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G = D45D4order 64 = 26

1st semidirect product of D4 and D4 acting through Inn(D4)

p-group, metabelian, nilpotent (class 2), monomial

Aliases: D45D4, C429C22, C22.40C24, C24.19C22, C23.43C23, C2.132+ 1+4, (C4×D4)⋊15C2, C4⋊C45C22, C4.36(C2×D4), C22≀C26C2, C4⋊D411C2, D42(C22⋊C4), (C22×D4)⋊9C2, (C2×D4)⋊6C22, C22⋊Q811C2, C22.3(C2×D4), C4.4D410C2, C222(C4○D4), C22⋊C47C22, (C2×C4).27C23, (C2×Q8)⋊12C22, C2.18(C22×D4), (C22×C4)⋊11C22, C22.D48C2, C22⋊C4(C2×D4), (C2×C4○D4)⋊6C2, C2.20(C2×C4○D4), (C2×C22⋊C4)⋊14C2, SmallGroup(64,227)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — D45D4
C1C2C22C23C24C22×D4 — D45D4
C1C22 — D45D4
C1C22 — D45D4
C1C22 — D45D4

Generators and relations for D45D4
 G = < a,b,c,d | a4=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 285 in 167 conjugacy classes, 83 normal (31 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×23], C2×C4 [×5], C2×C4 [×4], C2×C4 [×10], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], C23 [×10], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×4], C2×D4 [×3], C2×D4 [×6], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24 [×2], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, D45D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C22×D4, C2×C4○D4, 2+ 1+4, D45D4

Character table of D45D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L4A4B4C4D4E4F4G4H4I4J4K4L
 size 1111222222444222222444444
ρ11111111111111111111111111    trivial
ρ211111-1111-11-1-1-11-1111-1-1-11-11    linear of order 2
ρ31111-1-1-1-1-1-1111-11-111111-1-1-1-1    linear of order 2
ρ41111-11-1-1-111-1-1111111-1-11-11-1    linear of order 2
ρ51111-11-11111-11-1-1-1-1-1-1-11-1-111    linear of order 2
ρ61111-1-1-111-111-11-11-1-1-11-11-1-11    linear of order 2
ρ711111-11-1-1-11-111-11-1-1-1-1111-1-1    linear of order 2
ρ81111111-1-1111-1-1-1-1-1-1-11-1-111-1    linear of order 2
ρ91111-1-1-1-1-1-1-1111-1111-1-1-1-1111    linear of order 2
ρ101111-11-1-1-11-1-1-1-1-1-111-11111-11    linear of order 2
ρ111111111111-111-1-1-111-1-1-11-1-1-1    linear of order 2
ρ1211111-1111-1-1-1-11-1111-111-1-11-1    linear of order 2
ρ1311111-11-1-1-1-1-11-11-1-1-111-11-111    linear of order 2
ρ141111111-1-11-11-1111-1-11-11-1-1-11    linear of order 2
ρ151111-11-1111-1-11111-1-111-1-11-1-1    linear of order 2
ρ161111-1-1-111-1-11-1-11-1-1-11-11111-1    linear of order 2
ρ172-2-2220-2-2200000002-20000000    orthogonal lifted from D4
ρ182-2-2220-22-20000000-220000000    orthogonal lifted from D4
ρ192-2-22-202-220000000-220000000    orthogonal lifted from D4
ρ202-2-22-2022-200000002-20000000    orthogonal lifted from D4
ρ212-22-202000-20002i-2i-2i002i000000    complex lifted from C4○D4
ρ222-22-202000-2000-2i2i2i00-2i000000    complex lifted from C4○D4
ρ232-22-20-20002000-2i-2i2i002i000000    complex lifted from C4○D4
ρ242-22-20-200020002i2i-2i00-2i000000    complex lifted from C4○D4
ρ2544-4-4000000000000000000000    orthogonal lifted from 2+ 1+4

Permutation representations of D45D4
On 16 points - transitive group 16T115
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 11)(2 10)(3 9)(4 12)(5 13)(6 16)(7 15)(8 14)
(1 8 12 13)(2 5 9 14)(3 6 10 15)(4 7 11 16)
(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)

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

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

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

G:=TransitiveGroup(16,115);

D45D4 is a maximal subgroup of
D4×C4○D4  C22.64C25  C22.73C25  C22.74C25  C22.77C25  C22.78C25  C22.79C25  C22.83C25  C4⋊2+ 1+4  C22.89C25  C22.102C25  C22.110C25  C42⋊C23  C22.122C25  C22.123C25  C22.124C25  C22.134C25  C22.149C25  D42S4
 D4p⋊D4: D89D4  D810D4  D85D4  D812D4  D1223D4  D1220D4  D1221D4  D1210D4 ...
 C2p.2+ 1+4: SD166D4  SD167D4  SD162D4  SD1610D4  C42.461C23  C42.462C23  C42.41C23  C42.46C23 ...
D45D4 is a maximal quotient of
D4×C22⋊C4  C24.549C23  C23.224C24  C23.231C24  C23.234C24  C23.240C24  C23.241C24  C24.558C23  C24.217C23  C24.218C23  C24.220C23  C23.304C24  C24.244C23  C23.308C24  C248D4  C23.311C24  C24.249C23  C24.252C23  C23.318C24  C24.563C23  C24.254C23  C23.324C24  C24.258C23  C24.259C23  C23.327C24  C23.328C24  C24.262C23  C24.263C23  C23.335C24  C244Q8  C24.567C23  C24.267C23  C24.569C23  C23.344C24  C23.345C24  C24.271C23  C23.350C24  C23.351C24  C23.354C24  C23.356C24  C24.278C23  C24.279C23  C23.359C24  C23.360C24  C23.364C24  C24.285C23  C24.286C23  C23.367C24  C23.368C24  C24.289C23  C24.290C23  C23.372C24  C24.572C23  C23.374C24  C24.293C23  C23.377C24  C23.380C24  C24.573C23  C23.388C24  C23.390C24  C23.392C24  C24.311C23  C23.434C24  C4217D4  C23.439C24  C4219D4  C23.443C24  C4221D4  C23.449C24  C426Q8  C24.326C23  C24.327C23  C23.455C24  C23.456C24  C23.457C24  C23.458C24  C24.331C23  C24.332C23  C23.568C24  C23.570C24  C23.571C24  C23.572C24  C23.573C24  C23.576C24  C23.578C24  C23.581C24  C23.584C24  C23.585C24  C24.393C23  C24.394C23  C24.395C23  C23.589C24  C23.590C24  C23.591C24  C23.592C24  C23.593C24  C24.401C23  C23.595C24  C24.403C23  C23.597C24  C24.405C23  C24.406C23  C23.600C24  C24.407C23  C23.602C24  C23.603C24  C24.408C23  C23.605C24  C23.606C24  C23.607C24  C23.608C24  C24.411C23  C24.412C23  C23.611C24  C23.612C24  C23.613C24  C24.413C23  C23.615C24  C23.616C24
 C24.D2p: C24.94D4  C24.95D4  C24.38D6  C24.44D6  C24.53D6  C24.27D10  C24.33D10  C24.42D10 ...
 D4⋊D4p: D44D8  D45D12  D45D20  D45D28 ...
 C2p.2+ 1+4: D47SD16  C42.461C23  C42.462C23  D48SD16  D45Q16  C42.465C23  C42.466C23  C42.467C23 ...

Matrix representation of D45D4 in GL4(𝔽5) generated by

4000
0400
0013
0014
,
1000
0100
0013
0004
,
0100
4000
0021
0023
,
0100
1000
0021
0023
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,1,1,0,0,3,4],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,3,4],[0,4,0,0,1,0,0,0,0,0,2,2,0,0,1,3],[0,1,0,0,1,0,0,0,0,0,2,2,0,0,1,3] >;

D45D4 in GAP, Magma, Sage, TeX

D_4\rtimes_5D_4
% in TeX

G:=Group("D4:5D4");
// GroupNames label

G:=SmallGroup(64,227);
// by ID

G=gap.SmallGroup(64,227);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,96,217,650,297]);
// Polycyclic

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

Export

Character table of D45D4 in TeX

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