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G = 2+ 1+4:D5order 320 = 26·5

1st semidirect product of 2+ 1+4 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2+ 1+4:1D5, (C5xD4):15D4, (C5xQ8):15D4, C20:D4:8C2, C5:5(D4:4D4), D4:7(C5:D4), C4oD4.9D10, Q8:7(C5:D4), D4:D10:5C2, (C2xD4).82D10, C20.217(C2xD4), C10.80C22wrC2, C20.D4:11C2, (C2xD20):15C22, (C2xC20).21C23, (C4xDic5):8C22, (C22xC10).24D4, D4:2Dic5:11C2, C23.12(C5:D4), C4.Dic5:10C22, (C5x2+ 1+4):1C2, (D4xC10).107C22, C2.14(C24:2D5), C4.64(C2xC5:D4), (C2xC10).42(C2xD4), (C2xC4).21(C22xD5), C22.14(C2xC5:D4), (C5xC4oD4).19C22, SmallGroup(320,868)

Series: Derived Chief Lower central Upper central

C1C2xC20 — 2+ 1+4:D5
C1C5C10C2xC10C2xC20C2xD20D4:D10 — 2+ 1+4:D5
C5C10C2xC20 — 2+ 1+4:D5
C1C2C2xC42+ 1+4

Generators and relations for 2+ 1+4:D5
 G = < a,b,c,d,e,f | a4=b2=d2=e5=f2=1, c2=a2, bab=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=ab, dcd=fcf=a2c, ce=ec, de=ed, fdf=cd, fef=e-1 >

Subgroups: 670 in 168 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, D4, D4, Q8, C23, C23, D5, C10, C10, C42, M4(2), D8, SD16, C2xD4, C2xD4, C4oD4, C4oD4, Dic5, C20, C20, D10, C2xC10, C2xC10, C4.D4, C4wrC2, C4:1D4, C8:C22, 2+ 1+4, C5:2C8, D20, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C5xD4, C5xQ8, C22xD5, C22xC10, C22xC10, D4:4D4, C4.Dic5, C4xDic5, D4:D5, Q8:D5, C2xD20, C2xC5:D4, D4xC10, D4xC10, C5xC4oD4, C5xC4oD4, C20.D4, D4:2Dic5, C20:D4, D4:D10, C5x2+ 1+4, 2+ 1+4:D5
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, D10, C22wrC2, C5:D4, C22xD5, D4:4D4, C2xC5:D4, C24:2D5, 2+ 1+4:D5

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

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B8A8B10A10B10C···10T20A···20L
order122222224444445588101010···1020···20
size11244444022442020224040224···44···4

50 irreducible representations

dim11111122222222248
type++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5D10D10C5:D4C5:D4C5:D4D4:4D42+ 1+4:D5
kernel2+ 1+4:D5C20.D4D4:2Dic5C20:D4D4:D10C5x2+ 1+4C5xD4C5xQ8C22xC102+ 1+4C2xD4C4oD4D4Q8C23C5C1
# reps11212122222488822

Matrix representation of 2+ 1+4:D5 in GL6(F41)

4000000
0400000
0004000
001000
0000040
000010
,
4000000
010000
0000040
0000400
0004000
0040000
,
100000
010000
000100
0040000
0000040
000010
,
4000000
0400000
0000040
000010
000100
0040000
,
1000000
0370000
001000
000100
000010
000001
,
0180000
1600000
001000
0004000
000001
000010

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,40,0,0,0],[10,0,0,0,0,0,0,37,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

2+ 1+4:D5 in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes D_5
% in TeX

G:=Group("ES+(2,2):D5");
// GroupNames label

G:=SmallGroup(320,868);
// by ID

G=gap.SmallGroup(320,868);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,570,1684,851,438,102,12550]);
// Polycyclic

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

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