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## G = C2×D52order 200 = 23·52

### Direct product of C2, D5 and D5

Aliases: C2×D52, C52⋊C23, C101D10, C5⋊D5⋊C22, (C5×C10)⋊C22, (C5×D5)⋊C22, (D5×C10)⋊5C2, C51(C22×D5), (C2×C5⋊D5)⋊4C2, SmallGroup(200,49)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×D52
 Chief series C1 — C5 — C52 — C5×D5 — D52 — C2×D52
 Lower central C52 — C2×D52
 Upper central C1 — C2

Generators and relations for C2×D52
G = < a,b,c,d,e | a2=b5=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 436 in 74 conjugacy classes, 28 normal (6 characteristic)
C1, C2, C2, C22, C5, C5, C23, D5, D5, C10, C10, D10, D10, C2×C10, C52, C22×D5, C5×D5, C5⋊D5, C5×C10, D52, D5×C10, C2×C5⋊D5, C2×D52
Quotients: C1, C2, C22, C23, D5, D10, C22×D5, D52, C2×D52

Permutation representations of C2×D52
On 20 points - transitive group 20T59
Generators in S20
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 18)(2 17)(3 16)(4 20)(5 19)(6 13)(7 12)(8 11)(9 15)(10 14)
(1 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 18)(7 19)(8 20)(9 16)(10 17)

G:=sub<Sym(20)| (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,18)(2,17)(3,16)(4,20)(5,19)(6,13)(7,12)(8,11)(9,15)(10,14), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20), (1,13)(2,14)(3,15)(4,11)(5,12)(6,18)(7,19)(8,20)(9,16)(10,17)>;

G:=Group( (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,18)(2,17)(3,16)(4,20)(5,19)(6,13)(7,12)(8,11)(9,15)(10,14), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20), (1,13)(2,14)(3,15)(4,11)(5,12)(6,18)(7,19)(8,20)(9,16)(10,17) );

G=PermutationGroup([[(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,18),(2,17),(3,16),(4,20),(5,19),(6,13),(7,12),(8,11),(9,15),(10,14)], [(1,5,4,3,2),(6,10,9,8,7),(11,12,13,14,15),(16,17,18,19,20)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,18),(7,19),(8,20),(9,16),(10,17)]])

G:=TransitiveGroup(20,59);

C2×D52 is a maximal subgroup of
D5.D20  D10⋊F5  D52⋊C4  C20⋊D10  D10⋊D10
C2×D52 is a maximal quotient of
D205D5  D20⋊D5  Dic10⋊D5  D10.9D10  Dic105D5  C20⋊D10  Dic5.D10  D10.4D10  D10⋊D10

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P order 1 2 2 2 2 2 2 2 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 ··· 10 size 1 1 5 5 5 5 25 25 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 10 ··· 10

32 irreducible representations

 dim 1 1 1 1 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 D5 D10 D10 D52 C2×D52 kernel C2×D52 D52 D5×C10 C2×C5⋊D5 D10 D5 C10 C2 C1 # reps 1 4 2 1 4 8 4 4 4

Matrix representation of C2×D52 in GL4(𝔽11) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 3 10 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 8 1 0 0 3 3 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 3 10 0 0 1 0
,
 10 0 0 0 0 10 0 0 0 0 3 10 0 0 8 8
G:=sub<GL(4,GF(11))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[3,1,0,0,10,0,0,0,0,0,1,0,0,0,0,1],[8,3,0,0,1,3,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,3,1,0,0,10,0],[10,0,0,0,0,10,0,0,0,0,3,8,0,0,10,8] >;

C2×D52 in GAP, Magma, Sage, TeX

C_2\times D_5^2
% in TeX

G:=Group("C2xD5^2");
// GroupNames label

G:=SmallGroup(200,49);
// by ID

G=gap.SmallGroup(200,49);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,328,4004]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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