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G = C2×D4.4D4order 128 = 27

Direct product of C2 and D4.4D4

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

Aliases: C2×D4.4D4, M4(2).8C23, C4○D4.32D4, D4.12(C2×D4), (C2×C8).351D4, C8.112(C2×D4), Q8.12(C2×D4), (C2×D4).224D4, C8○D412C22, (C2×D8)⋊48C22, (C22×D8)⋊17C2, (C2×C4).16C24, (C2×Q8).179D4, C8⋊C2210C22, (C2×C8).261C23, C4○D4.28C23, (C2×D4).71C23, C4.163(C22×D4), C4.114(C4⋊D4), C8.C416C22, C4.D413C22, C23.317(C4○D4), C22.90(C4⋊D4), (C22×C4).991C23, (C22×C8).265C22, (C22×D4).353C22, (C2×M4(2)).59C22, (C2×C8○D4)⋊8C2, (C2×C8⋊C22)⋊20C2, C2.87(C2×C4⋊D4), (C2×C8.C4)⋊26C2, (C2×C4).1428(C2×D4), (C2×C4.D4)⋊11C2, C22.19(C2×C4○D4), (C2×C4).290(C4○D4), (C2×C4○D4).305C22, SmallGroup(128,1797)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×D4.4D4
C1C2C4C2×C4C22×C4C2×C4○D4C2×C8○D4 — C2×D4.4D4
C1C2C2×C4 — C2×D4.4D4
C1C22C22×C4 — C2×D4.4D4
C1C2C2C2×C4 — C2×D4.4D4

Generators and relations for C2×D4.4D4
 G = < a,b,c,d,e | a2=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=b2d3 >

Subgroups: 540 in 244 conjugacy classes, 100 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×2], C22 [×3], C22 [×22], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×5], D4 [×2], D4 [×17], Q8 [×2], Q8, C23, C23 [×13], C2×C8 [×2], C2×C8 [×4], C2×C8 [×7], M4(2) [×6], M4(2) [×7], D8 [×16], SD16 [×8], C22×C4, C22×C4, C2×D4, C2×D4 [×4], C2×D4 [×11], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24 [×2], C4.D4 [×8], C8.C4 [×4], C22×C8, C22×C8, C2×M4(2), C2×M4(2) [×2], C2×M4(2), C8○D4 [×4], C8○D4 [×2], C2×D8 [×4], C2×D8 [×6], C2×SD16 [×2], C8⋊C22 [×8], C8⋊C22 [×4], C22×D4 [×2], C2×C4○D4, C2×C4.D4 [×2], C2×C8.C4, D4.4D4 [×8], C2×C8○D4, C22×D8, C2×C8⋊C22 [×2], C2×D4.4D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, D4.4D4 [×2], C2×C4⋊D4, C2×D4.4D4

Smallest permutation representation of C2×D4.4D4
On 32 points
Generators in S32
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 13 5 9)(2 14 6 10)(3 15 7 11)(4 16 8 12)(17 27 21 31)(18 28 22 32)(19 29 23 25)(20 30 24 26)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(1 7)(2 6)(3 5)(9 11)(12 16)(13 15)(17 25)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)

G:=sub<Sym(32)| (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,13,5,9)(2,14,6,10)(3,15,7,11)(4,16,8,12)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,7)(2,6)(3,5)(9,11)(12,16)(13,15)(17,25)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)>;

G:=Group( (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,13,5,9)(2,14,6,10)(3,15,7,11)(4,16,8,12)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,7)(2,6)(3,5)(9,11)(12,16)(13,15)(17,25)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26) );

G=PermutationGroup([(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(1,13,5,9),(2,14,6,10),(3,15,7,11),(4,16,8,12),(17,27,21,31),(18,28,22,32),(19,29,23,25),(20,30,24,26)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,29),(10,30),(11,31),(12,32),(13,25),(14,26),(15,27),(16,28)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,7),(2,6),(3,5),(9,11),(12,16),(13,15),(17,25),(18,32),(19,31),(20,30),(21,29),(22,28),(23,27),(24,26)])

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F8A8B8C8D8E···8J8K8L8M8N
order12222222222244444488888···88888
size11112244888822224422224···48888

32 irreducible representations

dim11111112222224
type++++++++++++
imageC1C2C2C2C2C2C2D4D4D4D4C4○D4C4○D4D4.4D4
kernelC2×D4.4D4C2×C4.D4C2×C8.C4D4.4D4C2×C8○D4C22×D8C2×C8⋊C22C2×C8C2×D4C2×Q8C4○D4C2×C4C23C2
# reps12181124112224

Matrix representation of C2×D4.4D4 in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
0160000
000100
0016000
001112
000161616
,
1150000
0160000
001112
000010
000100
000161616
,
1380000
040000
00141400
0031400
003306
001401411
,
1600000
1610000
0001600
0016000
000010
00001616

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1,0,0,0,1,0,1,16,0,0,0,0,1,16,0,0,0,0,2,16],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,16,0,0,1,1,0,16,0,0,2,0,0,16],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,14,3,3,14,0,0,14,14,3,0,0,0,0,0,0,14,0,0,0,0,6,11],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;

C2×D4.4D4 in GAP, Magma, Sage, TeX

C_2\times D_4._4D_4
% in TeX

G:=Group("C2xD4.4D4");
// GroupNames label

G:=SmallGroup(128,1797);
// by ID

G=gap.SmallGroup(128,1797);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,2804,172,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=e^2=1,d^4=b^2,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,c*d=d*c,e*c*e=b*c,e*d*e=b^2*d^3>;
// generators/relations

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