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G = D205C8order 320 = 26·5

3rd semidirect product of D20 and C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D205C8, C42.201D10, C56(C8×D4), C41(C8×D5), C206(C2×C8), C4⋊C818D5, D105(C2×C8), C52C830D4, (C4×D20).7C2, C10.76(C4×D4), C4.205(D4×D5), (C2×D20).24C4, C20.364(C2×D4), (C2×C8).216D10, C4⋊Dic5.30C4, D101C822C2, C10.52(C8○D4), (C4×C20).60C22, C10.35(C22×C8), D10⋊C4.21C4, C20.334(C4○D4), C2.2(D208C4), (C2×C20).831C23, (C2×C40).210C22, C4.53(Q82D5), C2.4(D20.2C4), (D5×C2×C8)⋊22C2, (C5×C4⋊C8)⋊15C2, (C4×C52C8)⋊4C2, C2.13(D5×C2×C8), (C2×C4).72(C4×D5), C22.48(C2×C4×D5), (C2×C20).331(C2×C4), (C2×C4×D5).347C22, (C2×Dic5).98(C2×C4), (C22×D5).74(C2×C4), (C2×C4).773(C22×D5), (C2×C10).187(C22×C4), (C2×C52C8).314C22, SmallGroup(320,461)

Series: Derived Chief Lower central Upper central

C1C10 — D205C8
C1C5C10C20C2×C20C2×C4×D5C4×D20 — D205C8
C5C10 — D205C8
C1C2×C4C4⋊C8

Generators and relations for D205C8
 G = < a,b,c | a20=b2=c8=1, bab=a-1, cac-1=a11, cbc-1=a10b >

Subgroups: 446 in 134 conjugacy classes, 61 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×8], C5, C8 [×5], C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], D5 [×4], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×6], C22×C4 [×2], C2×D4, Dic5 [×2], C20 [×2], C20 [×2], C20, D10 [×4], D10 [×4], C2×C10, C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8 [×2], C52C8 [×2], C52C8, C40 [×2], C4×D5 [×4], D20 [×4], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C8×D4, C8×D5 [×4], C2×C52C8 [×2], C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5 [×2], C2×D20, C4×C52C8, D101C8 [×2], C5×C4⋊C8, C4×D20, D5×C2×C8 [×2], D205C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, D5, C2×C8 [×6], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C22×C8, C8○D4, C4×D5 [×2], C22×D5, C8×D4, C8×D5 [×2], C2×C4×D5, D4×D5, Q82D5, D208C4, D5×C2×C8, D20.2C4, D205C8

Smallest permutation representation of D205C8
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(98 100)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(118 120)(121 139)(122 138)(123 137)(124 136)(125 135)(126 134)(127 133)(128 132)(129 131)(141 151)(142 150)(143 149)(144 148)(145 147)(152 160)(153 159)(154 158)(155 157)
(1 72 102 46 123 154 21 87)(2 63 103 57 124 145 22 98)(3 74 104 48 125 156 23 89)(4 65 105 59 126 147 24 100)(5 76 106 50 127 158 25 91)(6 67 107 41 128 149 26 82)(7 78 108 52 129 160 27 93)(8 69 109 43 130 151 28 84)(9 80 110 54 131 142 29 95)(10 71 111 45 132 153 30 86)(11 62 112 56 133 144 31 97)(12 73 113 47 134 155 32 88)(13 64 114 58 135 146 33 99)(14 75 115 49 136 157 34 90)(15 66 116 60 137 148 35 81)(16 77 117 51 138 159 36 92)(17 68 118 42 139 150 37 83)(18 79 119 53 140 141 38 94)(19 70 120 44 121 152 39 85)(20 61 101 55 122 143 40 96)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A···8H8I···8P8Q8R8S8T10A···10F20A···20H20I···20P40A···40P
order12222222444444444444558···88···8888810···1020···2020···2040···40
size1111101010101111222210101010222···25···5101010102···22···24···44···4

80 irreducible representations

dim111111111122222222444
type++++++++++++
imageC1C2C2C2C2C2C4C4C4C8D4D5C4○D4D10D10C8○D4C4×D5C8×D5D4×D5Q82D5D20.2C4
kernelD205C8C4×C52C8D101C8C5×C4⋊C8C4×D20D5×C2×C8C4⋊Dic5D10⋊C4C2×D20D20C52C8C4⋊C8C20C42C2×C8C10C2×C4C4C4C4C2
# reps11211224216222244816224

Matrix representation of D205C8 in GL5(𝔽41)

10000
004000
01700
000329
00009
,
400000
00100
01000
000400
000391
,
30000
032000
003200
000329
000239

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,40,7,0,0,0,0,0,32,0,0,0,0,9,9],[40,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,40,39,0,0,0,0,1],[3,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32,23,0,0,0,9,9] >;

D205C8 in GAP, Magma, Sage, TeX

D_{20}\rtimes_5C_8
% in TeX

G:=Group("D20:5C8");
// GroupNames label

G:=SmallGroup(320,461);
// by ID

G=gap.SmallGroup(320,461);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,219,58,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b>;
// generators/relations

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