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G = C24.4Q8order 128 = 27

3rd non-split extension by C24 of Q8 acting via Q8/C2=C22

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

Aliases: C24.4Q8, C24.49D4, C25.1C22, C23.6C42, C24.37(C2×C4), C23.16(C4⋊C4), C243C4.1C2, C22.12(C23⋊C4), C2.9(C23.9D4), C23.144(C22⋊C4), C22.54(C2.C42), (C2×C22⋊C4)⋊1C4, SmallGroup(128,36)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.4Q8
C1C2C22C23C24C25C243C4 — C24.4Q8
C1C22C23 — C24.4Q8
C1C22C25 — C24.4Q8
C1C22C25 — C24.4Q8

Generators and relations for C24.4Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=be2, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, ebe-1=bc=cb, fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=acde-1 >

Subgroups: 600 in 198 conjugacy classes, 36 normal (6 characteristic)
C1, C2 [×3], C2 [×10], C4 [×6], C22, C22 [×6], C22 [×46], C2×C4 [×18], C23, C23 [×6], C23 [×46], C22⋊C4 [×18], C22×C4 [×6], C24, C24 [×6], C24 [×6], C2×C22⋊C4 [×6], C2×C22⋊C4 [×6], C25, C243C4 [×3], C24.4Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C23⋊C4 [×6], C23.9D4 [×3], C24.4Q8

Character table of C24.4Q8

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J4K4L
 size 11112222224444888888888888
ρ111111111111111111111111111    trivial
ρ211111111111111-111-1-1-1-11-1-1-11    linear of order 2
ρ311111111111111-1-1-1-1-1-11-1111-1    linear of order 2
ρ4111111111111111-1-1111-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-11-11-1-111iii-i-ii1-i1-1-1-i    linear of order 4
ρ61111-11-1-11-11-11-1i1-1-ii-ii1-ii-i-1    linear of order 4
ρ71111-11-1-11-11-11-1i-11-ii-i-i-1i-ii1    linear of order 4
ρ81111-1-1-11-11-1-111i-i-i-i-ii-1i-111i    linear of order 4
ρ911111-11-1-1-1-111-11-ii1-1-1ii-i-ii-i    linear of order 4
ρ101111-11-1-11-11-11-1-i1-1i-ii-i1i-ii-1    linear of order 4
ρ111111-1-1-11-11-1-111-iiiii-i-1-i-111-i    linear of order 4
ρ1211111-11-1-1-1-111-1-1-ii-111-iiii-i-i    linear of order 4
ρ1311111-11-1-1-1-111-1-1i-i-111i-i-i-iii    linear of order 4
ρ141111-1-1-11-11-1-111-i-i-iii-i1i1-1-1i    linear of order 4
ρ1511111-11-1-1-1-111-11i-i1-1-1-i-iii-ii    linear of order 4
ρ161111-11-1-11-11-11-1-i-11i-iii-1-ii-i1    linear of order 4
ρ172222-22-2222-22-2-2000000000000    orthogonal lifted from D4
ρ182222222-22-2-2-2-22000000000000    orthogonal lifted from D4
ρ1922222-222-222-2-2-2000000000000    orthogonal lifted from D4
ρ202222-2-2-2-2-2-222-22000000000000    symplectic lifted from Q8, Schur index 2
ρ214-4-4440-40000000000000000000    orthogonal lifted from C23⋊C4
ρ224-44-40400-400000000000000000    orthogonal lifted from C23⋊C4
ρ2344-4-4000-4040000000000000000    orthogonal lifted from C23⋊C4
ρ244-4-44-4040000000000000000000    orthogonal lifted from C23⋊C4
ρ2544-4-400040-40000000000000000    orthogonal lifted from C23⋊C4
ρ264-44-40-400400000000000000000    orthogonal lifted from C23⋊C4

Permutation representations of C24.4Q8
On 16 points - transitive group 16T323
Generators in S16
(1 3)(2 10)(4 12)(6 14)(8 16)(9 11)
(1 3)(5 13)(6 8)(7 15)(9 11)(14 16)
(1 3)(2 4)(5 15)(6 16)(7 13)(8 14)(9 11)(10 12)
(1 11)(2 12)(3 9)(4 10)(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 6)(2 5 4 15)(3 16)(7 12 13 10)(8 9)(11 14)

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

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

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

G:=TransitiveGroup(16,323);

Matrix representation of C24.4Q8 in GL8(ℤ)

10000000
0-1000000
00100000
000-10000
0000-1000
00000100
00000010
0000000-1
,
-10000000
01000000
00100000
000-10000
0000-1000
00000-100
00000010
00000001
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
01000000
10000000
00010000
00-100000
00000100
00001000
0000000-1
00000010
,
00100000
00010000
-10000000
01000000
00000010
00000001
0000-1000
00000-100

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0],[0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C24.4Q8 in GAP, Magma, Sage, TeX

C_2^4._4Q_8
% in TeX

G:=Group("C2^4.4Q8");
// GroupNames label

G:=SmallGroup(128,36);
// by ID

G=gap.SmallGroup(128,36);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,570,2804]);
// Polycyclic

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

Export

Character table of C24.4Q8 in TeX

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