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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, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C22×Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, 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)]])

C22×Q8 is a maximal subgroup of
C23.67C23  C23⋊Q8  C23.78C23  C23.38D4  Q8⋊D4  C22⋊Q16  C23.32C23  C23.38C23  Q85D4  Q8⋊A4
C22×Q8 is a maximal quotient of
C23.37C23  C232Q8  C23.41C23  D43Q8  Q83Q8

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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