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G = C2×C2≀C22order 128 = 27

Direct product of C2 and C2≀C22

direct product, p-group, metabelian, nilpotent (class 3), monomial, rational

Aliases: C2×C2≀C22, C244D4, C246C23, C253C22, C23.5C24, 2+ 1+47C22, C23⋊(C2×D4), (C2×D4)⋊24D4, (C22×C4)⋊5D4, C23⋊C45C22, C22⋊C41C23, (C2×D4).39C23, C22≀C223C22, C22.25C22≀C2, (C2×2+ 1+4)⋊5C2, C22.39(C22×D4), (C22×D4).332C22, (C2×C4)⋊(C2×D4), (C2×C23⋊C4)⋊16C2, (C2×C22≀C2)⋊19C2, C2.60(C2×C22≀C2), (C2×C22⋊C4)⋊36C22, SmallGroup(128,1755)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C2×C2≀C22
C1C2C22C23C24C22×D4C2×2+ 1+4 — C2×C2≀C22
C1C2C23 — C2×C2≀C22
C1C22C24 — C2×C2≀C22
C1C2C23 — C2×C2≀C22

Generators and relations for C2×C2≀C22
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf=bcd, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1236 in 509 conjugacy classes, 108 normal (8 characteristic)
C1, C2, C2 [×2], C2 [×16], C4 [×12], C22, C22 [×6], C22 [×71], C2×C4 [×6], C2×C4 [×30], D4 [×60], Q8 [×4], C23, C23 [×12], C23 [×64], C22⋊C4 [×6], C22⋊C4 [×15], C22×C4 [×3], C22×C4 [×6], C2×D4 [×12], C2×D4 [×63], C2×Q8, C4○D4 [×24], C24, C24 [×5], C24 [×11], C23⋊C4 [×12], C2×C22⋊C4 [×3], C2×C22⋊C4 [×3], C22≀C2 [×12], C22≀C2 [×6], C22×D4 [×3], C22×D4 [×6], C2×C4○D4 [×3], 2+ 1+4 [×4], 2+ 1+4 [×6], C25, C2×C23⋊C4 [×3], C2≀C22 [×8], C2×C22≀C2 [×3], C2×2+ 1+4, C2×C2≀C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2≀C22 [×2], C2×C22≀C2, C2×C2≀C22

Permutation representations of C2×C2≀C22
On 16 points - transitive group 16T245
Generators in S16
(1 8)(2 7)(3 6)(4 5)(9 14)(10 15)(11 16)(12 13)
(1 12)(2 14)(3 16)(4 10)(5 15)(6 11)(7 9)(8 13)
(1 5)(2 7)(3 6)(4 8)(9 14)(10 13)(11 16)(12 15)
(1 4)(2 3)(5 8)(6 7)(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 5)(2 6)(3 7)(4 8)(10 12)(13 15)

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

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

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

G:=TransitiveGroup(16,245);

On 16 points - transitive group 16T246
Generators in S16
(1 5)(2 6)(3 7)(4 8)(9 16)(10 13)(11 14)(12 15)
(1 9)(2 12)(3 13)(4 14)(5 16)(6 15)(7 10)(8 11)
(2 7)(3 6)(10 12)(13 15)
(1 8)(2 7)(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)
(9 11)(14 16)

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

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

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

G:=TransitiveGroup(16,246);

On 16 points - transitive group 16T271
Generators in S16
(1 4)(2 3)(5 8)(6 7)(9 15)(10 16)(11 13)(12 14)
(1 16)(2 14)(3 12)(4 10)(5 13)(6 15)(7 9)(8 11)
(1 5)(2 6)(3 7)(4 8)(9 12)(10 11)(13 16)(14 15)
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)
(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 4)(2 3)(5 8)(6 7)(9 16)(10 15)(11 14)(12 13)

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

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

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

G:=TransitiveGroup(16,271);

32 conjugacy classes

class 1 2A2B2C2D···2I2J···2S4A···4F4G···4L
order12222···22···24···44···4
size11112···24···44···48···8

32 irreducible representations

dim111112224
type+++++++++
imageC1C2C2C2C2D4D4D4C2≀C22
kernelC2×C2≀C22C2×C23⋊C4C2≀C22C2×C22≀C2C2×2+ 1+4C22×C4C2×D4C24C2
# reps138313634

Matrix representation of C2×C2≀C22 in GL6(ℤ)

-100000
0-10000
00-1000
000-100
0000-10
00000-1
,
010000
100000
000010
001112
001000
00-10-1-1
,
-100000
0-10000
000100
001000
001112
00-1-10-1
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
-100000
010000
001112
0000-10
00-1000
00000-1
,
-100000
010000
00-1000
000-100
001112
00000-1

G:=sub<GL(6,Integers())| [-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,-1,0,0,0,0,0,0,-1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,-1,0,0,0,1,0,0,0,0,1,1,0,-1,0,0,0,2,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,-1,0,0,1,0,1,-1,0,0,0,0,1,0,0,0,0,0,2,-1],[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,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,-1,0,0,0,1,0,0,0,0,0,1,-1,0,0,0,0,2,0,0,-1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,2,-1] >;

C2×C2≀C22 in GAP, Magma, Sage, TeX

C_2\times C_2\wr C_2^2
% in TeX

G:=Group("C2xC2wrC2^2");
// GroupNames label

G:=SmallGroup(128,1755);
// by ID

G=gap.SmallGroup(128,1755);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,718,2028]);
// Polycyclic

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

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