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## G = (C2×Q8).Q8order 128 = 27

### 2nd non-split extension by C2×Q8 of Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×Q8).Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22×Q8 — C23.67C23 — (C2×Q8).Q8
 Lower central C1 — C2 — C22 — C2×C4 — (C2×Q8).Q8
 Upper central C1 — C22 — C23 — C22×Q8 — (C2×Q8).Q8
 Jennings C1 — C2 — C22 — C22×Q8 — (C2×Q8).Q8

Generators and relations for (C2×Q8).Q8
G = < a,b,c,d,e | a2=b4=d4=1, c2=b2, e2=ab-1d2, dbd-1=ab=ba, ece-1=ac=ca, ad=da, eae-1=ab2, cbc-1=b-1, be=eb, cd=dc, ede-1=cd-1 >

Subgroups: 192 in 86 conjugacy classes, 32 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2×Q8, C2.C42, C4.10D4, C4.10D4, C2×C42, C2×C4⋊C4, C2×M4(2), C22×Q8, C23.67C23, C2×C4.10D4, (C2×Q8).Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C23⋊C4, C23.9D4, C42.C4, C42.3C4, (C2×Q8).Q8

Character table of (C2×Q8).Q8

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 4 8 8 8 8 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 i 1 -1 -1 1 i -1 -i -i 1 i -i 1 1 -i -i i i -1 -1 linear of order 4 ρ6 1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -i i i -i i -i -i i linear of order 4 ρ7 1 1 1 1 1 1 1 1 1 -1 -1 1 -1 1 1 -1 -1 -1 i -i i -i i -i i -i linear of order 4 ρ8 1 -1 1 -1 -1 1 -i 1 -1 1 -1 -i 1 i i -1 i -i -i i 1 -1 -1 1 i -i linear of order 4 ρ9 1 -1 1 -1 -1 1 i 1 -1 -1 1 i -1 -i -i 1 i -i -1 -1 i i -i -i 1 1 linear of order 4 ρ10 1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 i -i -i i -i i i -i linear of order 4 ρ11 1 1 1 1 1 1 1 1 1 -1 -1 1 -1 1 1 -1 -1 -1 -i i -i i -i i -i i linear of order 4 ρ12 1 -1 1 -1 -1 1 -i 1 -1 1 -1 -i 1 i i -1 i -i i -i -1 1 1 -1 -i i linear of order 4 ρ13 1 -1 1 -1 -1 1 -i 1 -1 -1 1 -i -1 i i 1 -i i -1 -1 -i -i i i 1 1 linear of order 4 ρ14 1 -1 1 -1 -1 1 i 1 -1 1 -1 i 1 -i -i -1 -i i i -i 1 -1 -1 1 -i i linear of order 4 ρ15 1 -1 1 -1 -1 1 -i 1 -1 -1 1 -i -1 i i 1 -i i 1 1 i i -i -i -1 -1 linear of order 4 ρ16 1 -1 1 -1 -1 1 i 1 -1 1 -1 i 1 -i -i -1 -i i -i i -1 1 1 -1 i -i linear of order 4 ρ17 2 2 2 2 2 2 0 -2 -2 2 2 0 -2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 2 2 0 -2 -2 -2 -2 0 2 0 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 -2 2 0 -2 2 -2 2 0 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 2 -2 -2 2 0 -2 2 2 -2 0 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 4 -4 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 4 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ23 4 4 -4 -4 0 0 -2 0 0 0 0 2 0 2 -2 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C42.3C4, Schur index 2 ρ24 4 4 -4 -4 0 0 2 0 0 0 0 -2 0 -2 2 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C42.3C4, Schur index 2 ρ25 4 -4 -4 4 0 0 2i 0 0 0 0 -2i 0 2i -2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42.C4 ρ26 4 -4 -4 4 0 0 -2i 0 0 0 0 2i 0 -2i 2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42.C4

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

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

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

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

Matrix representation of (C2×Q8).Q8 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 4 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 13
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 13 0 0 0 0 0 0 4 0 0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C2×Q8).Q8 in GAP, Magma, Sage, TeX

(C_2\times Q_8).Q_8
% in TeX

G:=Group("(C2xQ8).Q8");
// GroupNames label

G:=SmallGroup(128,126);
// by ID

G=gap.SmallGroup(128,126);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,352,1466,521,248,2804,4037]);
// Polycyclic

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

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