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G = C20.44D4order 160 = 25·5

1st non-split extension by C20 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.44D4, C10.1Q16, Dic10:5C4, C10.1SD16, C2.1Dic20, C22.7D20, C4.7(C4xD5), (C2xC8).2D5, (C2xC40).2C2, C20.38(C2xC4), C5:3(Q8:C4), (C2xC4).67D10, (C2xC10).12D4, C4:Dic5.1C2, C2.1(C40:C2), C4.19(C5:D4), (C2xC20).80C22, (C2xDic10).1C2, C2.7(D10:C4), C10.16(C22:C4), SmallGroup(160,23)

Series: Derived Chief Lower central Upper central

C1C20 — C20.44D4
C1C5C10C20C2xC20C4:Dic5 — C20.44D4
C5C10C20 — C20.44D4
C1C22C2xC4C2xC8

Generators and relations for C20.44D4
 G = < a,b,c | a20=b4=1, c2=a10, bab-1=cac-1=a-1, cbc-1=a5b-1 >

Subgroups: 136 in 42 conjugacy classes, 23 normal (21 characteristic)
Quotients: C1, C2, C4, C22, C2xC4, D4, D5, C22:C4, SD16, Q16, D10, Q8:C4, C4xD5, D20, C5:D4, C40:C2, Dic20, D10:C4, C20.44D4
10C4
10C4
20C4
2C8
5Q8
5Q8
10C2xC4
10Q8
10C2xC4
2Dic5
2Dic5
4Dic5
5C2xQ8
5C4:C4
2C2xDic5
2C2xDic5
2C40
2Dic10
5Q8:C4

Smallest permutation representation of C20.44D4
Regular action 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)
(1 102 32 64)(2 101 33 63)(3 120 34 62)(4 119 35 61)(5 118 36 80)(6 117 37 79)(7 116 38 78)(8 115 39 77)(9 114 40 76)(10 113 21 75)(11 112 22 74)(12 111 23 73)(13 110 24 72)(14 109 25 71)(15 108 26 70)(16 107 27 69)(17 106 28 68)(18 105 29 67)(19 104 30 66)(20 103 31 65)(41 126 93 155)(42 125 94 154)(43 124 95 153)(44 123 96 152)(45 122 97 151)(46 121 98 150)(47 140 99 149)(48 139 100 148)(49 138 81 147)(50 137 82 146)(51 136 83 145)(52 135 84 144)(53 134 85 143)(54 133 86 142)(55 132 87 141)(56 131 88 160)(57 130 89 159)(58 129 90 158)(59 128 91 157)(60 127 92 156)
(1 131 11 121)(2 130 12 140)(3 129 13 139)(4 128 14 138)(5 127 15 137)(6 126 16 136)(7 125 17 135)(8 124 18 134)(9 123 19 133)(10 122 20 132)(21 151 31 141)(22 150 32 160)(23 149 33 159)(24 148 34 158)(25 147 35 157)(26 146 36 156)(27 145 37 155)(28 144 38 154)(29 143 39 153)(30 142 40 152)(41 102 51 112)(42 101 52 111)(43 120 53 110)(44 119 54 109)(45 118 55 108)(46 117 56 107)(47 116 57 106)(48 115 58 105)(49 114 59 104)(50 113 60 103)(61 86 71 96)(62 85 72 95)(63 84 73 94)(64 83 74 93)(65 82 75 92)(66 81 76 91)(67 100 77 90)(68 99 78 89)(69 98 79 88)(70 97 80 87)

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

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

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

C20.44D4 is a maximal subgroup of
C20.14Q16  C4xC40:C2  C42.264D10  C4xDic20  C42.14D10  C42.16D10  C42.20D10  Dic20:9C4  C23.34D20  C23.35D20  C23.10D20  D20.32D4  D20:14D4  Dic10:14D4  C22:Dic20  D4.D5:5C4  Dic5:6SD16  C20:Q8:C2  (C8xDic5):C2  D4:(C4xD5)  D4:2D5:C4  D10:SD16  C5:2C8:D4  C5:Q16:5C4  Dic5:4Q16  Dic5.3Q16  C40:8C4.C2  D5xQ8:C4  (Q8xD5):C4  D10:Q16  C5:2C8.D4  Dic10.3Q8  C20:SD16  D20.19D4  C42.36D10  C4:Dic20  C20.7Q16  Dic10:4Q8  Dic10:Q8  Dic10.Q8  D10.12SD16  C20.(C4oD4)  Dic10:2Q8  Dic10.2Q8  D10.8Q16  C2.D8:7D5  C23.23D20  C40:30D4  C40.82D4  C23.46D20  C23.49D20  C40:2D4  C40.4D4  (C2xD8).D5  Dic10:D4  Dic5:3SD16  (C5xQ8).D4  D10:8SD16  Dic10.16D4  Dic5:3Q16  D10:5Q16  C6.Dic20  Dic30:12C4  Dic30:8C4
C20.44D4 is a maximal quotient of
Dic10:3C8  C23.30D20  C4.Dic20  C20.47D8  C20.39C42  C6.Dic20  Dic30:12C4  Dic30:8C4

46 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F5A5B8A8B8C8D10A···10F20A···20H40A···40P
order122244444455888810···1020···2040···40
size111122202020202222222···22···22···2

46 irreducible representations

dim1111122222222222
type+++++++-++-
imageC1C2C2C2C4D4D4D5SD16Q16D10C4xD5C5:D4D20C40:C2Dic20
kernelC20.44D4C4:Dic5C2xC40C2xDic10Dic10C20C2xC10C2xC8C10C10C2xC4C4C4C22C2C2
# reps1111411222244488

Matrix representation of C20.44D4 in GL5(F41)

400000
064000
01000
00090
0002332
,
320000
0382300
05300
0003939
000232
,
10000
0382300
05300
0003535
000136

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,6,1,0,0,0,40,0,0,0,0,0,0,9,23,0,0,0,0,32],[32,0,0,0,0,0,38,5,0,0,0,23,3,0,0,0,0,0,39,23,0,0,0,39,2],[1,0,0,0,0,0,38,5,0,0,0,23,3,0,0,0,0,0,35,13,0,0,0,35,6] >;

C20.44D4 in GAP, Magma, Sage, TeX

C_{20}._{44}D_4
% in TeX

G:=Group("C20.44D4");
// GroupNames label

G:=SmallGroup(160,23);
// by ID

G=gap.SmallGroup(160,23);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,73,79,362,86,4613]);
// Polycyclic

G:=Group<a,b,c|a^20=b^4=1,c^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^5*b^-1>;
// generators/relations

Export

Subgroup lattice of C20.44D4 in TeX

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