Copied to
clipboard

G = C20.D8order 320 = 26·5

19th non-split extension by C20 of D8 acting via D8/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.19D8, C20.19SD16, C42.224D10, C4⋊Q86D5, C4⋊C4.84D10, C4.7(D4⋊D5), C10.61(C2×D8), C54(C4.4D8), C4.5(Q8⋊D5), (C2×C20).158D4, D206C442C2, C20.83(C4○D4), C204D4.10C2, C10.78(C2×SD16), (C4×C20).135C22, (C2×C20).406C23, C4.16(Q82D5), C10.58(C4.4D4), (C2×D20).113C22, C2.11(C20.23D4), (C5×C4⋊Q8)⋊6C2, (C4×C52C8)⋊18C2, C2.16(C2×D4⋊D5), C2.16(C2×Q8⋊D5), (C2×C10).537(C2×D4), (C2×C4).137(C5⋊D4), (C5×C4⋊C4).131C22, (C2×C4).503(C22×D5), C22.209(C2×C5⋊D4), (C2×C52C8).271C22, SmallGroup(320,715)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20.D8
C1C5C10C20C2×C20C2×D20C204D4 — C20.D8
C5C10C2×C20 — C20.D8
C1C22C42C4⋊Q8

Generators and relations for C20.D8
 G = < a,b,c | a20=b8=c2=1, bab-1=a9, cac=a-1, cbc=a10b-1 >

Subgroups: 606 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C5, C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×2], C23 [×2], D5 [×2], C10 [×3], C42, C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×D4 [×4], C2×Q8, C20 [×6], C20 [×2], D10 [×6], C2×C10, C4×C8, D4⋊C4 [×4], C41D4, C4⋊Q8, C52C8 [×2], D20 [×8], C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C22×D5 [×2], C4.4D8, C2×C52C8 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×D20 [×2], C2×D20 [×2], Q8×C10, C4×C52C8, D206C4 [×4], C204D4, C5×C4⋊Q8, C20.D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], SD16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C2×D8, C2×SD16, C5⋊D4 [×2], C22×D5, C4.4D8, D4⋊D5 [×2], Q8⋊D5 [×2], Q82D5 [×2], C2×C5⋊D4, C2×D4⋊D5, C2×Q8⋊D5, C20.23D4, C20.D8

Smallest permutation representation of C20.D8
On 160 points
Generators in S160
(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)
(1 81 31 77 121 59 111 151)(2 90 32 66 122 48 112 160)(3 99 33 75 123 57 113 149)(4 88 34 64 124 46 114 158)(5 97 35 73 125 55 115 147)(6 86 36 62 126 44 116 156)(7 95 37 71 127 53 117 145)(8 84 38 80 128 42 118 154)(9 93 39 69 129 51 119 143)(10 82 40 78 130 60 120 152)(11 91 21 67 131 49 101 141)(12 100 22 76 132 58 102 150)(13 89 23 65 133 47 103 159)(14 98 24 74 134 56 104 148)(15 87 25 63 135 45 105 157)(16 96 26 72 136 54 106 146)(17 85 27 61 137 43 107 155)(18 94 28 70 138 52 108 144)(19 83 29 79 139 41 109 153)(20 92 30 68 140 50 110 142)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 101)(22 120)(23 119)(24 118)(25 117)(26 116)(27 115)(28 114)(29 113)(30 112)(31 111)(32 110)(33 109)(34 108)(35 107)(36 106)(37 105)(38 104)(39 103)(40 102)(41 65)(42 64)(43 63)(44 62)(45 61)(46 80)(47 79)(48 78)(49 77)(50 76)(51 75)(52 74)(53 73)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(81 141)(82 160)(83 159)(84 158)(85 157)(86 156)(87 155)(88 154)(89 153)(90 152)(91 151)(92 150)(93 149)(94 148)(95 147)(96 146)(97 145)(98 144)(99 143)(100 142)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)

G:=sub<Sym(160)| (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), (1,81,31,77,121,59,111,151)(2,90,32,66,122,48,112,160)(3,99,33,75,123,57,113,149)(4,88,34,64,124,46,114,158)(5,97,35,73,125,55,115,147)(6,86,36,62,126,44,116,156)(7,95,37,71,127,53,117,145)(8,84,38,80,128,42,118,154)(9,93,39,69,129,51,119,143)(10,82,40,78,130,60,120,152)(11,91,21,67,131,49,101,141)(12,100,22,76,132,58,102,150)(13,89,23,65,133,47,103,159)(14,98,24,74,134,56,104,148)(15,87,25,63,135,45,105,157)(16,96,26,72,136,54,106,146)(17,85,27,61,137,43,107,155)(18,94,28,70,138,52,108,144)(19,83,29,79,139,41,109,153)(20,92,30,68,140,50,110,142), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,101)(22,120)(23,119)(24,118)(25,117)(26,116)(27,115)(28,114)(29,113)(30,112)(31,111)(32,110)(33,109)(34,108)(35,107)(36,106)(37,105)(38,104)(39,103)(40,102)(41,65)(42,64)(43,63)(44,62)(45,61)(46,80)(47,79)(48,78)(49,77)(50,76)(51,75)(52,74)(53,73)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(81,141)(82,160)(83,159)(84,158)(85,157)(86,156)(87,155)(88,154)(89,153)(90,152)(91,151)(92,150)(93,149)(94,148)(95,147)(96,146)(97,145)(98,144)(99,143)(100,142)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)>;

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), (1,81,31,77,121,59,111,151)(2,90,32,66,122,48,112,160)(3,99,33,75,123,57,113,149)(4,88,34,64,124,46,114,158)(5,97,35,73,125,55,115,147)(6,86,36,62,126,44,116,156)(7,95,37,71,127,53,117,145)(8,84,38,80,128,42,118,154)(9,93,39,69,129,51,119,143)(10,82,40,78,130,60,120,152)(11,91,21,67,131,49,101,141)(12,100,22,76,132,58,102,150)(13,89,23,65,133,47,103,159)(14,98,24,74,134,56,104,148)(15,87,25,63,135,45,105,157)(16,96,26,72,136,54,106,146)(17,85,27,61,137,43,107,155)(18,94,28,70,138,52,108,144)(19,83,29,79,139,41,109,153)(20,92,30,68,140,50,110,142), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,101)(22,120)(23,119)(24,118)(25,117)(26,116)(27,115)(28,114)(29,113)(30,112)(31,111)(32,110)(33,109)(34,108)(35,107)(36,106)(37,105)(38,104)(39,103)(40,102)(41,65)(42,64)(43,63)(44,62)(45,61)(46,80)(47,79)(48,78)(49,77)(50,76)(51,75)(52,74)(53,73)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(81,141)(82,160)(83,159)(84,158)(85,157)(86,156)(87,155)(88,154)(89,153)(90,152)(91,151)(92,150)(93,149)(94,148)(95,147)(96,146)(97,145)(98,144)(99,143)(100,142)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132) );

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)], [(1,81,31,77,121,59,111,151),(2,90,32,66,122,48,112,160),(3,99,33,75,123,57,113,149),(4,88,34,64,124,46,114,158),(5,97,35,73,125,55,115,147),(6,86,36,62,126,44,116,156),(7,95,37,71,127,53,117,145),(8,84,38,80,128,42,118,154),(9,93,39,69,129,51,119,143),(10,82,40,78,130,60,120,152),(11,91,21,67,131,49,101,141),(12,100,22,76,132,58,102,150),(13,89,23,65,133,47,103,159),(14,98,24,74,134,56,104,148),(15,87,25,63,135,45,105,157),(16,96,26,72,136,54,106,146),(17,85,27,61,137,43,107,155),(18,94,28,70,138,52,108,144),(19,83,29,79,139,41,109,153),(20,92,30,68,140,50,110,142)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,101),(22,120),(23,119),(24,118),(25,117),(26,116),(27,115),(28,114),(29,113),(30,112),(31,111),(32,110),(33,109),(34,108),(35,107),(36,106),(37,105),(38,104),(39,103),(40,102),(41,65),(42,64),(43,63),(44,62),(45,61),(46,80),(47,79),(48,78),(49,77),(50,76),(51,75),(52,74),(53,73),(54,72),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66),(81,141),(82,160),(83,159),(84,158),(85,157),(86,156),(87,155),(88,154),(89,153),(90,152),(91,151),(92,150),(93,149),(94,148),(95,147),(96,146),(97,145),(98,144),(99,143),(100,142),(122,140),(123,139),(124,138),(125,137),(126,136),(127,135),(128,134),(129,133),(130,132)])

50 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H5A5B8A···8H10A···10F20A···20L20M···20T
order1222224···444558···810···1020···2020···20
size111140402···2882210···102···24···48···8

50 irreducible representations

dim1111122222222444
type+++++++++++++
imageC1C2C2C2C2D4D5D8SD16C4○D4D10D10C5⋊D4D4⋊D5Q8⋊D5Q82D5
kernelC20.D8C4×C52C8D206C4C204D4C5×C4⋊Q8C2×C20C4⋊Q8C20C20C20C42C4⋊C4C2×C4C4C4C4
# reps1141122444248444

Matrix representation of C20.D8 in GL6(𝔽41)

2090000
1210000
006100
005100
0000228
0000619
,
28120000
1520000
006100
0063500
0000710
00002834
,
2090000
24210000
006100
0063500
000010
00001540

G:=sub<GL(6,GF(41))| [20,1,0,0,0,0,9,21,0,0,0,0,0,0,6,5,0,0,0,0,1,1,0,0,0,0,0,0,22,6,0,0,0,0,8,19],[28,15,0,0,0,0,12,2,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,7,28,0,0,0,0,10,34],[20,24,0,0,0,0,9,21,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,1,15,0,0,0,0,0,40] >;

C20.D8 in GAP, Magma, Sage, TeX

C_{20}.D_8
% in TeX

G:=Group("C20.D8");
// GroupNames label

G:=SmallGroup(320,715);
// by ID

G=gap.SmallGroup(320,715);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,219,100,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=a^10*b^-1>;
// generators/relations

׿
×
𝔽