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G = C42.131D4order 128 = 27

113rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.131D4, (C2×C8).3D4, C4.72C22≀C2, (C2×D4).111D4, (C2×Q8).102D4, C4⋊M4(2)⋊1C2, C4.10C425C2, C4.148(C4⋊D4), D8⋊C22.4C2, C4.100(C4.4D4), C23.133(C4○D4), C22.28(C4⋊D4), (C2×C42).367C22, (C22×C4).729C23, C2.23(C23.10D4), (C2×M4(2)).232C22, M4(2).8C2213C2, C22.12(C22.D4), (C2×C4≀C2)⋊28C2, (C2×C4).263(C2×D4), (C2×C4).774(C4○D4), (C2×C4○D4).63C22, SmallGroup(128,782)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C42.131D4
C1C2C22C23C22×C4C2×M4(2)M4(2).8C22 — C42.131D4
C1C2C22×C4 — C42.131D4
C1C4C22×C4 — C42.131D4
C1C2C2C22×C4 — C42.131D4

Generators and relations for C42.131D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=a-1b2, dad=a-1b, cbc-1=b-1, bd=db, dcd=b2c3 >

Subgroups: 288 in 133 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×5], C8 [×5], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×8], Q8 [×4], C23, C23 [×2], C42 [×2], C42, C2×C8 [×2], C2×C8 [×3], M4(2) [×8], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4.D4 [×2], C4.10D4 [×2], C4≀C2 [×4], C4⋊C8 [×2], C2×C42, C2×M4(2) [×2], C2×M4(2) [×2], C4○D8 [×2], C8⋊C22 [×2], C8.C22 [×2], C2×C4○D4 [×2], C4.10C42, M4(2).8C22 [×2], C2×C4≀C2 [×2], C4⋊M4(2), D8⋊C22, C42.131D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C23.10D4, C42.131D4

Character table of C42.131D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H
 size 11222881122244448888888888
ρ111111111111111111111111111    trivial
ρ211111-1111111-1-1-1-11-11-1-11-11-11    linear of order 2
ρ311111-1-1111111111-1-11-11-1-111-1    linear of order 2
ρ4111111-111111-1-1-1-1-1111-1-111-1-1    linear of order 2
ρ511111-111111111111-1-11-1-11-1-1-1    linear of order 2
ρ6111111111111-1-1-1-111-1-11-1-1-11-1    linear of order 2
ρ7111111-1111111111-11-1-1-11-1-1-11    linear of order 2
ρ811111-1-111111-1-1-1-1-1-1-11111-111    linear of order 2
ρ9222-2-220-2-222-200000-200000000    orthogonal lifted from D4
ρ1022-22-202-2-2-2220000-2000000000    orthogonal lifted from D4
ρ1122-2-2200-2-22-222-22-20000000000    orthogonal lifted from D4
ρ1222-2-220022-22-200000000-200020    orthogonal lifted from D4
ρ1322-2-2200-2-22-22-22-220000000000    orthogonal lifted from D4
ρ14222-2-2-20-2-222-200000200000000    orthogonal lifted from D4
ρ1522-2-220022-22-2000000002000-20    orthogonal lifted from D4
ρ1622-22-20-2-2-2-22200002000000000    orthogonal lifted from D4
ρ1722-22-200222-2-2000000000-2i0002i    complex lifted from C4○D4
ρ18222-2-20022-2-220000000-2i002i000    complex lifted from C4○D4
ρ19222-2-20022-2-2200000002i00-2i000    complex lifted from C4○D4
ρ202222200-2-2-2-2-2000000-2i00002i00    complex lifted from C4○D4
ρ212222200-2-2-2-2-20000002i0000-2i00    complex lifted from C4○D4
ρ2222-22-200222-2-20000000002i000-2i    complex lifted from C4○D4
ρ234-400000-4i4i000-2-2i22i0000000000    complex faithful
ρ244-400000-4i4i00022i-2-2i0000000000    complex faithful
ρ254-4000004i-4i000-22i2-2i0000000000    complex faithful
ρ264-4000004i-4i0002-2i-22i0000000000    complex faithful

Permutation representations of C42.131D4
On 16 points - transitive group 16T360
Generators in S16
(1 5)(3 7)(9 15 13 11)(10 16 14 12)
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 10)(2 9)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)

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

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

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

G:=TransitiveGroup(16,360);

Matrix representation of C42.131D4 in GL4(𝔽5) generated by

4040
0101
3000
0401
,
4030
0004
1010
0100
,
0404
0040
0001
4040
,
0401
4040
0004
0040
G:=sub<GL(4,GF(5))| [4,0,3,0,0,1,0,4,4,0,0,0,0,1,0,1],[4,0,1,0,0,0,0,1,3,0,1,0,0,4,0,0],[0,0,0,4,4,0,0,0,0,4,0,4,4,0,1,0],[0,4,0,0,4,0,0,0,0,4,0,4,1,0,4,0] >;

C42.131D4 in GAP, Magma, Sage, TeX

C_4^2._{131}D_4
% in TeX

G:=Group("C4^2.131D4");
// GroupNames label

G:=SmallGroup(128,782);
// by ID

G=gap.SmallGroup(128,782);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2019,1018,248,1411,718,4037,1027]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a^-1*b,c*b*c^-1=b^-1,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations

Export

Character table of C42.131D4 in TeX

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