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G = C2×D4⋊4D4  order 128 = 27

Direct product of C2 and D4⋊4D4

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

Aliases: C2×D4⋊4D4, C42⋊6C23, C24.42D4, M4(2)⋊1C23, 2+ (1+4)⋊5C22, C4○D4⋊5D4, D4⋊9(C2×D4), Q8⋊9(C2×D4), (C2×D4)⋊51D4, (C2×Q8)⋊38D4, C4≀C2⋊7C22, (C2×D4)⋊3C23, (C2×C4).6C24, C4.43C22≀C2, C8⋊C22⋊4C22, C4○D4.1C23, C23.19(C2×D4), C4.51(C22×D4), C4⋊1D4⋊30C22, (C2×C42)⋊36C22, C4.D4⋊6C22, (C22×D4)⋊19C22, (C2×M4(2))⋊7C22, (C2×2+ (1+4))⋊3C2, C22.30(C22×D4), C22.118C22≀C2, (C22×C4).965C23, (C2×C4≀C2)⋊4C2, (C2×C4⋊1D4)⋊14C2, (C2×C8⋊C22)⋊11C2, (C2×C4.D4)⋊7C2, C2.51(C2×C22≀C2), (C2×C4).1095(C2×D4), (C2×C4○D4).104C22, SmallGroup(128,1746)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C2×D4⋊4D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C2×2+ (1+4) — C2×D4⋊4D4
Lower central
C1 — C2 — C2×C4 — C2×D4⋊4D4
Upper central
C1 — C22 — C22×C4 — C2×D4⋊4D4
Jennings
C1 — C2 — C2 — C2×C4 — C2×D4⋊4D4

Subgroups: 980 in 422 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×4], C4 [×8], C22 [×3], C22 [×42], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×24], D4 [×4], D4 [×56], Q8 [×4], Q8 [×2], C23, C23 [×4], C23 [×30], C42 [×2], C42, C2×C8 [×2], M4(2) [×4], M4(2) [×2], D8 [×8], SD16 [×8], C22×C4, C22×C4 [×5], C2×D4 [×8], C2×D4 [×63], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×20], C24 [×2], C24 [×3], C4.D4 [×4], C4≀C2 [×8], C2×C42, C4⋊1D4 [×4], C4⋊1D4 [×2], C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C8⋊C22 [×4], C22×D4 [×2], C22×D4 [×6], C2×C4○D4 [×2], C2×C4○D4 [×2], 2+ (1+4) [×4], 2+ (1+4) [×6], C2×C4.D4, C2×C4≀C2 [×2], D4⋊4D4 [×8], C2×C4⋊1D4, C2×C8⋊C22 [×2], C2×2+ (1+4), C2×D4⋊4D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], D4⋊4D4 [×2], C2×C22≀C2, C2×D4⋊4D4

Generators and relations
 G = < a,b,c,d,e | a2=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b-1c, ece=bc, ede=d-1 >

Permutation representations
►On 16 points - transitive group 16T265
Generators in S16
(1 5)(2 6)(3 7)(4 8)(9 15)(10 16)(11 13)(12 14)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 16)(2 15)(3 14)(4 13)(5 10)(6 9)(7 12)(8 11)

(9 12 11 10)(13 16 15 14)

(1 7)(2 6)(3 5)(4 8)(9 14)(10 13)(11 16)(12 15)
 
G:=sub<Sym(16)| (1,5)(2,6)(3,7)(4,8)(9,15)(10,16)(11,13)(12,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,10)(6,9)(7,12)(8,11), (9,12,11,10)(13,16,15,14), (1,7)(2,6)(3,5)(4,8)(9,14)(10,13)(11,16)(12,15)>;
 
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,15)(10,16)(11,13)(12,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,10)(6,9)(7,12)(8,11), (9,12,11,10)(13,16,15,14), (1,7)(2,6)(3,5)(4,8)(9,14)(10,13)(11,16)(12,15) );
 
G=PermutationGroup([(1,5),(2,6),(3,7),(4,8),(9,15),(10,16),(11,13),(12,14)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,16),(2,15),(3,14),(4,13),(5,10),(6,9),(7,12),(8,11)], [(9,12,11,10),(13,16,15,14)], [(1,7),(2,6),(3,5),(4,8),(9,14),(10,13),(11,16),(12,15)])
 
G:=TransitiveGroup(16,265);
 

Matrix representation ►G ⊆ GL6(ℤ)

-100000
0-10000
001000
000100
000010
000001
,
100000
010000
000100
00-1000
00000-1
000010
,
100000
010000
000001
0000-10
000-100
001000
,
0-10000
100000
000100
00-1000
0000-10
00000-1
,
-100000
010000
00-1000
000100
00000-1
0000-10

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],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0] >;
 

32 conjugacy classes

class 1 2A2B2C2D2E2F···2M2N2O4A4B4C4D4E···4L8A8B8C8D
order1222222···22244444···48888
size1111224···48822224···48888

32 irreducible representations

dim111111122224
type++++++++++++
imageC1C2C2C2C2C2C2D4D4D4D4D4⋊4D4
kernelC2×D4⋊4D4C2×C4.D4C2×C4≀C2D4⋊4D4C2×C4⋊1D4C2×C8⋊C22C2×2+ (1+4)C2×D4C2×Q8C4○D4C24C2
# reps112812142424

In GAP, Magma, Sage, TeX 

C_2\times D_4\rtimes_4D_4
 
% in TeX
 
G:=Group("C2xD4:4D4");
 
// GroupNames label
 
G:=SmallGroup(128,1746);
 
// by ID
 
G=gap.SmallGroup(128,1746);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2804,1411,718,172,2028]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
 
// generators/relations
 

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