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## G = C22×Q8order 32 = 25

### Direct product of C22 and Q8

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

Aliases: C22×Q8, C2.2C24, C4.7C23, C22.9C23, C23.14C22, (C22×C4).7C2, (C2×C4).30C22, SmallGroup(32,47)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C22×Q8
G = < a,b,c,d | a2=b2=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 78, all normal (4 characteristic)
C1, C2, C2 [×6], C4 [×12], C22 [×7], C2×C4 [×18], Q8 [×16], C23, C22×C4 [×3], C2×Q8 [×12], C22×Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C24, C22×Q8

Character table of C22×Q8

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L size 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 linear of order 2 ρ17 2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ18 2 2 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ19 2 -2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ20 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2

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

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

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

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

Matrix representation of C22×Q8 in GL4(𝔽5) generated by

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

C22×Q8 in GAP, Magma, Sage, TeX

C_2^2\times Q_8
% in TeX

G:=Group("C2^2xQ8");
// GroupNames label

G:=SmallGroup(32,47);
// by ID

G=gap.SmallGroup(32,47);
# by ID

G:=PCGroup([5,-2,2,2,2,-2,80,181,86]);
// Polycyclic

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

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