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G = D40:7C2  order 160 = 25·5

The semidirect product of D40 and C2 acting through Inn(D40)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D40:7C2, C4.20D20, C20.35D4, C8.17D10, Dic20:7C2, C22.1D20, C40.17C22, C20.30C23, D20.7C22, Dic10.6C22, (C2xC8):4D5, (C2xC40):6C2, C5:1(C4oD8), C4oD20:1C2, C40:C2:7C2, C2.13(C2xD20), C10.11(C2xD4), (C2xC10).18D4, (C2xC4).81D10, C4.28(C22xD5), (C2xC20).99C22, SmallGroup(160,125)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — D40:7C2
Chief series
C1 — C5 — C10 — C20 — D20 — C4oD20 — D40:7C2
Lower central
C5 — C10 — C20 — D40:7C2
Upper central
C1 — C4 — C2xC4 — C2xC8

Generators and relations for D40:7C2
 G = < a,b,c | a40=b2=c2=1, bab=a-1, ac=ca, cbc=a20b >

Subgroups: 232 in 62 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, D4, Q8, D5, C10, C10, C2xC8, D8, SD16, Q16, C4oD4, Dic5, C20, D10, C2xC10, C4oD8, C40, Dic10, C4xD5, D20, C5:D4, C2xC20, C40:C2, D40, Dic20, C2xC40, C4oD20, D40:7C2
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, D10, C4oD8, D20, C22xD5, C2xD20, D40:7C2

Smallest permutation representation of D40:7C2
►On 80 points
Generators in S80
(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)

(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 51)(42 50)(43 49)(44 48)(45 47)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)

(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
 
G:=sub<Sym(80)| (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,51)(42,50)(43,49)(44,48)(45,47)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)>;
 
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,51)(42,50)(43,49)(44,48)(45,47)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48) );
 
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(41,51),(42,50),(43,49),(44,48),(45,47),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,72),(61,71),(62,70),(63,69),(64,68),(65,67)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)]])
 

D40:7C2 is a maximal subgroup of
 D40:14C4  D40.5C4  D40.3C4  D40:8C4  D40:17C4  D40:10C4  D40:16C4  D40:13C4  D80:7C2  D80:C2  C16.D10  D8.D10  Q16.D10  C40.30C23  C40.9C23  D4.11D20  D4.12D20  D4.13D20  D8:13D10  D20.29D4  D20.30D4  D5xC4oD8  Q16:D10  D6.1D20  D40:7S3  D120:5C2  D20.31D6  C40.69D6
D40:7C2 is a maximal quotient of
 C40.13Q8  C4xC40:C2  C4xD40  C8.8D20  C42.264D10  C4xDic20  C23.10D20  D20.32D4  D20:14D4  C23.13D20  Dic10.3Q8  D20.19D4  C42.36D10  D20.3Q8  C23.22D20  C23.23D20  C40:30D4  C40:29D4  C40.82D4  D6.1D20  D40:7S3  D120:5C2  D20.31D6  C40.69D6

46 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B8A8B8C8D10A···10F20A···20H40A···40P
order122224444455888810···1020···2040···40
size112202011220202222222···22···22···2

46 irreducible representations

dim111111222222222
type+++++++++++++
imageC1C2C2C2C2C2D4D4D5D10D10C4oD8D20D20D40:7C2
kernelD40:7C2C40:C2D40Dic20C2xC40C4oD20C20C2xC10C2xC8C8C2xC4C5C4C22C1
# reps1211121124244416

Matrix representation of D40:7C2 ►in GL2(F41) generated by

2621
236
,
275
214
,
2434
617
G:=sub<GL(2,GF(41))| [26,23,21,6],[27,2,5,14],[24,6,34,17] >;
 

D40:7C2 in GAP, Magma, Sage, TeX 

D_{40}\rtimes_7C_2
 
% in TeX
 
G:=Group("D40:7C2");
 
// GroupNames label
 
G:=SmallGroup(160,125);
 
// by ID
 
G=gap.SmallGroup(160,125);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,50,579,69,4613]);
 
// Polycyclic
 
G:=Group<a,b,c|a^40=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^20*b>;
 
// generators/relations
 

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