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## G = (C2×C8).2D4order 128 = 27

### 2nd non-split extension by C2×C8 of D4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C8).2D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — D8⋊C22 — (C2×C8).2D4
 Lower central C1 — C2 — C22×C4 — (C2×C8).2D4
 Upper central C1 — C4 — C22×C4 — (C2×C8).2D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8).2D4

Generators and relations for (C2×C8).2D4
G = < a,b,c,d | a2=b8=d2=1, c4=b4, ab=ba, cac-1=ab4, ad=da, cbc-1=ab-1, dbd=ab5, dcd=b4c3 >

Subgroups: 376 in 175 conjugacy classes, 44 normal (8 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C22⋊C8, C22×C8, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C4.10C42, (C22×C8)⋊C2, D8⋊C22, (C2×C8).2D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C41D4, C232D4, (C2×C8).2D4

Character table of (C2×C8).2D4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 2 2 2 8 8 8 1 1 2 2 2 8 8 8 4 4 4 4 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 1 -1 1 1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 -1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ5 1 1 1 1 1 -1 1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ7 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ8 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ9 2 2 2 -2 -2 2 0 0 -2 -2 -2 2 2 0 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 2 0 0 0 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 2 -2 orthogonal lifted from D4 ρ11 2 2 -2 -2 2 0 0 -2 -2 -2 2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 2 0 0 2 -2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 -2 -2 -2 0 0 -2 -2 -2 2 2 0 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 -2 2 0 0 0 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 -2 2 orthogonal lifted from D4 ρ15 2 2 -2 2 -2 0 2 0 -2 -2 2 -2 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 2 -2 -2 0 0 0 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 -2 2 0 0 orthogonal lifted from D4 ρ17 2 2 -2 2 -2 0 0 0 2 2 -2 2 -2 0 0 0 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 2 -2 0 0 0 2 2 -2 2 -2 0 0 0 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 -2 -2 0 0 0 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 2 -2 0 0 orthogonal lifted from D4 ρ20 2 2 -2 2 -2 0 -2 0 -2 -2 2 -2 2 0 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 2 2 2 0 0 0 -2 -2 -2 -2 -2 0 0 0 2i -2i -2i 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 2 2 0 0 0 -2 -2 -2 -2 -2 0 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 -4 0 0 0 0 0 0 4i -4i 0 0 0 0 0 0 2ζ87 2ζ85 2ζ8 2ζ83 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 0 0 0 0 -4i 4i 0 0 0 0 0 0 2ζ8 2ζ83 2ζ87 2ζ85 0 0 0 0 0 0 complex faithful ρ25 4 -4 0 0 0 0 0 0 -4i 4i 0 0 0 0 0 0 2ζ85 2ζ87 2ζ83 2ζ8 0 0 0 0 0 0 complex faithful ρ26 4 -4 0 0 0 0 0 0 4i -4i 0 0 0 0 0 0 2ζ83 2ζ8 2ζ85 2ζ87 0 0 0 0 0 0 complex faithful

Smallest permutation representation of (C2×C8).2D4
On 32 points
Generators in S32
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)
(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 29 20 11 5 25 24 15)(2 16 17 26 6 12 21 30)(3 27 22 9 7 31 18 13)(4 14 19 32 8 10 23 28)
(1 16)(2 29)(3 10)(4 31)(5 12)(6 25)(7 14)(8 27)(9 19)(11 21)(13 23)(15 17)(18 28)(20 30)(22 32)(24 26)

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

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

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

Matrix representation of (C2×C8).2D4 in GL4(𝔽17) generated by

 0 13 0 0 4 0 0 0 0 0 0 4 0 0 13 0
,
 0 0 0 1 0 0 1 0 1 0 0 0 0 16 0 0
,
 15 0 0 0 0 2 0 0 0 0 8 0 0 0 0 9
,
 0 0 15 0 0 0 0 2 8 0 0 0 0 9 0 0
G:=sub<GL(4,GF(17))| [0,4,0,0,13,0,0,0,0,0,0,13,0,0,4,0],[0,0,1,0,0,0,0,16,0,1,0,0,1,0,0,0],[15,0,0,0,0,2,0,0,0,0,8,0,0,0,0,9],[0,0,8,0,0,0,0,9,15,0,0,0,0,2,0,0] >;

(C2×C8).2D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._2D_4
% in TeX

G:=Group("(C2xC8).2D4");
// GroupNames label

G:=SmallGroup(128,749);
// by ID

G=gap.SmallGroup(128,749);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,2019,1018,248,2804,718,172,2028,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=d^2=1,c^4=b^4,a*b=b*a,c*a*c^-1=a*b^4,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d=a*b^5,d*c*d=b^4*c^3>;
// generators/relations

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