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G = D108⋊5C2  order 432 = 24·33

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D108⋊5C2, C4.12D54, C36.65D6, Dic54⋊5C2, C12.65D18, C54.4C23, C22.2D54, D54.1C22, C108.12C22, Dic27.2C22, (C2×C4)⋊3D27, (C2×C108)⋊4C2, (C4×D27)⋊4C2, C27⋊1(C4○D4), C27⋊D4⋊3C2, C9.(C4○D12), (C2×C36).11S3, (C2×C18).32D6, (C2×C12).11D9, (C2×C6).32D18, C3.(D36⋊5C2), 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

Derived series
C1 — C54 — D108⋊5C2
Chief series
C1 — C3 — C9 — C27 — C54 — D54 — C4×D27 — D108⋊5C2
Lower central
C27 — C54 — D108⋊5C2
Upper central
C1 — C4 — C2×C4

Generators and relations for D108⋊5C2
 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, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C9, Dic3, C12, D6, C2×C6, C4○D4, D9, C18, C18, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C27, Dic9, C36, D18, C2×C18, C4○D12, D27, C54, C54, Dic18, C4×D9, D36, C9⋊D4, C2×C36, Dic27, C108, D54, C2×C54, D36⋊5C2, Dic54, C4×D27, D108, C27⋊D4, C2×C108, D108⋊5C2
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, C4○D12, D27, C22×D9, D54, D36⋊5C2, C22×D27, D108⋊5C2

Smallest permutation representation of D108⋊5C2
►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)(109 114)(110 113)(111 112)(115 216)(116 215)(117 214)(118 213)(119 212)(120 211)(121 210)(122 209)(123 208)(124 207)(125 206)(126 205)(127 204)(128 203)(129 202)(130 201)(131 200)(132 199)(133 198)(134 197)(135 196)(136 195)(137 194)(138 193)(139 192)(140 191)(141 190)(142 189)(143 188)(144 187)(145 186)(146 185)(147 184)(148 183)(149 182)(150 181)(151 180)(152 179)(153 178)(154 177)(155 176)(156 175)(157 174)(158 173)(159 172)(160 171)(161 170)(162 169)(163 168)(164 167)(165 166)

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

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○D12D27D54D54D36⋊5C2D108⋊5C2
kernelD108⋊5C2Dic54C4×D27D108C27⋊D4C2×C108C2×C36C36C2×C18C27C2×C12C12C2×C6C9C2×C4C4C22C3C1
# reps1121211212363491891236

Matrix representation of D108⋊5C2 ►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] >;
 

D108⋊5C2 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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