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## G = C2×C24⋊D5order 320 = 26·5

### Direct product of C2 and C24⋊D5

Aliases: C2×C24⋊D5, C2D5, C25⋊D5, C24⋊D10, C24⋊C5⋊C22, (C2×C24⋊C5)⋊C2, SmallGroup(320,1636)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C24⋊C5 — C2×C24⋊D5
 Chief series C1 — C24 — C24⋊C5 — C24⋊D5 — C2×C24⋊D5
 Lower central C24⋊C5 — C2×C24⋊D5
 Upper central C1 — C2

Generators and relations for C2×C24⋊D5
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f5=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, geg=bc=cb, bd=db, fcf-1=gcg=be=eb, fbf-1=e, bg=gb, cd=dc, ce=ec, de=ed, fdf-1=bce, gdg=cde, fef-1=bcde, gfg=f-1 >

Subgroups: 1434 in 193 conjugacy classes, 9 normal (7 characteristic)
C1, C2, C2 [×8], C4 [×6], C22 [×35], C5, C2×C4 [×12], D4 [×24], C23 [×33], D5 [×2], C10, C22⋊C4 [×12], C22×C4 [×3], C2×D4 [×24], C24, C24 [×7], D10, C2×C22⋊C4 [×3], C22≀C2 [×8], C22×D4 [×3], C25, C2×C22≀C2, C24⋊C5, C24⋊D5 [×2], C2×C24⋊C5, C2×C24⋊D5
Quotients: C1, C2 [×3], C22, D5, D10, C24⋊D5, C2×C24⋊D5

Character table of C2×C24⋊D5

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 5A 5B 10A 10B size 1 1 5 5 5 5 5 5 20 20 20 20 20 20 20 20 32 32 32 32 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ4 1 -1 -1 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 1 1 -1 -1 linear of order 2 ρ5 2 -2 -2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 orthogonal lifted from D10 ρ6 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ7 2 -2 -2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 orthogonal lifted from D10 ρ8 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ9 5 -5 -1 1 1 -3 3 -1 -1 1 1 -1 1 -1 -1 1 0 0 0 0 orthogonal faithful ρ10 5 -5 3 1 -3 1 -1 -1 -1 1 -1 1 -1 1 -1 1 0 0 0 0 orthogonal faithful ρ11 5 5 1 -3 1 1 1 -3 1 1 -1 -1 -1 -1 1 1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ12 5 5 1 -3 1 1 1 -3 -1 -1 1 1 1 1 -1 -1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ13 5 -5 3 1 -3 1 -1 -1 1 -1 1 -1 1 -1 1 -1 0 0 0 0 orthogonal faithful ρ14 5 -5 -1 1 1 -3 3 -1 1 -1 -1 1 -1 1 1 -1 0 0 0 0 orthogonal faithful ρ15 5 5 1 1 1 -3 -3 1 -1 -1 -1 -1 1 1 1 1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ16 5 5 -3 1 -3 1 1 1 -1 -1 1 1 -1 -1 1 1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ17 5 -5 -1 -3 1 1 -1 3 1 -1 1 -1 -1 1 -1 1 0 0 0 0 orthogonal faithful ρ18 5 -5 -1 -3 1 1 -1 3 -1 1 -1 1 1 -1 1 -1 0 0 0 0 orthogonal faithful ρ19 5 5 -3 1 -3 1 1 1 1 1 -1 -1 1 1 -1 -1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ20 5 5 1 1 1 -3 -3 1 1 1 1 1 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from C24⋊D5

Permutation representations of C2×C24⋊D5
On 10 points - transitive group 10T23
Generators in S10
(1 10)(2 6)(3 7)(4 8)(5 9)
(1 10)(2 6)(4 8)(5 9)
(3 7)(4 8)
(3 7)(5 9)
(1 10)(3 7)(4 8)(5 9)
(1 2 3 4 5)(6 7 8 9 10)
(1 9)(2 8)(3 7)(4 6)(5 10)

G:=sub<Sym(10)| (1,10)(2,6)(3,7)(4,8)(5,9), (1,10)(2,6)(4,8)(5,9), (3,7)(4,8), (3,7)(5,9), (1,10)(3,7)(4,8)(5,9), (1,2,3,4,5)(6,7,8,9,10), (1,9)(2,8)(3,7)(4,6)(5,10)>;

G:=Group( (1,10)(2,6)(3,7)(4,8)(5,9), (1,10)(2,6)(4,8)(5,9), (3,7)(4,8), (3,7)(5,9), (1,10)(3,7)(4,8)(5,9), (1,2,3,4,5)(6,7,8,9,10), (1,9)(2,8)(3,7)(4,6)(5,10) );

G=PermutationGroup([(1,10),(2,6),(3,7),(4,8),(5,9)], [(1,10),(2,6),(4,8),(5,9)], [(3,7),(4,8)], [(3,7),(5,9)], [(1,10),(3,7),(4,8),(5,9)], [(1,2,3,4,5),(6,7,8,9,10)], [(1,9),(2,8),(3,7),(4,6),(5,10)])

G:=TransitiveGroup(10,23);

On 20 points - transitive group 20T71
Generators in S20
(1 18)(2 19)(3 20)(4 16)(5 17)(6 11)(7 12)(8 13)(9 14)(10 15)
(1 13)(2 14)(4 11)(5 12)(6 16)(7 17)(8 18)(9 19)
(3 15)(4 11)(6 16)(10 20)
(3 15)(5 12)(7 17)(10 20)
(1 13)(3 15)(4 11)(5 12)(6 16)(7 17)(8 18)(10 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 12)(2 11)(3 15)(4 14)(5 13)(6 19)(7 18)(8 17)(9 16)(10 20)

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

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

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

G:=TransitiveGroup(20,71);

On 20 points - transitive group 20T73
Generators in S20
(1 18)(2 19)(3 20)(4 16)(5 17)(6 11)(7 12)(8 13)(9 14)(10 15)
(1 13)(5 12)(7 17)(8 18)
(2 14)(5 12)(7 17)(9 19)
(1 13)(2 14)(3 15)(5 12)(7 17)(8 18)(9 19)(10 20)
(4 11)(5 12)(6 16)(7 17)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 5)(2 4)(6 9)(7 8)(11 14)(12 13)(16 19)(17 18)

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

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

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

G:=TransitiveGroup(20,73);

On 20 points - transitive group 20T76
Generators in S20
(1 14)(2 15)(3 11)(4 12)(5 13)(6 20)(7 16)(8 17)(9 18)(10 19)
(1 6)(2 15)(4 12)(5 10)(7 16)(9 18)(13 19)(14 20)
(2 16)(3 11)(4 12)(5 19)(7 15)(8 17)(9 18)(10 13)
(1 20)(2 16)(3 8)(5 10)(6 14)(7 15)(11 17)(13 19)
(1 14)(3 11)(4 9)(5 10)(6 20)(8 17)(12 18)(13 19)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 19)(2 18)(3 17)(4 16)(5 20)(6 13)(7 12)(8 11)(9 15)(10 14)

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

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

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

G:=TransitiveGroup(20,76);

On 20 points - transitive group 20T81
Generators in S20
(1 15)(2 11)(3 12)(4 13)(5 14)(6 17)(7 18)(8 19)(9 20)(10 16)
(2 11)(3 12)(4 13)(5 14)(6 17)(7 18)(8 19)(9 20)
(1 15)(2 11)(6 17)(10 16)
(1 15)(3 12)(7 18)(10 16)
(1 15)(2 11)(3 12)(4 13)(6 17)(7 18)(8 19)(10 16)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 16)(2 20)(3 19)(4 18)(5 17)(6 14)(7 13)(8 12)(9 11)(10 15)

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

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

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

G:=TransitiveGroup(20,81);

On 20 points - transitive group 20T85
Generators in S20
(1 11)(2 12)(3 13)(4 14)(5 15)(6 18)(7 19)(8 20)(9 16)(10 17)
(1 6)(2 19)(4 16)(5 10)(7 12)(9 14)(11 18)(15 17)
(2 12)(3 20)(4 16)(5 15)(7 19)(8 13)(9 14)(10 17)
(1 11)(2 12)(3 8)(5 10)(6 18)(7 19)(13 20)(15 17)
(1 18)(3 20)(4 9)(5 10)(6 11)(8 13)(14 16)(15 17)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 5)(2 4)(6 10)(7 9)(11 15)(12 14)(16 19)(17 18)

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

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

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

G:=TransitiveGroup(20,85);

On 20 points - transitive group 20T87
Generators in S20
(1 20)(2 16)(3 17)(4 18)(5 19)(6 11)(7 12)(8 13)(9 14)(10 15)
(1 15)(2 6)(4 13)(5 9)(8 18)(10 20)(11 16)(14 19)
(1 20)(2 16)(3 7)(4 13)(5 19)(6 11)(8 18)(9 14)(10 15)(12 17)
(3 12)(4 18)(5 9)(7 17)(8 13)(14 19)
(1 10)(3 12)(4 8)(5 14)(7 17)(9 19)(13 18)(15 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 5)(2 4)(6 13)(7 12)(8 11)(9 15)(10 14)(16 18)(19 20)

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

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

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

G:=TransitiveGroup(20,87);

Polynomial with Galois group C2×C24⋊D5 over ℚ
actionf(x)Disc(f)
10T23x10+5x9-2x8-38x7-30x6+64x5+70x4-21x3-26x2+3x+1315·83·4014

Matrix representation of C2×C24⋊D5 in GL5(ℤ)

 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1
,
 1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 -1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 0 0 0 0 -1 0 0 0 -1 0 0 0 -1 0 0 0 -1 0 0 0 -1 0 0 0 0

G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1],[1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0],[0,0,0,0,-1,0,0,0,-1,0,0,0,-1,0,0,0,-1,0,0,0,-1,0,0,0,0] >;

C2×C24⋊D5 in GAP, Magma, Sage, TeX

C_2\times C_2^4\rtimes D_5
% in TeX

G:=Group("C2xC2^4:D5");
// GroupNames label

G:=SmallGroup(320,1636);
// by ID

G=gap.SmallGroup(320,1636);
# by ID

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

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

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