direct product, metabelian, supersoluble, monomial
Aliases: C5×C5⋊D4, C10≀C2, C52⋊6D4, Dic5⋊C10, D10⋊2C10, C102⋊2C2, C10.21D10, C5⋊2(C5×D4), C22⋊(C5×D5), (C2×C10)⋊1D5, (C2×C10)⋊2C10, (D5×C10)⋊4C2, C2.5(D5×C10), C10.5(C2×C10), (C5×Dic5)⋊4C2, (C5×C10).10C22, SmallGroup(200,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C5⋊D4
G = < a,b,c,d | a5=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Permutation representations of C5×C5⋊D4
►On 20 points - transitive group 20T53
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 3 5 2 4)(6 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)
(1 17 7 12)(2 18 8 13)(3 19 9 14)(4 20 10 15)(5 16 6 11)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 16)(7 17)(8 18)(9 19)(10 20)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18), (1,17,7,12)(2,18,8,13)(3,19,9,14)(4,20,10,15)(5,16,6,11), (1,12)(2,13)(3,14)(4,15)(5,11)(6,16)(7,17)(8,18)(9,19)(10,20)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18), (1,17,7,12)(2,18,8,13)(3,19,9,14)(4,20,10,15)(5,16,6,11), (1,12)(2,13)(3,14)(4,15)(5,11)(6,16)(7,17)(8,18)(9,19)(10,20) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,3,5,2,4),(6,8,10,7,9),(11,14,12,15,13),(16,19,17,20,18)], [(1,17,7,12),(2,18,8,13),(3,19,9,14),(4,20,10,15),(5,16,6,11)], [(1,12),(2,13),(3,14),(4,15),(5,11),(6,16),(7,17),(8,18),(9,19),(10,20)]])
G:=TransitiveGroup(20,53);
C5×C5⋊D4 is a maximal subgroup of
Dic5.D10 D10.4D10 D10⋊D10 C5×D4×D5
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | 10B | 10C | 10D | 10E | ··· | 10AL | 10AM | 10AN | 10AO | 10AP | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 2 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | D5 | D10 | C5⋊D4 | C5×D4 | C5×D5 | D5×C10 | C5×C5⋊D4 |
| kernel | C5×C5⋊D4 | C5×Dic5 | D5×C10 | C102 | C5⋊D4 | Dic5 | D10 | C2×C10 | C52 | C2×C10 | C10 | C5 | C5 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C5×C5⋊D4 ►in GL2(𝔽11) generated by
| 9 | 0 |
| 0 | 9 |
| 8 | 1 |
| 2 | 10 |
| 2 | 4 |
| 7 | 9 |
| 1 | 1 |
| 0 | 10 |
G:=sub<GL(2,GF(11))| [9,0,0,9],[8,2,1,10],[2,7,4,9],[1,0,1,10] >;
C5×C5⋊D4 in GAP, Magma, Sage, TeX
C_5\times C_5\rtimes D_4
% in TeX
G:=Group("C5xC5:D4"); // GroupNames label
G:=SmallGroup(200,31);
// by ID
G=gap.SmallGroup(200,31);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-5,221,4004]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
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