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G = C42.5D4order 128 = 27

5th non-split extension by C42 of D4 acting faithfully

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

Aliases: C42.5D4, (C2×D8)⋊4C4, C4.3(C4×D4), (C2×SD16)⋊3C4, (C2×D4).78D4, C4.9C422C2, C22.55(C4×D4), D4.7(C22⋊C4), C22⋊C4.120D4, C23.129(C2×D4), C4.137(C4⋊D4), M4(2)⋊4C45C2, C22.11C243C2, C22.31C22≀C2, (C22×C4).31C23, C42⋊C2214C2, C23.37D424C2, C22.51(C4⋊D4), C23.C235C2, (C22×D4).30C22, (C2×M4(2)).9C22, C42⋊C2.29C22, C4.10(C22.D4), C2.43(C23.23D4), (C2×C8)⋊(C2×C4), (C2×D4)⋊10(C2×C4), (C2×Q8)⋊10(C2×C4), (C2×C4).240(C2×D4), (C2×C8⋊C22).3C2, C4.20(C2×C22⋊C4), (C2×C4).326(C4○D4), (C2×C4).191(C22×C4), (C2×C4○D4).25C22, SmallGroup(128,636)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.5D4
C1C2C4C2×C4C22×C4C22×D4C22.11C24 — C42.5D4
C1C2C2×C4 — C42.5D4
C1C2C22×C4 — C42.5D4
C1C2C2C22×C4 — C42.5D4

Generators and relations for C42.5D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b, ab=ba, cac-1=dad-1=a-1b-1, cbc-1=b-1, bd=db, dcd-1=bc-1 >

Subgroups: 412 in 173 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2 [×8], C4 [×4], C4 [×6], C22 [×3], C22 [×15], C8 [×3], C2×C4 [×6], C2×C4 [×12], D4 [×4], D4 [×10], Q8 [×2], C23, C23 [×9], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×3], C2×C8 [×2], C2×C8, M4(2) [×4], D8 [×4], SD16 [×4], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×6], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24, C23⋊C4 [×2], D4⋊C4 [×2], C4≀C2 [×2], C2×C22⋊C4 [×2], C42⋊C2 [×3], C4×D4 [×4], C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×4], C22×D4, C2×C4○D4, C4.9C42, M4(2)⋊4C4, C23.C23, C23.37D4, C42⋊C22, C22.11C24, C2×C8⋊C22, C42.5D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C42.5D4

Character table of C42.5D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D
 size 11222444482222444444448888888
ρ111111111111111111111111111111    trivial
ρ211111-1-1-1-111111-1-1-1-11111111-1-1-1-1    linear of order 2
ρ3111111111-11111-1-1-1-1-1-1-1-111-111-1-1    linear of order 2
ρ411111-1-1-1-1-111111111-1-1-1-111-1-1-111    linear of order 2
ρ511111-1-1-1-1-11111-1-1-1-11111-1-1-11111    linear of order 2
ρ6111111111-1111111111111-1-1-1-1-1-1-1    linear of order 2
ρ711111-1-1-1-1111111111-1-1-1-1-1-1111-1-1    linear of order 2
ρ811111111111111-1-1-1-1-1-1-1-1-1-11-1-111    linear of order 2
ρ9111-1-1-11-1111-11-1-ii-ii-ii-ii-ii-1i-i1-1    linear of order 4
ρ10111-1-11-11-111-11-1i-ii-i-ii-ii-ii-1-ii-11    linear of order 4
ρ11111-1-1-11-1111-11-1i-ii-ii-ii-ii-i-1-ii1-1    linear of order 4
ρ12111-1-11-11-111-11-1-ii-iii-ii-ii-i-1i-i-11    linear of order 4
ρ13111-1-1-11-11-11-11-1-ii-ii-ii-iii-i1-ii-11    linear of order 4
ρ14111-1-11-11-1-11-11-1i-ii-i-ii-iii-i1i-i1-1    linear of order 4
ρ15111-1-1-11-11-11-11-1i-ii-ii-ii-i-ii1i-i-11    linear of order 4
ρ16111-1-11-11-1-11-11-1-ii-iii-ii-i-ii1-ii1-1    linear of order 4
ρ172222200000-2-2-2-2000022-2-20000000    orthogonal lifted from D4
ρ1822-2-22-222-20-222-2000000000000000    orthogonal lifted from D4
ρ1922-22-2-2-2220-2-222000000000000000    orthogonal lifted from D4
ρ202222200000-2-2-2-20000-2-2220000000    orthogonal lifted from D4
ρ2122-2-22000002-2-22-2-22200000000000    orthogonal lifted from D4
ρ2222-22-222-2-20-2-222000000000000000    orthogonal lifted from D4
ρ2322-2-222-2-220-222-2000000000000000    orthogonal lifted from D4
ρ2422-2-22000002-2-2222-2-200000000000    orthogonal lifted from D4
ρ2522-22-20000022-2-22i-2i-2i2i00000000000    complex lifted from C4○D4
ρ26222-2-200000-22-2200002i-2i-2i2i0000000    complex lifted from C4○D4
ρ2722-22-20000022-2-2-2i2i2i-2i00000000000    complex lifted from C4○D4
ρ28222-2-200000-22-220000-2i2i2i-2i0000000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    orthogonal faithful

Permutation representations of C42.5D4
On 16 points - transitive group 16T221
Generators in S16
(1 9 5 13)(2 12)(3 11 7 15)(4 14)(6 16)(8 10)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)
(1 10 13 6)(2 5 14 9)(3 16 15 4)(7 12 11 8)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,221);

On 16 points - transitive group 16T307
Generators in S16
(1 11)(2 10 6 14)(3 13)(4 12 8 16)(5 15)(7 9)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)
(1 4 5 8)(2 7 6 3)(9 12)(10 11)(13 16)(14 15)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,307);

Matrix representation of C42.5D4 in GL8(ℤ)

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

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,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0],[0,-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],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C42.5D4 in GAP, Magma, Sage, TeX

C_4^2._5D_4
% in TeX

G:=Group("C4^2.5D4");
// GroupNames label

G:=SmallGroup(128,636);
// by ID

G=gap.SmallGroup(128,636);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,521,248,1411,718,172,1027,124]);
// Polycyclic

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

Export

Character table of C42.5D4 in TeX

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