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G = Q82D27order 432 = 24·33

The semidirect product of Q8 and D27 acting via D27/C27=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q82D27, C36.4D6, C4.4D54, C273SD16, C12.4D18, C54.10D4, D108.2C2, C108.4C22, C27⋊C83C2, (Q8×C27)⋊1C2, (Q8×C9).6S3, (C3×Q8).6D9, C3.(Q82D9), C9.(Q82S3), C6.19(C9⋊D4), C2.7(C27⋊D4), C18.19(C3⋊D4), SmallGroup(432,18)

Series: Derived Chief Lower central Upper central

C1C108 — Q82D27
C1C3C9C27C54C108D108 — Q82D27
C27C54C108 — Q82D27
C1C2C4Q8

Generators and relations for Q82D27
 G = < a,b,c,d | a4=c27=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >

108C2
2C4
54C22
36S3
27D4
27C8
2C12
18D6
12D9
27SD16
9D12
9C3⋊C8
2C36
6D18
4D27
9Q82S3
3C9⋊C8
3D36
2D54
2C108
3Q82D9

Smallest permutation representation of Q82D27
On 216 points
Generators in S216
(1 89 34 73)(2 90 35 74)(3 91 36 75)(4 92 37 76)(5 93 38 77)(6 94 39 78)(7 95 40 79)(8 96 41 80)(9 97 42 81)(10 98 43 55)(11 99 44 56)(12 100 45 57)(13 101 46 58)(14 102 47 59)(15 103 48 60)(16 104 49 61)(17 105 50 62)(18 106 51 63)(19 107 52 64)(20 108 53 65)(21 82 54 66)(22 83 28 67)(23 84 29 68)(24 85 30 69)(25 86 31 70)(26 87 32 71)(27 88 33 72)(109 179 146 210)(110 180 147 211)(111 181 148 212)(112 182 149 213)(113 183 150 214)(114 184 151 215)(115 185 152 216)(116 186 153 190)(117 187 154 191)(118 188 155 192)(119 189 156 193)(120 163 157 194)(121 164 158 195)(122 165 159 196)(123 166 160 197)(124 167 161 198)(125 168 162 199)(126 169 136 200)(127 170 137 201)(128 171 138 202)(129 172 139 203)(130 173 140 204)(131 174 141 205)(132 175 142 206)(133 176 143 207)(134 177 144 208)(135 178 145 209)
(1 154 34 117)(2 155 35 118)(3 156 36 119)(4 157 37 120)(5 158 38 121)(6 159 39 122)(7 160 40 123)(8 161 41 124)(9 162 42 125)(10 136 43 126)(11 137 44 127)(12 138 45 128)(13 139 46 129)(14 140 47 130)(15 141 48 131)(16 142 49 132)(17 143 50 133)(18 144 51 134)(19 145 52 135)(20 146 53 109)(21 147 54 110)(22 148 28 111)(23 149 29 112)(24 150 30 113)(25 151 31 114)(26 152 32 115)(27 153 33 116)(55 200 98 169)(56 201 99 170)(57 202 100 171)(58 203 101 172)(59 204 102 173)(60 205 103 174)(61 206 104 175)(62 207 105 176)(63 208 106 177)(64 209 107 178)(65 210 108 179)(66 211 82 180)(67 212 83 181)(68 213 84 182)(69 214 85 183)(70 215 86 184)(71 216 87 185)(72 190 88 186)(73 191 89 187)(74 192 90 188)(75 193 91 189)(76 194 92 163)(77 195 93 164)(78 196 94 165)(79 197 95 166)(80 198 96 167)(81 199 97 168)
(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)
(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 107)(56 106)(57 105)(58 104)(59 103)(60 102)(61 101)(62 100)(63 99)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(81 108)(109 168)(110 167)(111 166)(112 165)(113 164)(114 163)(115 189)(116 188)(117 187)(118 186)(119 185)(120 184)(121 183)(122 182)(123 181)(124 180)(125 179)(126 178)(127 177)(128 176)(129 175)(130 174)(131 173)(132 172)(133 171)(134 170)(135 169)(136 209)(137 208)(138 207)(139 206)(140 205)(141 204)(142 203)(143 202)(144 201)(145 200)(146 199)(147 198)(148 197)(149 196)(150 195)(151 194)(152 193)(153 192)(154 191)(155 190)(156 216)(157 215)(158 214)(159 213)(160 212)(161 211)(162 210)

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

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

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

72 conjugacy classes

class 1 2A2B 3 4A4B 6 8A8B9A9B9C12A12B12C18A18B18C27A···27I36A···36I54A···54I108A···108AA
order12234468899912121218181827···2736···3654···54108···108
size11108224254542224442222···24···42···24···4

72 irreducible representations

dim111122222222222444
type++++++++++++++
imageC1C2C2C2S3D4D6SD16D9C3⋊D4D18D27C9⋊D4D54C27⋊D4Q82S3Q82D9Q82D27
kernelQ82D27C27⋊C8D108Q8×C27Q8×C9C54C36C27C3×Q8C18C12Q8C6C4C2C9C3C1
# reps1111111232396918139

Matrix representation of Q82D27 in GL4(𝔽433) generated by

1000
0100
00432431
0011
,
1000
0100
00069
002510
,
28818200
25110600
0010
0001
,
1000
43243200
0010
00432432
G:=sub<GL(4,GF(433))| [1,0,0,0,0,1,0,0,0,0,432,1,0,0,431,1],[1,0,0,0,0,1,0,0,0,0,0,251,0,0,69,0],[288,251,0,0,182,106,0,0,0,0,1,0,0,0,0,1],[1,432,0,0,0,432,0,0,0,0,1,432,0,0,0,432] >;

Q82D27 in GAP, Magma, Sage, TeX

Q_8\rtimes_2D_{27}
% in TeX

G:=Group("Q8:2D27");
// GroupNames label

G:=SmallGroup(432,18);
// by ID

G=gap.SmallGroup(432,18);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,64,254,135,58,2804,557,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^27=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of Q82D27 in TeX

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