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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

C1C2 — C22×Q8
C1C2C22C23C22×C4 — C22×Q8
C1C2 — C22×Q8
C1C23 — C22×Q8
C1C2 — 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 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L
 size 11111111222222222222
ρ111111111111111111111    trivial
ρ211-111-1-1-11-1-11-111-1-111-1    linear of order 2
ρ31-1-1-111-11-1-11111-1-1-1-111    linear of order 2
ρ41-11-11-11-1-11-11-11-111-11-1    linear of order 2
ρ511111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ61-11-11-11-11-11-11-11-11-11-1    linear of order 2
ρ711-111-1-1-1-111-11-1-11-111-1    linear of order 2
ρ81-1-1-111-1111-1-1-1-111-1-111    linear of order 2
ρ9111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ101-11-11-11-1-11-111-11-1-11-11    linear of order 2
ρ1111-111-1-1-11-1-111-1-111-1-11    linear of order 2
ρ121-1-1-111-11-1-111-1-11111-1-1    linear of order 2
ρ1311-111-1-1-1-111-1-111-11-1-11    linear of order 2
ρ141-1-1-111-1111-1-111-1-111-1-1    linear of order 2
ρ1511111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ161-11-11-11-11-11-1-11-11-11-11    linear of order 2
ρ1722-2-2-2-222000000000000    symplectic lifted from Q8, Schur index 2
ρ18222-2-22-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ192-222-2-2-22000000000000    symplectic lifted from Q8, Schur index 2
ρ202-2-22-222-2000000000000    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 23)(6 24)(7 21)(8 22)(13 17)(14 18)(15 19)(16 20)(25 32)(26 29)(27 30)(28 31)
(1 22)(2 23)(3 24)(4 21)(5 10)(6 11)(7 12)(8 9)(13 26)(14 27)(15 28)(16 25)(17 29)(18 30)(19 31)(20 32)
(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 19 3 17)(2 18 4 20)(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)(5,23)(6,24)(7,21)(8,22)(13,17)(14,18)(15,19)(16,20)(25,32)(26,29)(27,30)(28,31), (1,22)(2,23)(3,24)(4,21)(5,10)(6,11)(7,12)(8,9)(13,26)(14,27)(15,28)(16,25)(17,29)(18,30)(19,31)(20,32), (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), (1,19,3,17)(2,18,4,20)(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)(5,23)(6,24)(7,21)(8,22)(13,17)(14,18)(15,19)(16,20)(25,32)(26,29)(27,30)(28,31), (1,22)(2,23)(3,24)(4,21)(5,10)(6,11)(7,12)(8,9)(13,26)(14,27)(15,28)(16,25)(17,29)(18,30)(19,31)(20,32), (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), (1,19,3,17)(2,18,4,20)(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),(5,23),(6,24),(7,21),(8,22),(13,17),(14,18),(15,19),(16,20),(25,32),(26,29),(27,30),(28,31)], [(1,22),(2,23),(3,24),(4,21),(5,10),(6,11),(7,12),(8,9),(13,26),(14,27),(15,28),(16,25),(17,29),(18,30),(19,31),(20,32)], [(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)], [(1,19,3,17),(2,18,4,20),(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

4000
0100
0040
0004
,
4000
0400
0010
0001
,
4000
0100
0013
0014
,
1000
0100
0034
0002
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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Character table of C22×Q8 in TeX

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