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
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
(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 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 10A | 10B | 10C | ··· | 10T | 20A | ··· | 20L |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 40 | 40 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
50 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | C5:D4 | C5:D4 | C5:D4 | D4:4D4 | 2+ 1+4:D5 |
kernel | 2+ 1+4:D5 | C20.D4 | D4:2Dic5 | C20:D4 | D4:D10 | C5x2+ 1+4 | C5xD4 | C5xQ8 | C22xC10 | 2+ 1+4 | C2xD4 | C4oD4 | D4 | Q8 | C23 | C5 | C1 |
# reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 2 | 2 |
Matrix representation of 2+ 1+4:D5 ►in GL6(F41)
40 | 0 | 0 | 0 | 0 | 0 |
0 | 40 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
0 | 0 | 0 | 0 | 1 | 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 | 1 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
0 | 0 | 0 | 0 | 1 | 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 | 18 | 0 | 0 | 0 | 0 |
16 | 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 |
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