metabelian, supersoluble, monomial
Aliases: C52:4SD16, Dic10:1D5, C10.14D20, C20.13D10, C4.3D52, C5:2C8:3D5, C5:1(Q8:D5), C5:2(C40:C2), (C5xC10).10D4, (C5xDic10):2C2, C20:D5.2C2, C10.3(C5:D4), (C5xC20).5C22, C2.6(C5:D20), (C5xC5:2C8):3C2, SmallGroup(400,68)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52:4SD16
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=dad=a-1, bc=cb, dbd=b-1, dcd=c3 >
Subgroups: 500 in 56 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C8, D4, Q8, D5, C10, C10, SD16, Dic5, C20, C20, D10, C52, C5:2C8, C40, Dic10, D20, C5xQ8, C5:D5, C5xC10, C40:C2, Q8:D5, C5xDic5, C5xC20, C2xC5:D5, C5xC5:2C8, C5xDic10, C20:D5, C52:4SD16
Quotients: C1, C2, C22, D4, D5, SD16, D10, D20, C5:D4, C40:C2, Q8:D5, D52, C5:D20, C52:4SD16
►On 40 points
(1 37 20 14 31)(2 32 15 21 38)(3 39 22 16 25)(4 26 9 23 40)(5 33 24 10 27)(6 28 11 17 34)(7 35 18 12 29)(8 30 13 19 36)
(1 31 14 20 37)(2 32 15 21 38)(3 25 16 22 39)(4 26 9 23 40)(5 27 10 24 33)(6 28 11 17 34)(7 29 12 18 35)(8 30 13 19 36)
(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 3)(2 6)(5 7)(9 23)(10 18)(11 21)(12 24)(13 19)(14 22)(15 17)(16 20)(25 37)(26 40)(27 35)(28 38)(29 33)(30 36)(31 39)(32 34)
G:=sub<Sym(40)| (1,37,20,14,31)(2,32,15,21,38)(3,39,22,16,25)(4,26,9,23,40)(5,33,24,10,27)(6,28,11,17,34)(7,35,18,12,29)(8,30,13,19,36), (1,31,14,20,37)(2,32,15,21,38)(3,25,16,22,39)(4,26,9,23,40)(5,27,10,24,33)(6,28,11,17,34)(7,29,12,18,35)(8,30,13,19,36), (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), (1,3)(2,6)(5,7)(9,23)(10,18)(11,21)(12,24)(13,19)(14,22)(15,17)(16,20)(25,37)(26,40)(27,35)(28,38)(29,33)(30,36)(31,39)(32,34)>;
G:=Group( (1,37,20,14,31)(2,32,15,21,38)(3,39,22,16,25)(4,26,9,23,40)(5,33,24,10,27)(6,28,11,17,34)(7,35,18,12,29)(8,30,13,19,36), (1,31,14,20,37)(2,32,15,21,38)(3,25,16,22,39)(4,26,9,23,40)(5,27,10,24,33)(6,28,11,17,34)(7,29,12,18,35)(8,30,13,19,36), (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), (1,3)(2,6)(5,7)(9,23)(10,18)(11,21)(12,24)(13,19)(14,22)(15,17)(16,20)(25,37)(26,40)(27,35)(28,38)(29,33)(30,36)(31,39)(32,34) );
G=PermutationGroup([[(1,37,20,14,31),(2,32,15,21,38),(3,39,22,16,25),(4,26,9,23,40),(5,33,24,10,27),(6,28,11,17,34),(7,35,18,12,29),(8,30,13,19,36)], [(1,31,14,20,37),(2,32,15,21,38),(3,25,16,22,39),(4,26,9,23,40),(5,27,10,24,33),(6,28,11,17,34),(7,29,12,18,35),(8,30,13,19,36)], [(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)], [(1,3),(2,6),(5,7),(9,23),(10,18),(11,21),(12,24),(13,19),(14,22),(15,17),(16,20),(25,37),(26,40),(27,35),(28,38),(29,33),(30,36),(31,39),(32,34)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20N | 20O | 20P | 20Q | 20R | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 100 | 2 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 20 | 20 | 20 | 20 | 10 | ··· | 10 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | D4 | D5 | D5 | SD16 | D10 | D20 | C5:D4 | C40:C2 | Q8:D5 | D52 | C5:D20 | C52:4SD16 |
| kernel | C52:4SD16 | C5xC5:2C8 | C5xDic10 | C20:D5 | C5xC10 | C5:2C8 | Dic10 | C52 | C20 | C10 | C10 | C5 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 2 | 4 | 4 | 8 |
Matrix representation of C52:4SD16 ►in GL4(F41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 34 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 34 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 14 | 12 |
| 0 | 0 | 29 | 16 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 32 | 11 |
| 0 | 0 | 30 | 9 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,34,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,34],[0,40,0,0,40,0,0,0,0,0,14,29,0,0,12,16],[0,1,0,0,1,0,0,0,0,0,32,30,0,0,11,9] >;
C52:4SD16 in GAP, Magma, Sage, TeX
C_5^2\rtimes_4{\rm SD}_{16} % in TeX
G:=Group("C5^2:4SD16"); // GroupNames label
G:=SmallGroup(400,68);
// by ID
G=gap.SmallGroup(400,68);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,73,31,218,50,970,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^3>;
// generators/relations