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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

Aliases: C2×C22⋊Q8, C233Q8, C23.34C23, C22.17C24, C24.30C22, C4.61(C2×D4), C221(C2×Q8), C4⋊C410C22, (C2×C4).134D4, (C22×Q8)⋊3C2, (C2×Q8)⋊8C22, C2.6(C22×D4), C2.3(C22×Q8), (C2×C4).11C23, (C23×C4).10C2, C22.60(C2×D4), C22.30(C4○D4), C22⋊C4.11C22, (C22×C4).122C22, (C2×C4⋊C4)⋊15C2, C2.6(C2×C4○D4), (C2×C22⋊C4).11C2, SmallGroup(64,204)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C22⋊Q8
C1C2C22C23C24C23×C4 — C2×C22⋊Q8
C1C22 — C2×C22⋊Q8
C1C23 — C2×C22⋊Q8
C1C22 — 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 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×10], C22 [×12], C2×C4 [×16], C2×C4 [×18], Q8 [×8], C23, C23 [×6], C23 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C24, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C22⋊Q8 [×8], C23×C4, C22×Q8, C2×C22⋊Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8

Character table of C2×C22⋊Q8

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 1111111122222222222244444444
ρ11111111111111111111111111111    trivial
ρ21111-1-1-1-1-11-111-1-111-1-11-111-1-111-1    linear of order 2
ρ311111111-1-1-1-1-11-1-111-11-1-111-1-111    linear of order 2
ρ41111-1-1-1-11-11-1-1-11-11-1111-11-11-11-1    linear of order 2
ρ51111-1-1-1-1-11-111-1-111-1-111-1-111-1-11    linear of order 2
ρ611111111-1-1-1-1-11-1-111-1111-1-111-1-1    linear of order 2
ρ711111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ81111-1-1-1-11-11-1-1-11-11-111-11-11-11-11    linear of order 2
ρ91111-1-1-1-11-11-111-11-11-1-1-11-111-11-1    linear of order 2
ρ10111111111111-1-1-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ1111111111-1-1-1-11-111-1-11-111-1-1-1-111    linear of order 2
ρ121111-1-1-1-1-11-11-111-1-111-11-1-11-111-1    linear of order 2
ρ131111-1-1-1-11-11-111-11-11-1-11-11-1-11-11    linear of order 2
ρ14111111111111-1-1-1-1-1-1-1-11111-1-1-1-1    linear of order 2
ρ1511111111-1-1-1-11-111-1-11-1-1-11111-1-1    linear of order 2
ρ161111-1-1-1-1-11-11-111-1-111-1-111-11-1-11    linear of order 2
ρ172-2-222-22-200000200-2-20200000000    orthogonal lifted from D4
ρ182-2-222-22-200000-200220-200000000    orthogonal lifted from D4
ρ192-2-22-22-22000002002-20-200000000    orthogonal lifted from D4
ρ202-2-22-22-2200000-200-220200000000    orthogonal lifted from D4
ρ212-22-2-2-2222-2-220000000000000000    symplectic lifted from Q8, Schur index 2
ρ222-22-222-2-222-2-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ232-22-2-2-222-222-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ242-22-222-2-2-2-2220000000000000000    symplectic lifted from Q8, Schur index 2
ρ2522-2-22-2-220000-2i02i2i00-2i000000000    complex lifted from C4○D4
ρ2622-2-2-222-20000-2i0-2i2i002i000000000    complex lifted from C4○D4
ρ2722-2-2-222-200002i02i-2i00-2i000000000    complex lifted from C4○D4
ρ2822-2-22-2-2200002i0-2i-2i002i000000000    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)])

C2×C22⋊Q8 is a maximal subgroup of
C24.45(C2×C4)  C24.55D4  C24.57D4  C24.61D4  C24.160D4  C24.73D4  C24.135D4  C24.75D4  M4(2).45D4  C24.176C23  C233SD16  C232Q16  C24.85D4  C24.86D4  C42.159D4  C23.211C24  C23.214C24  C24.558C23  C23.244C24  C23.250C24  C24.227C23  C24.244C23  C23.309C24  C23.315C24  C24.252C23  C23.321C24  C23.323C24  C24.259C23  C23.327C24  C23.329C24  C24.264C23  C23.334C24  C23.335C24  C244Q8  C24.567C23  C24.568C23  C23.349C24  C23.350C24  C23.352C24  C24.282C23  C24.283C23  C24.285C23  C23.372C24  C23.377C24  C23.388C24  C24.301C23  C23.392C24  C24.308C23  C23.402C24  C24.579C23  C42.165D4  C42.166D4  C4219D4  C42.167D4  C23.449C24  C23.456C24  C24.332C23  C23.461C24  C24.583C23  C23.483C24  C24.361C23  C4228D4  C42.186D4  C23.525C24  C245Q8  C23.527C24  C24.374C23  C24.592C23  C23.559C24  C24.378C23  C23.572C24  C23.574C24  C23.576C24  C24.385C23  C23.580C24  C23.581C24  C23.583C24  C24.394C23  C23.589C24  C23.590C24  C23.592C24  C24.403C23  C24.405C23  C23.600C24  C23.602C24  C24.408C23  C23.620C24  C24.418C23  C24.420C23  C24.421C23  C23.630C24  C23.632C24  C23.714C24  C23.716C24  C24.462C23  C248Q8  C4246D4  C24.599C23  C42.440D4  C24.106D4  M4(2)⋊15D4  C24.118D4  C234SD16  C233Q16  C24.123D4  C24.126D4  C24.128D4  C24.129D4  C2×D4×Q8  C22.78C25  C22.84C25  C22.90C25  C22.94C25  C23.144C24  C22.124C25  C22.125C25  C22.127C25  C22.130C25
C2×C22⋊Q8 is a maximal quotient of
C42.162D4  C23.309C24  C24.252C23  C244Q8  C24.567C23  C24.267C23  C24.568C23  C24.268C23  C24.569C23  C23.349C24  C23.350C24  C23.351C24  C23.352C24  C23.353C24  C23.354C24  C24.300C23  C42.166D4  C42.167D4  C42.173D4  C24.583C23  C42.174D4  C42.175D4  C42.176D4  C42.177D4  C23.479C24  C42.178D4  C42.179D4  C42.180D4  C245Q8  C23.527C24  C42.187D4  C42.188D4  C248Q8  C42.439D4  C24.599C23  C42.440D4  C42.447D4  C42.219D4  C42.220D4  C42.448D4  C42.449D4  C42.20C23  C42.21C23  C42.22C23  C42.23C23

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

400000
040000
001000
000100
000040
000004
,
400000
110000
004000
001100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
300000
220000
003000
002200
000001
000040
,
120000
440000
004300
001100
000003
000030

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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Character table of C2×C22⋊Q8 in TeX

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