metabelian, supersoluble, monomial
Aliases: C102⋊4C4, C5⋊D5.8D4, (C2×C10)⋊2F5, C5⋊3(C22⋊F5), C22⋊(C52⋊C4), C10.27(C2×F5), C52⋊7(C22⋊C4), (C2×C5⋊D5)⋊7C4, (C2×C52⋊C4)⋊3C2, C2.7(C2×C52⋊C4), (C5×C10).40(C2×C4), (C22×C5⋊D5).5C2, (C2×C5⋊D5).26C22, SmallGroup(400,162)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C102⋊4C4
G = < a,b,c | a10=b10=c4=1, ab=ba, cac-1=a7b5, cbc-1=b3 >
Subgroups: 908 in 100 conjugacy classes, 20 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C5, C5, C2×C4, C23, D5, C10, C10, C22⋊C4, F5, D10, C2×C10, C2×C10, C52, C2×F5, C22×D5, C5⋊D5, C5⋊D5, C5×C10, C5×C10, C22⋊F5, C52⋊C4, C2×C5⋊D5, C2×C5⋊D5, C102, C2×C52⋊C4, C22×C5⋊D5, C102⋊4C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C2×F5, C22⋊F5, C52⋊C4, C2×C52⋊C4, C102⋊4C4
►On 20 points - transitive group 20T103
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 10 4 8 2 6 5 9 3 7)(11 12 13 14 15 16 17 18 19 20)
(1 15 5 17)(2 13 4 19)(3 11)(6 20 10 12)(7 18 9 14)(8 16)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,10,4,8,2,6,5,9,3,7)(11,12,13,14,15,16,17,18,19,20), (1,15,5,17)(2,13,4,19)(3,11)(6,20,10,12)(7,18,9,14)(8,16)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,10,4,8,2,6,5,9,3,7)(11,12,13,14,15,16,17,18,19,20), (1,15,5,17)(2,13,4,19)(3,11)(6,20,10,12)(7,18,9,14)(8,16) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,10,4,8,2,6,5,9,3,7),(11,12,13,14,15,16,17,18,19,20)], [(1,15,5,17),(2,13,4,19),(3,11),(6,20,10,12),(7,18,9,14),(8,16)]])
G:=TransitiveGroup(20,103);
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | ··· | 5F | 10A | ··· | 10R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | ··· | 5 | 10 | ··· | 10 |
| size | 1 | 1 | 2 | 25 | 25 | 50 | 50 | 50 | 50 | 50 | 4 | ··· | 4 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C4 | C4 | D4 | F5 | C2×F5 | C22⋊F5 | C52⋊C4 | C2×C52⋊C4 | C102⋊4C4 |
| kernel | C102⋊4C4 | C2×C52⋊C4 | C22×C5⋊D5 | C2×C5⋊D5 | C102 | C5⋊D5 | C2×C10 | C10 | C5 | C22 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C102⋊4C4 ►in GL4(𝔽41) generated by
| 7 | 34 | 0 | 0 |
| 7 | 40 | 0 | 0 |
| 29 | 22 | 6 | 40 |
| 38 | 30 | 36 | 1 |
| 1 | 34 | 0 | 0 |
| 7 | 34 | 0 | 0 |
| 1 | 35 | 6 | 40 |
| 6 | 39 | 36 | 1 |
| 0 | 0 | 34 | 1 |
| 1 | 1 | 39 | 40 |
| 1 | 35 | 40 | 0 |
| 0 | 6 | 34 | 0 |
G:=sub<GL(4,GF(41))| [7,7,29,38,34,40,22,30,0,0,6,36,0,0,40,1],[1,7,1,6,34,34,35,39,0,0,6,36,0,0,40,1],[0,1,1,0,0,1,35,6,34,39,40,34,1,40,0,0] >;
C102⋊4C4 in GAP, Magma, Sage, TeX
C_{10}^2\rtimes_4C_4 % in TeX
G:=Group("C10^2:4C4"); // GroupNames label
G:=SmallGroup(400,162);
// by ID
G=gap.SmallGroup(400,162);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,1444,496,5765,2897]);
// Polycyclic
G:=Group<a,b,c|a^10=b^10=c^4=1,a*b=b*a,c*a*c^-1=a^7*b^5,c*b*c^-1=b^3>;
// generators/relations