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## G = C2×C22⋊Q8order 64 = 26

### Direct product of C2 and C22⋊Q8

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C22⋊Q8
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C22⋊Q8
 Lower central C1 — C22 — C2×C22⋊Q8
 Upper central C1 — C23 — C2×C22⋊Q8
 Jennings C1 — C22 — C2×C22⋊Q8

Generators and relations for C2×C22⋊Q8
G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 225 in 161 conjugacy classes, 97 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, C2×C22⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8

Character table of C2×C22⋊Q8

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P size 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 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 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 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 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 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 -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 -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 -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 -1 1 linear of order 2 ρ9 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 1 -1 linear of order 2 ρ10 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 1 1 linear of order 2 ρ11 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 1 1 linear of order 2 ρ12 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 1 -1 linear of order 2 ρ13 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 -1 1 linear of order 2 ρ14 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 -1 -1 linear of order 2 ρ15 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 -1 -1 linear of order 2 ρ16 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 -1 1 linear of order 2 ρ17 2 -2 -2 2 2 -2 2 -2 0 0 0 0 0 2 0 0 -2 -2 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 -2 2 2 -2 2 -2 0 0 0 0 0 -2 0 0 2 2 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 -2 2 -2 2 -2 2 0 0 0 0 0 2 0 0 2 -2 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 -2 2 -2 2 -2 2 0 0 0 0 0 -2 0 0 -2 2 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 -2 2 -2 -2 -2 2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ22 2 -2 2 -2 2 2 -2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ23 2 -2 2 -2 -2 -2 2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ24 2 -2 2 -2 2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ25 2 2 -2 -2 2 -2 -2 2 0 0 0 0 -2i 0 2i 2i 0 0 -2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 2 -2 -2 -2 2 2 -2 0 0 0 0 -2i 0 -2i 2i 0 0 2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 2 -2 -2 -2 2 2 -2 0 0 0 0 2i 0 2i -2i 0 0 -2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 2 -2 -2 2 -2 -2 2 0 0 0 0 2i 0 -2i -2i 0 0 2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4

Smallest permutation representation of C2×C22⋊Q8
On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 17)(14 18)(15 19)(16 20)(25 30)(26 31)(27 32)(28 29)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)(21 23)(22 24)
(1 21)(2 22)(3 23)(4 24)(5 10)(6 11)(7 12)(8 9)(13 27)(14 28)(15 25)(16 26)(17 32)(18 29)(19 30)(20 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 20 3 18)(2 19 4 17)(5 25 7 27)(6 28 8 26)(9 16 11 14)(10 15 12 13)(21 31 23 29)(22 30 24 32)

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

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

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

Matrix representation of C2×C22⋊Q8 in GL6(𝔽5)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 1 1 0 0 0 0 0 0 4 0 0 0 0 0 1 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 3 0 0 0 0 0 2 2 0 0 0 0 0 0 3 0 0 0 0 0 2 2 0 0 0 0 0 0 0 1 0 0 0 0 4 0
,
 1 2 0 0 0 0 4 4 0 0 0 0 0 0 4 3 0 0 0 0 1 1 0 0 0 0 0 0 0 3 0 0 0 0 3 0

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

C2×C22⋊Q8 in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes Q_8
% in TeX

G:=Group("C2xC2^2:Q8");
// GroupNames label

G:=SmallGroup(64,204);
// by ID

G=gap.SmallGroup(64,204);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,96,217,103,650]);
// Polycyclic

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

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