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G = D407C2order 160 = 25·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D407C2, C4.20D20, C20.35D4, C8.17D10, Dic207C2, C22.1D20, C40.17C22, C20.30C23, D20.7C22, Dic10.6C22, (C2×C8)⋊4D5, (C2×C40)⋊6C2, C51(C4○D8), C4○D201C2, C40⋊C27C2, C2.13(C2×D20), C10.11(C2×D4), (C2×C10).18D4, (C2×C4).81D10, C4.28(C22×D5), (C2×C20).99C22, SmallGroup(160,125)

Series: Derived Chief Lower central Upper central

C1C20 — D407C2
C1C5C10C20D20C4○D20 — D407C2
C5C10C20 — D407C2
C1C4C2×C4C2×C8

Generators and relations for D407C2
 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 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C5, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], D5 [×2], C10, C10, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C4○D8, C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C2×C20, C40⋊C2 [×2], D40, Dic20, C2×C40, C4○D20 [×2], D407C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, D10 [×3], C4○D8, D20 [×2], C22×D5, C2×D20, D407C2

Smallest permutation representation of D407C2
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 47)(42 46)(43 45)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)

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

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

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

D407C2 is a maximal subgroup of
D4014C4  D40.5C4  D40.3C4  D408C4  D4017C4  D4010C4  D4016C4  D4013C4  D807C2  D80⋊C2  C16.D10  D8.D10  Q16.D10  C40.30C23  C40.9C23  D4.11D20  D4.12D20  D4.13D20  D813D10  D20.29D4  D20.30D4  D5×C4○D8  Q16⋊D10  D6.1D20  D407S3  D1205C2  D20.31D6  C40.69D6
D407C2 is a maximal quotient of
C40.13Q8  C4×C40⋊C2  C4×D40  C8.8D20  C42.264D10  C4×Dic20  C23.10D20  D20.32D4  D2014D4  C23.13D20  Dic10.3Q8  D20.19D4  C42.36D10  D20.3Q8  C23.22D20  C23.23D20  C4030D4  C4029D4  C40.82D4  D6.1D20  D407S3  D1205C2  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+++++++++++++
imageC1C2C2C2C2C2D4D4D5D10D10C4○D8D20D20D407C2
kernelD407C2C40⋊C2D40Dic20C2×C40C4○D20C20C2×C10C2×C8C8C2×C4C5C4C22C1
# reps1211121124244416

Matrix representation of D407C2 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] >;

D407C2 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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