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G = (C2×Q8).D4order 128 = 27

6th non-split extension by C2×Q8 of D4 acting faithfully

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

Aliases: (C4×C8).4C4, C4⋊Q8.4C4, (C2×Q8).6D4, C4⋊Q8.3C22, C42.18(C2×C4), C4⋊Q16.2C2, C42.3C4.1C2, C2.11(C42⋊C4), C22.21(C23⋊C4), (C2×C4).37(C22⋊C4), SmallGroup(128,143)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — (C2×Q8).D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8 — C4⋊Q8 — C4⋊Q16 — (C2×Q8).D4
 Lower central C1 — C2 — C22 — C2×C4 — C42 — (C2×Q8).D4
 Upper central C1 — C2 — C22 — C2×C4 — C4⋊Q8 — (C2×Q8).D4
 Jennings C1 — C2 — C2 — C22 — C2×C4 — C4⋊Q8 — (C2×Q8).D4

Generators and relations for (C2×Q8).D4
G = < a,b,c,d,e | a2=b4=1, c2=d4=b2, e2=ca=ac, dbd-1=ab=ba, dad-1=eae-1=ab2, cbc-1=b-1, ebe-1=ab-1, dcd-1=bc, ece-1=b2c, ede-1=acd3 >

Character table of (C2×Q8).D4

 class 1 2A 2B 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 4 4 4 8 8 16 4 4 4 4 16 16 16 16 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 i i -i -i linear of order 4 ρ6 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -i i i -i linear of order 4 ρ7 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 -i -i i i linear of order 4 ρ8 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 i -i -i i linear of order 4 ρ9 2 2 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 -4 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 orthogonal lifted from C42⋊C4 ρ12 4 4 -4 0 0 0 0 0 0 -2 -2 2 2 0 0 0 0 orthogonal lifted from C42⋊C4 ρ13 4 4 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ14 4 -4 0 2 0 -2 0 0 0 -2√2 2√2 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ15 4 -4 0 -2 0 2 0 0 0 0 0 2√2 -2√2 0 0 0 0 symplectic faithful, Schur index 2 ρ16 4 -4 0 2 0 -2 0 0 0 2√2 -2√2 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ17 4 -4 0 -2 0 2 0 0 0 0 0 -2√2 2√2 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of (C2×Q8).D4
On 32 points
Generators in S32
(1 5)(3 7)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)
(1 27 5 31)(2 32 6 28)(3 25 7 29)(4 30 8 26)(9 18 13 22)(10 19 14 23)(11 24 15 20)(12 17 16 21)
(1 3 5 7)(2 30 6 26)(4 28 8 32)(9 24 13 20)(10 12 14 16)(11 22 15 18)(17 19 21 23)(25 27 29 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)
(1 13 7 20 5 9 3 24)(2 14 30 12 6 10 26 16)(4 17 28 23 8 21 32 19)(11 27 22 25 15 31 18 29)

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

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

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

Matrix representation of (C2×Q8).D4 in GL4(𝔽7) generated by

 0 6 3 2 6 0 4 2 0 0 6 0 0 0 0 1
,
 2 4 1 2 3 2 3 4 1 1 5 5 5 2 1 5
,
 2 3 0 3 0 1 5 5 4 4 4 6 3 4 2 0
,
 6 0 6 2 2 3 0 1 0 0 0 4 1 1 1 5
,
 1 0 5 6 4 0 2 5 4 3 5 3 3 3 5 1
G:=sub<GL(4,GF(7))| [0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1],[2,3,1,5,4,2,1,2,1,3,5,1,2,4,5,5],[2,0,4,3,3,1,4,4,0,5,4,2,3,5,6,0],[6,2,0,1,0,3,0,1,6,0,0,1,2,1,4,5],[1,4,4,3,0,0,3,3,5,2,5,5,6,5,3,1] >;

(C2×Q8).D4 in GAP, Magma, Sage, TeX

(C_2\times Q_8).D_4
% in TeX

G:=Group("(C2xQ8).D4");
// GroupNames label

G:=SmallGroup(128,143);
// by ID

G=gap.SmallGroup(128,143);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,456,422,1059,520,794,745,248,1684,1411,375,172,4037]);
// Polycyclic

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

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