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## G = (C2×D4).D4order 128 = 27

### 4th non-split extension by C2×D4 of D4 acting faithfully

p-group, metabelian, nilpotent (class 5), monomial

Aliases: C4⋊Q81C4, C8⋊C42C4, (C2×D4).4D4, (C2×Q8).4D4, C42.4(C2×C4), C8.2D4.1C2, C423C4.1C2, C2.7(C42⋊C4), C42.C4.1C2, C4.4D4.4C22, C22.17(C23⋊C4), (C2×C4).33(C22⋊C4), SmallGroup(128,139)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — (C2×D4).D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×D4 — C4.4D4 — C8.2D4 — (C2×D4).D4
 Lower central C1 — C2 — C22 — C2×C4 — C42 — (C2×D4).D4
 Upper central C1 — C2 — C22 — C2×C4 — C4.4D4 — (C2×D4).D4
 Jennings C1 — C2 — C2 — C22 — C2×C4 — C4.4D4 — (C2×D4).D4

Generators and relations for (C2×D4).D4
G = < a,b,c,d,e | a2=b4=c2=1, d4=b2, e2=c, ebe-1=ab=ba, ac=ca, dad-1=eae-1=ab2, cbc=b-1, dbd-1=ab-1, dcd-1=ab-1c, ce=ec, ede-1=cd3 >

Character table of (C2×D4).D4

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D size 1 1 2 8 4 8 8 16 16 16 8 8 16 16 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 1 -1 1 -1 1 -i i -1 1 1 i -i linear of order 4 ρ6 1 1 1 -1 1 -1 1 -i i 1 -1 -1 -i i linear of order 4 ρ7 1 1 1 -1 1 -1 1 i -i 1 -1 -1 i -i linear of order 4 ρ8 1 1 1 -1 1 -1 1 i -i -1 1 1 -i i linear of order 4 ρ9 2 2 2 2 2 -2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 2 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 4 0 -4 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ12 4 4 -4 0 0 0 0 0 0 0 2 -2 0 0 orthogonal lifted from C42⋊C4 ρ13 4 4 -4 0 0 0 0 0 0 0 -2 2 0 0 orthogonal lifted from C42⋊C4 ρ14 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of (C2×D4).D4 in GL8(𝔽17)

 1 15 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 15 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 15 0 0 0 0 0 0 0 16 0 0 0 16 0 16 0 1 0 1 0 16 0 16 0 1 1 0
,
 0 0 16 2 0 0 0 0 0 0 0 1 0 0 0 0 1 15 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 16 0 1 0 0 2 0 0 0 0 0 0 16 1 1 16 0 1 16 1 0 16 1 16 0 1 16 0 0 16
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 2 0 0 0 0 0 0 0 1 0 0 0 0 16 0 1 16 0 1 0 0 16 0 1 16 1 0
,
 7 16 7 16 11 16 1 10 7 13 7 13 11 3 14 14 10 7 1 10 0 16 16 7 10 4 1 13 0 3 3 3 16 1 16 10 8 16 7 1 16 14 16 14 8 13 4 4 0 3 1 3 10 3 14 3 1 6 8 0 10 16 10 16
,
 1 0 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 16 2 0 0 0 0 0 0 16 1 0 0 0 0 1 0 1 0 16 0 0 15 1 0 1 0 16 0 16 16 0 0 16 1 0 0 0 1 0 0 16 1 1 16 0 1

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,15,16,0,0,0,0,16,16,0,0,1,0,0,0,0,0,0,0,15,16,0,0,16,16,0,0,0,0,1,0,0,0,0,0,0,0,15,16,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,1,0,16,0,1,1,0,0,15,16,0,0,16,16,16,0,0,0,16,0,0,0,2,1,0,0,0,0,1,1,0,0,0,0,1,0,16,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,2,1,16,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,16,16,0,0,0,16,0,0,0,0,0,0,0,0,16,0,1,1,0,0,0,0,2,1,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[7,7,10,10,16,16,0,1,16,13,7,4,1,14,3,6,7,7,1,1,16,16,1,8,16,13,10,13,10,14,3,0,11,11,0,0,8,8,10,10,16,3,16,3,16,13,3,16,1,14,16,3,7,4,14,10,10,14,7,3,1,4,3,16],[1,1,0,0,1,1,0,0,0,16,0,0,0,0,0,0,0,0,16,16,1,1,16,16,0,0,2,1,0,0,1,1,0,0,0,0,16,16,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,15,16,1,1] >;

(C2×D4).D4 in GAP, Magma, Sage, TeX

(C_2\times D_4).D_4
% in TeX

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

G:=SmallGroup(128,139);
// by ID

G=gap.SmallGroup(128,139);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,456,422,387,184,1690,745,1684,1411,375,172,4037]);
// Polycyclic

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

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