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## G = C2×D4.8D4order 128 = 27

### Direct product of C2 and D4.8D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×D4.8D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×Q8 — C2×2- 1+4 — C2×D4.8D4
 Lower central C1 — C2 — C2×C4 — C2×D4.8D4
 Upper central C1 — C22 — C22×C4 — C2×D4.8D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×D4.8D4

Generators and relations for C2×D4.8D4
G = < a,b,c,d,e | a2=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, dcd-1=bc, ece=b-1c, ede=b2d3 >

Subgroups: 708 in 364 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C2×C8, M4(2), M4(2), D8, SD16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C4.10D4, C4≀C2, C2×C42, C2×C22⋊C4, C4.4D4, C4.4D4, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C8⋊C22, C22×D4, C22×Q8, C22×Q8, C2×C4○D4, C2×C4○D4, 2- 1+4, 2- 1+4, C2×C4.10D4, C2×C4≀C2, D4.8D4, C2×C4.4D4, C2×C8⋊C22, C2×2- 1+4, C2×D4.8D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, D4.8D4, C2×C22≀C2, C2×D4.8D4

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E ··· 4P 8A 8B 8C 8D order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 8 8 8 8 size 1 1 1 1 2 2 4 4 4 4 8 8 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 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4.8D4 kernel C2×D4.8D4 C2×C4.10D4 C2×C4≀C2 D4.8D4 C2×C4.4D4 C2×C8⋊C22 C2×2- 1+4 C22×C4 C2×D4 C2×Q8 C4○D4 C2 # reps 1 1 2 8 1 2 1 2 2 4 4 4

Matrix representation of C2×D4.8D4 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 8 8 0 1 0 0 8 9 16 0
,
 1 15 0 0 0 0 0 16 0 0 0 0 0 0 2 2 0 13 0 0 0 0 13 0 0 0 0 4 0 0 0 0 5 1 15 15
,
 1 15 0 0 0 0 1 16 0 0 0 0 0 0 0 0 1 0 0 0 9 9 0 16 0 0 0 1 0 0 0 0 14 13 9 8
,
 1 0 0 0 0 0 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 8 8 0 1 0 0 9 8 1 0

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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,8,8,0,0,1,0,8,9,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,2,0,0,5,0,0,2,0,4,1,0,0,0,13,0,15,0,0,13,0,0,15],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,0,9,0,14,0,0,0,9,1,13,0,0,1,0,0,9,0,0,0,16,0,8],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,8,9,0,0,0,16,8,8,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×D4.8D4 in GAP, Magma, Sage, TeX

C_2\times D_4._8D_4
% in TeX

G:=Group("C2xD4.8D4");
// GroupNames label

G:=SmallGroup(128,1748);
// by ID

G=gap.SmallGroup(128,1748);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,2804,1411,718,172,2028]);
// 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=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b*c,e*c*e=b^-1*c,e*d*e=b^2*d^3>;
// generators/relations

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