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G = C41D4.C4order 128 = 27

5th non-split extension by C41D4 of C4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C41D4.5C4, (C2×D4).136D4, C42.13(C2×C4), C4.20(C23⋊C4), C42.C47C2, (C22×D4).16C4, (C22×C4).101D4, (C2×Q8).14C23, C42⋊C2.11C4, C4.10D418C22, C23.26(C22⋊C4), C22.29C24.9C2, C4.4D4.17C22, M4(2).8C2217C2, (C2×C4).11(C2×D4), (C2×D4).42(C2×C4), C2.46(C2×C23⋊C4), (C22×C4).35(C2×C4), (C2×C4).103(C22×C4), (C2×C4○D4).78C22, C22.70(C2×C22⋊C4), (C2×C4).146(C22⋊C4), SmallGroup(128,866)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C41D4.C4
C1C2C22C2×C4C2×Q8C2×C4○D4C22.29C24 — C41D4.C4
C1C2C22C2×C4 — C41D4.C4
C1C2C2×C4C2×C4○D4 — C41D4.C4
C1C2C22C2×Q8 — C41D4.C4

Generators and relations for C41D4.C4
 G = < a,b,c,d | a4=b4=c2=1, d4=a2b2, ab=ba, cac=a-1, dad-1=cbc=b-1, dbd-1=a-1b2, dcd-1=a-1bc >

Subgroups: 364 in 131 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×5], C22, C22 [×16], C8 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×8], D4 [×11], Q8, C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×5], C4⋊C4, C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×4], C2×D4 [×7], C2×Q8, C4○D4 [×2], C24, C4.D4 [×2], C4.10D4 [×4], C42⋊C2, C22≀C2 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4 [×2], C2×M4(2) [×2], C22×D4, C2×C4○D4, C42.C4 [×4], M4(2).8C22 [×2], C22.29C24, C41D4.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C2×C23⋊C4, C41D4.C4

Character table of C41D4.C4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H
 size 11244488224448888888888
ρ111111111111111111111111    trivial
ρ2111-1-1-1-11-1-11111-1-11-1-1111-1    linear of order 2
ρ3111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111-1-1-1-11-1-11111-11-111-1-1-11    linear of order 2
ρ5111-1-1-11-1-1-1111-1111-11-1-11-1    linear of order 2
ρ6111111-1-111111-1-1-111-1-1-111    linear of order 2
ρ7111-1-1-11-1-1-1111-11-1-11-111-11    linear of order 2
ρ8111111-1-111111-1-11-1-1111-1-1    linear of order 2
ρ9111-11-11111-11-1-1-1i-i-i-i-iiii    linear of order 4
ρ101111-11-11-1-1-11-1-11-i-iii-iii-i    linear of order 4
ρ11111-11-11111-11-1-1-1-iiiii-i-i-i    linear of order 4
ρ121111-11-11-1-1-11-1-11ii-i-ii-i-ii    linear of order 4
ρ131111-111-1-1-1-11-11-1i-ii-ii-ii-i    linear of order 4
ρ14111-11-1-1-111-11-111-i-i-iii-iii    linear of order 4
ρ151111-111-1-1-1-11-11-1-ii-ii-ii-ii    linear of order 4
ρ16111-11-1-1-111-11-111iii-i-ii-i-i    linear of order 4
ρ17222-22200-2-22-2-20000000000    orthogonal lifted from D4
ρ182222-2-200222-2-20000000000    orthogonal lifted from D4
ρ19222-2-220022-2-220000000000    orthogonal lifted from D4
ρ2022222-200-2-2-2-220000000000    orthogonal lifted from D4
ρ2144-400000-440000000000000    orthogonal lifted from C23⋊C4
ρ2244-4000004-40000000000000    orthogonal lifted from C23⋊C4
ρ238-8000000000000000000000    orthogonal faithful

Permutation representations of C41D4.C4
On 16 points - transitive group 16T226
Generators in S16
(1 11 5 15)(3 13 7 9)(4 8)(10 14)
(2 16 6 12)(3 7)(4 10 8 14)(9 13)
(1 11)(2 6)(3 9)(4 8)(5 15)(7 13)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,226);

On 16 points - transitive group 16T285
Generators in S16
(1 9 5 13)(2 4)(3 11 7 15)(6 8)(10 12)(14 16)
(1 3)(2 14 6 10)(4 16 8 12)(5 7)(9 11)(13 15)
(1 11)(2 6)(3 9)(4 8)(5 15)(7 13)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,285);

Matrix representation of C41D4.C4 in GL8(ℤ)

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

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

C41D4.C4 in GAP, Magma, Sage, TeX

C_4\rtimes_1D_4.C_4
% in TeX

G:=Group("C4:1D4.C4");
// GroupNames label

G:=SmallGroup(128,866);
// by ID

G=gap.SmallGroup(128,866);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,352,1123,1018,248,1971,375,172,4037]);
// Polycyclic

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

Export

Character table of C41D4.C4 in TeX

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