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G = C25.D5order 320 = 26·5

The non-split extension by C25 of D5 acting faithfully

non-abelian, soluble, monomial

Aliases: C25.D5, C24⋊Dic5, C24⋊C52C4, C2.1(C24⋊D5), (C2×C24⋊C5).C2, SmallGroup(320,1583)

Series: Derived Chief Lower central Upper central

C1C24C24⋊C5 — C25.D5
C1C24C24⋊C5C2×C24⋊C5 — C25.D5
C24⋊C5 — C25.D5
C1C2

Generators and relations for C25.D5
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f5=1, g2=a, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, geg-1=bc=cb, bd=db, fcf-1=gcg-1=be=eb, fbf-1=e, bg=gb, cd=dc, ce=ec, de=ed, fdf-1=bce, gdg-1=cde, fef-1=bcde, gfg-1=f-1 >

Subgroups: 800 in 111 conjugacy classes, 7 normal (all characteristic)
C1, C2, C2 [×6], C4 [×4], C22 [×25], C5, C2×C4 [×12], C23 [×25], C10, C22⋊C4 [×12], C22×C4 [×4], C24, C24 [×6], Dic5, C2×C22⋊C4 [×6], C25, C243C4, C24⋊C5, C2×C24⋊C5, C25.D5
Quotients: C1, C2, C4, D5, Dic5, C24⋊D5, C25.D5

Character table of C25.D5

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H5A5B10A10B
 size 11555555202020202020202032323232
ρ111111111111111111111    trivial
ρ211111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ31-1111-1-1-1-iiii-i-i-ii11-1-1    linear of order 4
ρ41-1111-1-1-1i-i-i-iiii-i11-1-1    linear of order 4
ρ52222222200000000-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ62222222200000000-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ72-2222-2-2-200000000-1-5/2-1+5/21+5/21-5/2    symplectic lifted from Dic5, Schur index 2
ρ82-2222-2-2-200000000-1+5/2-1-5/21-5/21+5/2    symplectic lifted from Dic5, Schur index 2
ρ955-3111-311-1-111-1-110000    orthogonal lifted from C24⋊D5
ρ1055-3111-31-111-1-111-10000    orthogonal lifted from C24⋊D5
ρ115511-3-3111-111-1-11-10000    orthogonal lifted from C24⋊D5
ρ12551-3111-311-11-11-1-10000    orthogonal lifted from C24⋊D5
ρ13551-3111-3-1-11-11-1110000    orthogonal lifted from C24⋊D5
ρ145511-3-311-11-1-111-110000    orthogonal lifted from C24⋊D5
ρ155-51-31-1-13-ii-iii-ii-i0000    complex faithful
ρ165-511-33-1-1-i-iiiii-i-i0000    complex faithful
ρ175-5-311-13-1iii-ii-i-i-i0000    complex faithful
ρ185-51-31-1-13i-ii-i-ii-ii0000    complex faithful
ρ195-511-33-1-1ii-i-i-i-iii0000    complex faithful
ρ205-5-311-13-1-i-i-ii-iiii0000    complex faithful

Permutation representations of C25.D5
On 20 points - transitive group 20T82
Generators in S20
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)
(1 6)(3 13)(5 20)(8 18)(10 15)(11 16)
(1 11)(2 7)(3 13)(4 14)(5 20)(6 16)(8 18)(9 19)(10 15)(12 17)
(1 6)(2 17)(3 18)(5 10)(7 12)(8 13)(11 16)(15 20)
(2 12)(4 19)(5 10)(7 17)(9 14)(15 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 10 11 20)(2 9 12 19)(3 8 13 18)(4 7 14 17)(5 6 15 16)

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

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

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

G:=TransitiveGroup(20,82);

On 20 points - transitive group 20T84
Generators in S20
(1 11)(2 12)(3 13)(4 14)(5 15)(6 19)(7 20)(8 16)(9 17)(10 18)
(2 12)(3 13)(7 20)(8 16)
(2 12)(4 14)(7 20)(9 17)
(2 12)(3 13)(4 14)(5 15)(7 20)(8 16)(9 17)(10 18)
(1 11)(2 12)(6 19)(7 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 9 11 17)(2 8 12 16)(3 7 13 20)(4 6 14 19)(5 10 15 18)

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

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

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

G:=TransitiveGroup(20,84);

Matrix representation of C25.D5 in GL5(𝔽41)

400000
040000
004000
000400
000040
,
402000
01000
0234000
0210400
0260040
,
400000
040000
018100
119010
3422001
,
139000
040000
018100
020010
396252340
,
400000
040000
018100
0182400
0227040
,
00100
09100
40034390
00071
245303532
,
0234000
032000
139000
331231132
393182340

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,2,1,23,21,26,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,1,34,0,40,18,19,22,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,39,39,40,18,20,6,0,0,1,0,25,0,0,0,1,23,0,0,0,0,40],[40,0,0,0,0,0,40,18,18,22,0,0,1,2,7,0,0,0,40,0,0,0,0,0,40],[0,0,40,0,24,0,9,0,0,5,1,1,34,0,30,0,0,39,7,35,0,0,0,1,32],[0,0,1,33,39,23,32,39,12,3,40,0,0,31,18,0,0,0,1,23,0,0,0,32,40] >;

C25.D5 in GAP, Magma, Sage, TeX

C_2^5.D_5
% in TeX

G:=Group("C2^5.D5");
// GroupNames label

G:=SmallGroup(320,1583);
// by ID

G=gap.SmallGroup(320,1583);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,2,2,2,14,338,1683,437,1068,9245,2539,4906,265]);
// Polycyclic

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

Export

Character table of C25.D5 in TeX

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