metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5:C4wrC2, (D4xD5):8C4, C4oD4:1F5, D4:5(C2xF5), Q8:5(C2xF5), (Q8xD5):8C4, C4oD20:1C4, D20:5(C2xC4), D4:F5:7C2, Q8:2F5:7C2, (C4xF5):7C22, C4.F5:5C22, Dic10:5(C2xC4), D10.10(C2xD4), (C4xD5).121D4, D5:M4(2):5C2, C4.21(C22xF5), C20.21(C22xC4), (C4xD5).43C23, (C22xD5).71D4, C4.46(C22:F5), C20.46(C22:C4), (C2xDic5).123D4, Dic5.114(C2xD4), C22.4(C22:F5), D10.48(C22:C4), D4:2D5.13C22, Q8:2D5.13C22, Dic5.15(C22:C4), C5:3(C2xC4wrC2), (C2xC4xF5):4C2, (C5xC4oD4):1C4, (C5xD4):5(C2xC4), (C5xQ8):5(C2xC4), (D5xC4oD4).5C2, (C2xC4).89(C2xF5), (C2xC20).66(C2xC4), (C4xD5).27(C2xC4), C2.34(C2xC22:F5), C10.33(C2xC22:C4), (C2xC4xD5).210C22, (C2xC10).4(C22:C4), SmallGroup(320,1130)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5:C4wrC2
G = < a,b,c,d,e | a5=b2=c4=d2=e4=1, bab=a-1, ac=ca, ad=da, eae-1=a3, bc=cb, bd=db, ebe-1=a2b, dcd=c-1, ce=ec, ede-1=c-1d >
Subgroups: 730 in 170 conjugacy classes, 50 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C42, C2xC8, M4(2), C22xC4, C2xD4, C2xQ8, C4oD4, C4oD4, Dic5, Dic5, C20, C20, F5, D10, D10, C2xC10, C2xC10, C4wrC2, C2xC42, C2xM4(2), C2xC4oD4, C5:C8, Dic10, Dic10, C4xD5, C4xD5, D20, D20, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C5xD4, C5xQ8, C2xF5, C22xD5, C22xD5, C2xC4wrC2, D5:C8, C4.F5, C4xF5, C4xF5, C22.F5, C2xC4xD5, C2xC4xD5, C4oD20, C4oD20, D4xD5, D4xD5, D4:2D5, D4:2D5, Q8xD5, Q8:2D5, C5xC4oD4, C22xF5, D4:F5, Q8:2F5, D5:M4(2), C2xC4xF5, D5xC4oD4, D5:C4wrC2
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, F5, C4wrC2, C2xC22:C4, C2xF5, C2xC4wrC2, C22:F5, C22xF5, C2xC22:F5, D5:C4wrC2
(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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)
(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 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 9)(2 6 5 7)(3 8 4 10)(11 18 12 20)(13 17 15 16)(14 19)(21 38 27 35)(22 40 26 33)(23 37 30 31)(24 39 29 34)(25 36 28 32)
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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39), (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), (1,34)(2,35)(3,31)(4,32)(5,33)(6,36)(7,37)(8,38)(9,39)(10,40)(11,26)(12,27)(13,28)(14,29)(15,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,9)(2,6,5,7)(3,8,4,10)(11,18,12,20)(13,17,15,16)(14,19)(21,38,27,35)(22,40,26,33)(23,37,30,31)(24,39,29,34)(25,36,28,32)>;
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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39), (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), (1,34)(2,35)(3,31)(4,32)(5,33)(6,36)(7,37)(8,38)(9,39)(10,40)(11,26)(12,27)(13,28)(14,29)(15,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,9)(2,6,5,7)(3,8,4,10)(11,18,12,20)(13,17,15,16)(14,19)(21,38,27,35)(22,40,26,33)(23,37,30,31)(24,39,29,34)(25,36,28,32) );
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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39)], [(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)], [(1,34),(2,35),(3,31),(4,32),(5,33),(6,36),(7,37),(8,38),(9,39),(10,40),(11,26),(12,27),(13,28),(14,29),(15,30),(16,21),(17,22),(18,23),(19,24),(20,25)], [(1,9),(2,6,5,7),(3,8,4,10),(11,18,12,20),(13,17,15,16),(14,19),(21,38,27,35),(22,40,26,33),(23,37,30,31),(24,39,29,34),(25,36,28,32)]])
38 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4O | 4P | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 |
size | 1 | 1 | 2 | 4 | 5 | 5 | 10 | 20 | 1 | 1 | 2 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | 4 | 20 | 20 | 20 | 20 | 4 | 8 | 8 | 8 | 4 | 4 | 8 | 8 | 8 |
38 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D4 | D4 | C4wrC2 | F5 | C2xF5 | C2xF5 | C2xF5 | C22:F5 | C22:F5 | D5:C4wrC2 |
kernel | D5:C4wrC2 | D4:F5 | Q8:2F5 | D5:M4(2) | C2xC4xF5 | D5xC4oD4 | C4oD20 | D4xD5 | Q8xD5 | C5xC4oD4 | C4xD5 | C2xDic5 | C22xD5 | D5 | C4oD4 | C2xC4 | D4 | Q8 | C4 | C22 | C1 |
# reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of D5:C4wrC2 ►in GL6(F41)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 40 | 40 | 40 | 40 |
40 | 0 | 0 | 0 | 0 | 0 |
0 | 40 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 40 | 40 | 40 | 40 |
0 | 0 | 0 | 0 | 0 | 1 |
32 | 32 | 0 | 0 | 0 | 0 |
0 | 9 | 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 |
9 | 9 | 0 | 0 | 0 | 0 |
23 | 32 | 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 |
40 | 36 | 0 | 0 | 0 | 0 |
0 | 9 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 40 | 40 | 40 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,40,0,0,0,1,0,40,0,0,0,0,0,40,0,0,0,0,0,40,1],[32,0,0,0,0,0,32,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,23,0,0,0,0,9,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,36,9,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;
D5:C4wrC2 in GAP, Magma, Sage, TeX
D_5\rtimes C_4\wr C_2
% in TeX
G:=Group("D5:C4wrC2");
// GroupNames label
G:=SmallGroup(320,1130);
// by ID
G=gap.SmallGroup(320,1130);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,297,438,102,6278,1595]);
// 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,e*a*e^-1=a^3,b*c=c*b,b*d=d*b,e*b*e^-1=a^2*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=c^-1*d>;
// generators/relations