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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×D4.4D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — 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

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 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E ··· 8J 8K 8L 8M 8N order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 8 8 8 8 8 ··· 8 8 8 8 8 size 1 1 1 1 2 2 4 4 8 8 8 8 2 2 2 2 4 4 2 2 2 2 4 ··· 4 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 C4○D4 C4○D4 D4.4D4 kernel C2×D4.4D4 C2×C4.D4 C2×C8.C4 D4.4D4 C2×C8○D4 C22×D8 C2×C8⋊C22 C2×C8 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C2 # reps 1 2 1 8 1 1 2 4 1 1 2 2 2 4

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

 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 1 0 0 0 0 16 0 0 0 0 0 1 1 1 2 0 0 0 16 16 16
,
 1 15 0 0 0 0 0 16 0 0 0 0 0 0 1 1 1 2 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 16 16 16
,
 13 8 0 0 0 0 0 4 0 0 0 0 0 0 14 14 0 0 0 0 3 14 0 0 0 0 3 3 0 6 0 0 14 0 14 11
,
 16 0 0 0 0 0 16 1 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 0 16 16

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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