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G = C242Q8order 128 = 27

1st semidirect product of C24 and Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C242Q8, C25.12C22, C24.179C23, C2.17C2≀C22, C232Q82C2, (C22×C4).79D4, C23.52(C2×Q8), C243C4.6C2, C23.587(C2×D4), C23.9D413C2, C22.213C22≀C2, C23.128(C4○D4), C2.10(C23⋊Q8), C22.33(C22⋊Q8), C22.25(C4.4D4), (C2×C22⋊C4).17C22, SmallGroup(128,761)

Series: Derived Chief Lower central Upper central Jennings

C1C24 — C242Q8
C1C2C22C23C24C25C243C4 — C242Q8
C1C2C24 — C242Q8
C1C22C24 — C242Q8
C1C2C24 — C242Q8

Generators and relations for C242Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=e2, faf-1=ab=ba, eae-1=ac=ca, ad=da, bc=cb, ebe-1=bd=db, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 648 in 233 conjugacy classes, 42 normal (8 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×9], C22, C22 [×6], C22 [×46], C2×C4 [×21], Q8 [×2], C23, C23 [×6], C23 [×46], C22⋊C4 [×24], C4⋊C4 [×6], C22×C4 [×6], C22×C4 [×3], C2×Q8 [×2], C24, C24 [×2], C24 [×10], C2×C22⋊C4 [×6], C2×C22⋊C4 [×6], C22⋊Q8 [×6], C25, C23.9D4 [×3], C243C4 [×3], C232Q8, C242Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], C23⋊Q8, C2≀C22 [×2], C242Q8

Character table of C242Q8

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J4K4L
 size 11112222224444888888888888
ρ111111111111111111111111111    trivial
ρ21111111111111111-1-1-11-1-11-1-1-1    linear of order 2
ρ31111111111-1-1-1-1-1-11-1-11-111-111    linear of order 2
ρ41111111111-1-1-1-1-1-1-11111-111-1-1    linear of order 2
ρ51111111111-1-1-1-1111-1-1-11-1-111-1    linear of order 2
ρ61111111111-1-1-1-111-111-1-11-1-1-11    linear of order 2
ρ711111111111111-1-1111-1-1-1-1-11-1    linear of order 2
ρ811111111111111-1-1-1-1-1-111-11-11    linear of order 2
ρ922222-22-2-2-200002-20000000000    orthogonal lifted from D4
ρ1022222-22-2-2-20000-220000000000    orthogonal lifted from D4
ρ112222-22-2-2-22000000-2000000020    orthogonal lifted from D4
ρ122222-22-2-2-2200000020000000-20    orthogonal lifted from D4
ρ132222-2-2-222-20000000000200-200    orthogonal lifted from D4
ρ142222-2-2-222-20000000000-200200    orthogonal lifted from D4
ρ152-2-22-2-22-222-2-222000000000000    symplectic lifted from Q8, Schur index 2
ρ162-2-22-2-22-22222-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ172-2-2222-2-22-200000000000-2i0002i    complex lifted from C4○D4
ρ182-2-22-2222-2-2000000000-2i002i000    complex lifted from C4○D4
ρ192-2-22-2222-2-20000000002i00-2i000    complex lifted from C4○D4
ρ202-2-222-2-22-2200000002i-2i0000000    complex lifted from C4○D4
ρ212-2-2222-2-22-2000000000002i000-2i    complex lifted from C4○D4
ρ222-2-222-2-22-220000000-2i2i0000000    complex lifted from C4○D4
ρ2344-4-40000002-2-22000000000000    orthogonal lifted from C2≀C22
ρ244-44-40000002-22-2000000000000    orthogonal lifted from C2≀C22
ρ2544-4-4000000-222-2000000000000    orthogonal lifted from C2≀C22
ρ264-44-4000000-22-22000000000000    orthogonal lifted from C2≀C22

Permutation representations of C242Q8
On 16 points - transitive group 16T333
Generators in S16
(1 3)(2 4)(5 13)(6 8)(7 15)(9 11)(10 12)(14 16)
(2 10)(4 12)(5 15)(7 13)
(5 15)(6 16)(7 13)(8 14)
(1 9)(2 10)(3 11)(4 12)(5 15)(6 16)(7 13)(8 14)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16 3 14)(2 15 4 13)(5 12 7 10)(6 11 8 9)

G:=sub<Sym(16)| (1,3)(2,4)(5,13)(6,8)(7,15)(9,11)(10,12)(14,16), (2,10)(4,12)(5,15)(7,13), (5,15)(6,16)(7,13)(8,14), (1,9)(2,10)(3,11)(4,12)(5,15)(6,16)(7,13)(8,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16,3,14)(2,15,4,13)(5,12,7,10)(6,11,8,9)>;

G:=Group( (1,3)(2,4)(5,13)(6,8)(7,15)(9,11)(10,12)(14,16), (2,10)(4,12)(5,15)(7,13), (5,15)(6,16)(7,13)(8,14), (1,9)(2,10)(3,11)(4,12)(5,15)(6,16)(7,13)(8,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16,3,14)(2,15,4,13)(5,12,7,10)(6,11,8,9) );

G=PermutationGroup([(1,3),(2,4),(5,13),(6,8),(7,15),(9,11),(10,12),(14,16)], [(2,10),(4,12),(5,15),(7,13)], [(5,15),(6,16),(7,13),(8,14)], [(1,9),(2,10),(3,11),(4,12),(5,15),(6,16),(7,13),(8,14)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,16,3,14),(2,15,4,13),(5,12,7,10),(6,11,8,9)])

G:=TransitiveGroup(16,333);

On 16 points - transitive group 16T347
Generators in S16
(1 3)(5 13)(6 8)(7 15)(9 11)(14 16)
(2 10)(4 12)(5 13)(7 15)
(1 3)(2 4)(5 15)(6 16)(7 13)(8 14)(9 11)(10 12)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 14 3 16)(2 13 4 15)(5 12 7 10)(6 11 8 9)

G:=sub<Sym(16)| (1,3)(5,13)(6,8)(7,15)(9,11)(14,16), (2,10)(4,12)(5,13)(7,15), (1,3)(2,4)(5,15)(6,16)(7,13)(8,14)(9,11)(10,12), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,3,16)(2,13,4,15)(5,12,7,10)(6,11,8,9)>;

G:=Group( (1,3)(5,13)(6,8)(7,15)(9,11)(14,16), (2,10)(4,12)(5,13)(7,15), (1,3)(2,4)(5,15)(6,16)(7,13)(8,14)(9,11)(10,12), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,3,16)(2,13,4,15)(5,12,7,10)(6,11,8,9) );

G=PermutationGroup([(1,3),(5,13),(6,8),(7,15),(9,11),(14,16)], [(2,10),(4,12),(5,13),(7,15)], [(1,3),(2,4),(5,15),(6,16),(7,13),(8,14),(9,11),(10,12)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,14,3,16),(2,13,4,15),(5,12,7,10),(6,11,8,9)])

G:=TransitiveGroup(16,347);

Matrix representation of C242Q8 in GL6(𝔽5)

400000
040000
004000
000400
000010
000004
,
100000
010000
004000
000100
000040
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
040000
100000
000100
001000
000001
000010
,
030000
300000
000010
000001
001000
000100

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

C242Q8 in GAP, Magma, Sage, TeX

C_2^4\rtimes_2Q_8
% in TeX

G:=Group("C2^4:2Q8");
// GroupNames label

G:=SmallGroup(128,761);
// by ID

G=gap.SmallGroup(128,761);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,1411,4037]);
// Polycyclic

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

Export

Character table of C242Q8 in TeX

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