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G = C25.C22order 128 = 27

9th non-split extension by C25 of C22 acting faithfully

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

Aliases: C24.25D4, C25.9C22, C24.171C23, C247(C2×C4), (C22×C4)⋊1D4, C22≀C24C4, C2.3C2≀C22, (C2×D4).75D4, C243C42C2, C22.38(C4×D4), C232(C22⋊C4), C23.564(C2×D4), C23.9D46C2, C22.11C242C2, C23.67(C22×C4), C22.103C22≀C2, C23.118(C4○D4), C22.45(C4⋊D4), (C22×D4).22C22, C2.28(C23.23D4), C22.52(C22.D4), C22⋊C45(C2×C4), (C2×C23⋊C4)⋊4C2, (C2×C4)⋊2(C22⋊C4), (C2×D4).78(C2×C4), (C2×C22≀C2).1C2, C22.44(C2×C22⋊C4), (C2×C22⋊C4).10C22, SmallGroup(128,621)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C25.C22
C1C2C22C23C24C22×D4C22.11C24 — C25.C22
C1C2C23 — C25.C22
C1C22C24 — C25.C22
C1C2C24 — C25.C22

Generators and relations for C25.C22
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=g2=1, f2=d, faf-1=ab=ba, ac=ca, ad=da, ae=ea, gag=ace, bc=cb, bd=db, gbg=be=eb, bf=fb, cd=dc, fcf-1=ce=ec, cg=gc, gdg=de=ed, df=fd, ef=fe, eg=ge, gfg=cef >

Subgroups: 772 in 289 conjugacy classes, 56 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×10], C22 [×3], C22 [×4], C22 [×53], C2×C4 [×2], C2×C4 [×22], D4 [×16], C23 [×3], C23 [×6], C23 [×50], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×23], C4⋊C4 [×2], C22×C4, C22×C4 [×4], C22×C4 [×5], C2×D4 [×4], C2×D4 [×12], C24 [×2], C24 [×2], C24 [×10], C23⋊C4 [×2], C2×C22⋊C4, C2×C22⋊C4 [×4], C2×C22⋊C4 [×6], C42⋊C2, C4×D4 [×4], C22≀C2 [×4], C22≀C2 [×2], C22×D4, C22×D4, C25, C23.9D4 [×2], C243C4 [×2], C2×C23⋊C4, C22.11C24, C2×C22≀C2, C25.C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C2≀C22 [×2], C25.C22

Permutation representations of C25.C22
On 16 points - transitive group 16T208
Generators in S16
(6 12)(8 10)
(5 11)(6 12)(7 9)(8 10)
(2 14)(4 16)(5 11)(7 9)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 13)(2 14)(3 15)(4 16)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 8)(2 11)(3 12)(4 7)(5 14)(6 15)(9 16)(10 13)

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

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

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

G:=TransitiveGroup(16,208);

On 16 points - transitive group 16T210
Generators in S16
(2 3)(5 7)(9 13)(10 12)(11 15)(14 16)
(1 4)(2 3)(5 7)(6 8)(9 15)(10 16)(11 13)(12 14)
(2 5)(3 7)(9 11)(13 15)
(9 11)(10 12)(13 15)(14 16)
(1 6)(2 5)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 12)(2 11)(3 15)(4 16)(5 9)(6 10)(7 13)(8 14)

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

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

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

G:=TransitiveGroup(16,210);

On 16 points - transitive group 16T217
Generators in S16
(5 7)(6 11)(8 9)(10 12)
(5 12)(6 9)(7 10)(8 11)
(1 3)(2 16)(4 14)(5 10)(6 8)(7 12)(9 11)(13 15)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 12)(2 11)(3 7)(4 6)(5 13)(8 14)(9 16)(10 15)

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

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

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

G:=TransitiveGroup(16,217);

On 16 points - transitive group 16T247
Generators in S16
(1 8)(2 14)(3 6)(4 16)(5 12)(7 10)(9 15)(11 13)
(1 11)(2 12)(3 9)(4 10)(5 14)(6 15)(7 16)(8 13)
(2 7)(4 5)(10 14)(12 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 6)(2 7)(3 8)(4 5)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 9)(2 14)(3 15)(4 12)(5 16)(6 13)(7 10)(8 11)

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

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

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

G:=TransitiveGroup(16,247);

32 conjugacy classes

class 1 2A2B2C2D···2I2J···2O4A···4J4K···4P
order12222···22···24···44···4
size11112···24···44···48···8

32 irreducible representations

dim111111122224
type++++++++++
imageC1C2C2C2C2C2C4D4D4D4C4○D4C2≀C22
kernelC25.C22C23.9D4C243C4C2×C23⋊C4C22.11C24C2×C22≀C2C22≀C2C22×C4C2×D4C24C23C2
# reps122111852144

Matrix representation of C25.C22 in GL6(𝔽5)

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

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

C25.C22 in GAP, Magma, Sage, TeX

C_2^5.C_2^2
% in TeX

G:=Group("C2^5.C2^2");
// GroupNames label

G:=SmallGroup(128,621);
// by ID

G=gap.SmallGroup(128,621);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,521,2804,1411,1027]);
// Polycyclic

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

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