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G = C25⋊C4order 128 = 27

1st semidirect product of C25 and C4 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C251C4, C24.64D4, C25.7C22, C24.164C23, (C23×C4)⋊6C4, C243C41C2, (C2×D4).259D4, (C22×D4)⋊15C4, (D4×C23).2C2, C222(C23⋊C4), C233(C22⋊C4), C24.112(C2×C4), C22.7C22≀C2, C23.543(C2×D4), C2.6(C243C4), C23.180(C22×C4), (C22×D4).448C22, (C2×C23⋊C4)⋊1C2, (C2×C4)⋊3(C22⋊C4), C2.24(C2×C23⋊C4), (C2×C22⋊C4)⋊1C22, (C22×C4).72(C2×C4), C22.28(C2×C22⋊C4), SmallGroup(128,513)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C25⋊C4
C1C2C22C23C24C25D4×C23 — C25⋊C4
C1C2C23 — C25⋊C4
C1C22C25 — C25⋊C4
C1C2C24 — C25⋊C4

Generators and relations for C25⋊C4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, faf-1=ace, bc=cb, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 1108 in 440 conjugacy classes, 72 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×16], C4 [×8], C22, C22 [×10], C22 [×80], C2×C4 [×4], C2×C4 [×24], D4 [×32], C23, C23 [×14], C23 [×88], C22⋊C4 [×16], C22×C4 [×2], C22×C4 [×10], C2×D4 [×8], C2×D4 [×52], C24 [×3], C24 [×6], C24 [×18], C23⋊C4 [×8], C2×C22⋊C4 [×4], C2×C22⋊C4 [×4], C23×C4, C22×D4 [×4], C22×D4 [×12], C25 [×2], C243C4 [×2], C2×C23⋊C4 [×4], D4×C23, C25⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, C22⋊C4 [×12], C22×C4, C2×D4 [×6], C23⋊C4 [×4], C2×C22⋊C4 [×3], C22≀C2 [×4], C243C4, C2×C23⋊C4 [×2], C25⋊C4

Permutation representations of C25⋊C4
On 16 points - transitive group 16T240
Generators in S16
(1 7)(3 12)(4 14)(5 13)(6 9)(10 15)
(1 15)(2 11)(3 13)(4 9)(5 12)(6 14)(7 10)(8 16)
(1 10)(2 8)(3 12)(4 6)(5 13)(7 15)(9 14)(11 16)
(1 7)(2 8)(3 5)(4 6)(9 14)(10 15)(11 16)(12 13)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

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

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

G:=TransitiveGroup(16,240);

On 16 points - transitive group 16T275
Generators in S16
(1 11)(2 12)(3 16)(4 15)(5 13)(6 14)(7 9)(8 10)
(1 5)(4 7)(9 15)(11 13)
(1 7)(4 5)(9 11)(13 15)
(1 5)(2 6)(3 8)(4 7)(9 15)(10 16)(11 13)(12 14)
(1 7)(2 8)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

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

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

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

G:=TransitiveGroup(16,275);

32 conjugacy classes

class 1 2A2B2C2D···2M2N···2S4A4B4C4D4E···4L
order12222···22···244444···4
size11112···24···444448···8

32 irreducible representations

dim1111111224
type+++++++
imageC1C2C2C2C4C4C4D4D4C23⋊C4
kernelC25⋊C4C243C4C2×C23⋊C4D4×C23C23×C4C22×D4C25C2×D4C24C22
# reps1241242844

Matrix representation of C25⋊C4 in GL6(𝔽5)

100000
010000
004000
001100
000040
000011
,
010000
100000
001000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000040
000004
,
100000
010000
004000
000400
000040
000004
,
300000
020000
000010
000001
004300
000100

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C25⋊C4 in GAP, Magma, Sage, TeX

C_2^5\rtimes C_4
% in TeX

G:=Group("C2^5:C4");
// GroupNames label

G:=SmallGroup(128,513);
// by ID

G=gap.SmallGroup(128,513);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,2028]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a*c*e,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f^-1=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations

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