Copied to
clipboard

G = D5×C4≀C2order 320 = 26·5

Direct product of D5 and C4≀C2

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5×C4≀C2, C4231D10, M4(2)⋊15D10, (D4×D5)⋊3C4, D47(C4×D5), Q87(C4×D5), (Q8×D5)⋊3C4, D42D53C4, D2015(C2×C4), (C4×C20)⋊9C22, Q82D53C4, (C4×D5).71D4, C4.200(D4×D5), (D5×C42)⋊1C2, D207C45C2, D204C43C2, C4○D4.18D10, C20.359(C2×D4), (D5×M4(2))⋊8C2, C22.27(D4×D5), D42Dic51C2, Dic1014(C2×C4), C20.52(C22×C4), C4○D20.9C22, C4.Dic52C22, (C2×C20).260C23, (C2×Dic5).215D4, (C4×Dic5)⋊61C22, (C22×D5).117D4, D10.57(C22⋊C4), (C5×M4(2))⋊13C22, Dic5.27(C22⋊C4), C54(C2×C4≀C2), (C5×C4≀C2)⋊5C2, C4.17(C2×C4×D5), (C5×D4)⋊15(C2×C4), (C5×Q8)⋊14(C2×C4), (D5×C4○D4).1C2, (C4×D5).52(C2×C4), (C2×C10).24(C2×D4), C2.25(D5×C22⋊C4), C10.65(C2×C22⋊C4), (C5×C4○D4).1C22, (C2×C4×D5).300C22, (C2×C4).367(C22×D5), SmallGroup(320,447)

Series: Derived Chief Lower central Upper central

C1C20 — D5×C4≀C2
C1C5C10C20C2×C20C2×C4×D5D5×C4○D4 — D5×C4≀C2
C5C10C20 — D5×C4≀C2
C1C4C2×C4C4≀C2

Generators and relations for D5×C4≀C2
 G = < a,b,c,d,e | a5=b2=c4=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec, ede-1=c-1d >

Subgroups: 686 in 170 conjugacy classes, 53 normal (51 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×8], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×16], D4, D4 [×6], Q8, Q8 [×2], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C42, C42 [×2], C2×C8, M4(2), M4(2) [×2], C22×C4 [×3], C2×D4 [×2], C2×Q8, C4○D4, C4○D4 [×5], Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×3], D10 [×2], D10 [×5], C2×C10, C2×C10, C4≀C2, C4≀C2 [×3], C2×C42, C2×M4(2), C2×C4○D4, C52C8, C40, Dic10, Dic10, C4×D5 [×4], C4×D5 [×7], D20, D20, C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C22×D5, C22×D5, C2×C4≀C2, C8×D5, C8⋊D5, C4.Dic5, C4×Dic5, C4×Dic5, C4×C20, C5×M4(2), C2×C4×D5, C2×C4×D5 [×2], C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, D204C4, D207C4, D42Dic5, C5×C4≀C2, D5×C42, D5×M4(2), D5×C4○D4, D5×C4≀C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C4≀C2 [×2], C2×C22⋊C4, C4×D5 [×2], C22×D5, C2×C4≀C2, C2×C4×D5, D4×D5 [×2], D5×C22⋊C4, D5×C4≀C2

Smallest permutation representation of D5×C4≀C2
On 40 points
Generators in S40
(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)
(1 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)
(1 14 9 19)(2 15 10 20)(3 11 6 16)(4 12 7 17)(5 13 8 18)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(1 39)(2 40)(3 36)(4 37)(5 38)(6 31)(7 32)(8 33)(9 34)(10 35)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)
(1 14 9 19)(2 15 10 20)(3 11 6 16)(4 12 7 17)(5 13 8 18)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)

G:=sub<Sym(40)| (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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (1,39)(2,40)(3,36)(4,37)(5,38)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)>;

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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (1,39)(2,40)(3,36)(4,37)(5,38)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40) );

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)], [(1,8),(2,7),(3,6),(4,10),(5,9),(11,16),(12,20),(13,19),(14,18),(15,17),(21,26),(22,30),(23,29),(24,28),(25,27),(31,36),(32,40),(33,39),(34,38),(35,37)], [(1,14,9,19),(2,15,10,20),(3,11,6,16),(4,12,7,17),(5,13,8,18),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35)], [(1,39),(2,40),(3,36),(4,37),(5,38),(6,31),(7,32),(8,33),(9,34),(10,35),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30)], [(1,14,9,19),(2,15,10,20),(3,11,6,16),(4,12,7,17),(5,13,8,18),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)])

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C···4G4H4I4J4K···4O4P5A5B8A8B8C8D10A10B10C10D10E10F20A20B20C20D20E···20N20O20P40A40B40C40D
order12222222444···44444···445588881010101010102020202020···20202040404040
size1124551020112···245510···10202244202022448822224···4888888

56 irreducible representations

dim1111111111112222222222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4D4D4D4D5D10D10D10C4≀C2C4×D5C4×D5D4×D5D4×D5D5×C4≀C2
kernelD5×C4≀C2D204C4D207C4D42Dic5C5×C4≀C2D5×C42D5×M4(2)D5×C4○D4D4×D5D42D5Q8×D5Q82D5C4×D5C2×Dic5C22×D5C4≀C2C42M4(2)C4○D4D5D4Q8C4C22C1
# reps1111111122222112222844228

Matrix representation of D5×C4≀C2 in GL4(𝔽41) generated by

1000
0100
00401
00337
,
40000
04000
00400
00331
,
32000
0900
0010
0001
,
03200
9000
00400
00040
,
32000
04000
0010
0001
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,33,0,0,1,7],[40,0,0,0,0,40,0,0,0,0,40,33,0,0,0,1],[32,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[0,9,0,0,32,0,0,0,0,0,40,0,0,0,0,40],[32,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1] >;

D5×C4≀C2 in GAP, Magma, Sage, TeX

D_5\times C_4\wr C_2
% in TeX

G:=Group("D5xC4wrC2");
// GroupNames label

G:=SmallGroup(320,447);
// by ID

G=gap.SmallGroup(320,447);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,58,136,851,438,102,12550]);
// Polycyclic

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

׿
×
𝔽