)(98 119)(99 120)(100 121)(101 122)(102 123)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)

G:=sub<Sym(216)| (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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216), (1,108)(2,107)(3,106)(4,105)(5,104)(6,103)(7,102)(8,101)(9,100)(10,99)(11,98)(12,97)(13,96)(14,95)(15,94)(16,93)(17,92)(18,91)(19,90)(20,89)(21,88)(22,87)(23,86)(24,85)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(109,204)(110,203)(111,202)(112,201)(113,200)(114,199)(115,198)(116,197)(117,196)(118,195)(119,194)(120,193)(121,192)(122,191)(123,190)(124,189)(125,188)(126,187)(127,186)(128,185)(129,184)(130,183)(131,182)(132,181)(133,180)(134,179)(135,178)(136,177)(137,176)(138,175)(139,174)(140,173)(141,172)(142,171)(143,170)(144,169)(145,168)(146,167)(147,166)(148,165)(149,164)(150,163)(151,162)(152,161)(153,160)(154,159)(155,158)(156,157)(205,216)(206,215)(207,214)(208,213)(209,212)(210,211), (1,130)(2,131)(3,132)(4,133)(5,134)(6,135)(7,136)(8,137)(9,138)(10,139)(11,140)(12,141)(13,142)(14,143)(15,144)(16,145)(17,146)(18,147)(19,148)(20,149)(21,150)(22,151)(23,152)(24,153)(25,154)(26,155)(27,156)(28,157)(29,158)(30,159)(31,160)(32,161)(33,162)(34,163)(35,164)(36,165)(37,166)(38,167)(39,168)(40,169)(41,170)(42,171)(43,172)(44,173)(45,174)(46,175)(47,176)(48,177)(49,178)(50,179)(51,180)(52,181)(53,182)(54,183)(55,184)(56,185)(57,186)(58,187)(59,188)(60,189)(61,190)(62,191)(63,192)(64,193)(65,194)(66,195)(67,196)(68,197)(69,198)(70,199)(71,200)(72,201)(73,202)(74,203)(75,204)(76,205)(77,206)(78,207)(79,208)(80,209)(81,210)(82,211)(83,212)(84,213)(85,214)(86,215)(87,216)(88,109)(89,110)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,118)(98,119)(99,120)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)>;

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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216), (1,108)(2,107)(3,106)(4,105)(5,104)(6,103)(7,102)(8,101)(9,100)(10,99)(11,98)(12,97)(13,96)(14,95)(15,94)(16,93)(17,92)(18,91)(19,90)(20,89)(21,88)(22,87)(23,86)(24,85)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(109,204)(110,203)(111,202)(112,201)(113,200)(114,199)(115,198)(116,197)(117,196)(118,195)(119,194)(120,193)(121,192)(122,191)(123,190)(124,189)(125,188)(126,187)(127,186)(128,185)(129,184)(130,183)(131,182)(132,181)(133,180)(134,179)(135,178)(136,177)(137,176)(138,175)(139,174)(140,173)(141,172)(142,171)(143,170)(144,169)(145,168)(146,167)(147,166)(148,165)(149,164)(150,163)(151,162)(152,161)(153,160)(154,159)(155,158)(156,157)(205,216)(206,215)(207,214)(208,213)(209,212)(210,211), (1,130)(2,131)(3,132)(4,133)(5,134)(6,135)(7,136)(8,137)(9,138)(10,139)(11,140)(12,141)(13,142)(14,143)(15,144)(16,145)(17,146)(18,147)(19,148)(20,149)(21,150)(22,151)(23,152)(24,153)(25,154)(26,155)(27,156)(28,157)(29,158)(30,159)(31,160)(32,161)(33,162)(34,163)(35,164)(36,165)(37,166)(38,167)(39,168)(40,169)(41,170)(42,171)(43,172)(44,173)(45,174)(46,175)(47,176)(48,177)(49,178)(50,179)(51,180)(52,181)(53,182)(54,183)(55,184)(56,185)(57,186)(58,187)(59,188)(60,189)(61,190)(62,191)(63,192)(64,193)(65,194)(66,195)(67,196)(68,197)(69,198)(70,199)(71,200)(72,201)(73,202)(74,203)(75,204)(76,205)(77,206)(78,207)(79,208)(80,209)(81,210)(82,211)(83,212)(84,213)(85,214)(86,215)(87,216)(88,109)(89,110)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,118)(98,119)(99,120)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129) );

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),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)], [(1,108),(2,107),(3,106),(4,105),(5,104),(6,103),(7,102),(8,101),(9,100),(10,99),(11,98),(12,97),(13,96),(14,95),(15,94),(16,93),(17,92),(18,91),(19,90),(20,89),(21,88),(22,87),(23,86),(24,85),(25,84),(26,83),(27,82),(28,81),(29,80),(30,79),(31,78),(32,77),(33,76),(34,75),(35,74),(36,73),(37,72),(38,71),(39,70),(40,69),(41,68),(42,67),(43,66),(44,65),(45,64),(46,63),(47,62),(48,61),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(109,204),(110,203),(111,202),(112,201),(113,200),(114,199),(115,198),(116,197),(117,196),(118,195),(119,194),(120,193),(121,192),(122,191),(123,190),(124,189),(125,188),(126,187),(127,186),(128,185),(129,184),(130,183),(131,182),(132,181),(133,180),(134,179),(135,178),(136,177),(137,176),(138,175),(139,174),(140,173),(141,172),(142,171),(143,170),(144,169),(145,168),(146,167),(147,166),(148,165),(149,164),(150,163),(151,162),(152,161),(153,160),(154,159),(155,158),(156,157),(205,216),(206,215),(207,214),(208,213),(209,212),(210,211)], [(1,130),(2,131),(3,132),(4,133),(5,134),(6,135),(7,136),(8,137),(9,138),(10,139),(11,140),(12,141),(13,142),(14,143),(15,144),(16,145),(17,146),(18,147),(19,148),(20,149),(21,150),(22,151),(23,152),(24,153),(25,154),(26,155),(27,156),(28,157),(29,158),(30,159),(31,160),(32,161),(33,162),(34,163),(35,164),(36,165),(37,166),(38,167),(39,168),(40,169),(41,170),(42,171),(43,172),(44,173),(45,174),(46,175),(47,176),(48,177),(49,178),(50,179),(51,180),(52,181),(53,182),(54,183),(55,184),(56,185),(57,186),(58,187),(59,188),(60,189),(61,190),(62,191),(63,192),(64,193),(65,194),(66,195),(67,196),(68,197),(69,198),(70,199),(71,200),(72,201),(73,202),(74,203),(75,204),(76,205),(77,206),(78,207),(79,208),(80,209),(81,210),(82,211),(83,212),(84,213),(85,214),(86,215),(87,216),(88,109),(89,110),(90,111),(91,112),(92,113),(93,114),(94,115),(95,116),(96,117),(97,118),(98,119),(99,120),(100,121),(101,122),(102,123),(103,124),(104,125),(105,126),(106,127),(107,128),(108,129)])

114 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E6A6B6C9A9B9C12A12B12C12D18A···18I27A···27I36A···36L54A···54AA108A···108AJ
order122223444446669991212121218···1827···2736···3654···54108···108
size11254542112545422222222222···22···22···22···22···2

114 irreducible representations

dim1111112222222222222
type+++++++++++++++
imageC1C2C2C2C2C2S3D6D6C4○D4D9D18D18C4○D12D27D54D54D365C2D1085C2
kernelD1085C2Dic54C4×D27D108C27⋊D4C2×C108C2×C36C36C2×C18C27C2×C12C12C2×C6C9C2×C4C4C22C3C1
# reps1121211212363491891236

Matrix representation of D1085C2 in GL2(𝔽109) generated by

668
4147
,
647
41103
,
10397
126
G:=sub<GL(2,GF(109))| [6,41,68,47],[6,41,47,103],[103,12,97,6] >;

D1085C2 in GAP, Magma, Sage, TeX

D_{108}\rtimes_5C_2
% in TeX

G:=Group("D108:5C2");
// GroupNames label

G:=SmallGroup(432,46);
// by ID

G=gap.SmallGroup(432,46);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,2804,557,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^108=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^54*b>;
// generators/relations

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