metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:3Dic5, (C4xC20):13C4, (D4xC10):14C4, (C2xD4):2Dic5, C4:1D4.2D5, C5:4(C42:C4), (C2xD4).10D10, C23:Dic5:8C2, (C22xC10).17D4, C23.8(C5:D4), C10.46(C23:C4), (D4xC10).173C22, C2.10(C23:Dic5), C22.16(C23.D5), (C5xC4:1D4).7C2, (C2xC4).3(C2xDic5), (C2xC20).183(C2xC4), (C2xC10).167(C22:C4), SmallGroup(320,103)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:3Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, cac-1=a-1, dad-1=a-1b, cbc-1=b-1, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 366 in 86 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2xC4, C2xC4, D4, C23, C23, C10, C10, C42, C22:C4, C2xD4, C2xD4, Dic5, C20, C2xC10, C2xC10, C23:C4, C4:1D4, C2xDic5, C2xC20, C2xC20, C5xD4, C22xC10, C22xC10, C42:C4, C23.D5, C4xC20, D4xC10, D4xC10, C23:Dic5, C5xC4:1D4, C42:3Dic5
Quotients: C1, C2, C4, C22, C2xC4, D4, D5, C22:C4, Dic5, D10, C23:C4, C2xDic5, C5:D4, C42:C4, C23.D5, C23:Dic5, C42:3Dic5
►On 40 points
(1 30 25 6)(2 7 26 21)(3 22 27 8)(4 9 28 23)(5 24 29 10)
(1 30 25 6)(2 7 26 21)(3 22 27 8)(4 9 28 23)(5 24 29 10)(11 16 35 40)(12 31 36 17)(13 18 37 32)(14 33 38 19)(15 20 39 34)
(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 35 6 40)(2 34 7 39)(3 33 8 38)(4 32 9 37)(5 31 10 36)(11 30 16 25)(12 29 17 24)(13 28 18 23)(14 27 19 22)(15 26 20 21)
G:=sub<Sym(40)| (1,30,25,6)(2,7,26,21)(3,22,27,8)(4,9,28,23)(5,24,29,10), (1,30,25,6)(2,7,26,21)(3,22,27,8)(4,9,28,23)(5,24,29,10)(11,16,35,40)(12,31,36,17)(13,18,37,32)(14,33,38,19)(15,20,39,34), (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), (1,35,6,40)(2,34,7,39)(3,33,8,38)(4,32,9,37)(5,31,10,36)(11,30,16,25)(12,29,17,24)(13,28,18,23)(14,27,19,22)(15,26,20,21)>;
G:=Group( (1,30,25,6)(2,7,26,21)(3,22,27,8)(4,9,28,23)(5,24,29,10), (1,30,25,6)(2,7,26,21)(3,22,27,8)(4,9,28,23)(5,24,29,10)(11,16,35,40)(12,31,36,17)(13,18,37,32)(14,33,38,19)(15,20,39,34), (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), (1,35,6,40)(2,34,7,39)(3,33,8,38)(4,32,9,37)(5,31,10,36)(11,30,16,25)(12,29,17,24)(13,28,18,23)(14,27,19,22)(15,26,20,21) );
G=PermutationGroup([[(1,30,25,6),(2,7,26,21),(3,22,27,8),(4,9,28,23),(5,24,29,10)], [(1,30,25,6),(2,7,26,21),(3,22,27,8),(4,9,28,23),(5,24,29,10),(11,16,35,40),(12,31,36,17),(13,18,37,32),(14,33,38,19),(15,20,39,34)], [(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)], [(1,35,6,40),(2,34,7,39),(3,33,8,38),(4,32,9,37),(5,31,10,36),(11,30,16,25),(12,29,17,24),(13,28,18,23),(14,27,19,22),(15,26,20,21)]])
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 40 | 40 | 40 | 40 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | D5 | Dic5 | Dic5 | D10 | C5:D4 | C23:C4 | C42:C4 | C23:Dic5 | C42:3Dic5 |
| kernel | C42:3Dic5 | C23:Dic5 | C5xC4:1D4 | C4xC20 | D4xC10 | C22xC10 | C4:1D4 | C42 | C2xD4 | C2xD4 | C23 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C42:3Dic5 ►in GL4(F41) generated by
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,1,0,0,0,0,0,0,40,0,0,1,0],[0,16,0,0,16,0,0,0,0,0,0,18,0,0,18,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;
C42:3Dic5 in GAP, Magma, Sage, TeX
C_4^2\rtimes_3{\rm Dic}_5 % in TeX
G:=Group("C4^2:3Dic5"); // GroupNames label
G:=SmallGroup(320,103);
// by ID
G=gap.SmallGroup(320,103);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,1571,570,297,136,1684,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c^5,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations