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G = D4○D8order 64 = 26

Central product of D4 and D8

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

Aliases: D4D8, Q8Q16, D4.11D4, D87C22, C4.8C24, C8.3C23, Q8.11D4, Q167C22, D4.5C23, Q8.5C23, SD164C22, M4(2)⋊6C22, 2+ 1+43C2, C4○D84C2, C8○D43C2, (C2×D8)⋊12C2, C8⋊C224C2, (C2×C8)⋊4C22, C4.41(C2×D4), C4○D41C22, (C2×D4)⋊8C22, C22.5(C2×D4), (C2×C4).43C23, C2.30(C22×D4), SmallGroup(64,257)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — D4○D8
C1C2C4C2×C4C4○D42+ 1+4 — D4○D8
C1C2C4 — D4○D8
C1C2C4○D4 — D4○D8
C1C2C2C4 — D4○D8

Generators and relations for D4○D8
 G = < a,b,c,d | a4=b2=d2=1, c4=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c3 >

Subgroups: 237 in 134 conjugacy classes, 79 normal (9 characteristic)
C1, C2, C2 [×9], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C8, C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×9], D4 [×12], Q8, Q8 [×2], C23 [×6], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×6], C2×D4 [×6], C4○D4, C4○D4 [×6], C4○D4 [×2], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ 1+4 [×2], D4○D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, D4○D8

Character table of D4○D8

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F8A8B8C8D8E
 size 1122244444422224422444
ρ11111111111111111111111    trivial
ρ211-1-111-11-11-11-11-1-11-1-111-1    linear of order 2
ρ311-1-11-11-11-111-11-11-1-1-111-1    linear of order 2
ρ411111-1-1-1-1-1-11111-1-111111    linear of order 2
ρ511-11-11-111-1111-1-1-1-111-11-1    linear of order 2
ρ6111-1-1111-1-1-11-1-111-1-1-1-111    linear of order 2
ρ7111-1-1-1-1-11111-1-11-11-1-1-111    linear of order 2
ρ811-11-1-11-1-11-111-1-11111-11-1    linear of order 2
ρ9111-1-111-111-11-1-11-1-1111-1-1    linear of order 2
ρ1011-11-11-1-1-11111-1-11-1-1-11-11    linear of order 2
ρ1111-11-1-1111-1-111-1-1-11-1-11-11    linear of order 2
ρ12111-1-1-1-11-1-111-1-1111111-1-1    linear of order 2
ρ1311-1-111-1-11-1-11-11-11111-1-11    linear of order 2
ρ141111111-1-1-111111-11-1-1-1-1-1    linear of order 2
ρ1511111-1-1111-111111-1-1-1-1-1-1    linear of order 2
ρ1611-1-11-111-1111-11-1-1-111-1-11    linear of order 2
ρ1722-222000000-2-2-220000000    orthogonal lifted from D4
ρ182222-2000000-2-22-20000000    orthogonal lifted from D4
ρ19222-22000000-22-2-20000000    orthogonal lifted from D4
ρ2022-2-2-2000000-22220000000    orthogonal lifted from D4
ρ214-400000000000000022-22000    orthogonal faithful
ρ224-4000000000000000-2222000    orthogonal faithful

Permutation representations of D4○D8
On 16 points - transitive group 16T80
Generators in S16
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)

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

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

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

G:=TransitiveGroup(16,80);

D4○D8 is a maximal subgroup of
D8○SD16  D8⋊C23  Q16.A4  Q8.7S4
 D8⋊D2p: D811D4  D8○D8  D813D6  D815D6  D85D6  D813D10  D815D10  D85D10 ...
 D4.D4p: Q16.10D4  D8.3D4  D4○D16  D4○SD32  D4.12D12  D4.12D20  D4.12D28 ...
 D4p.C23: C8.C24  D12.32C23  D20.32C23  D28.32C23 ...
D4○D8 is a maximal quotient of
2+ 1+45C4  C4○D4.8Q8  C42.275C23  C42.277C23  C42.280C23  C42.14C23  (C2×C8)⋊12D4  (C2×C8)⋊14D4  C42.20C23  C42.22C23  (C2×D4).301D4  (C2×D4).303D4  C42.352C23  C42.353C23  C42.356C23  C42.358C23  C42.366C23  M4(2)⋊4Q8  C42.387C23  C42.388C23  C42.391C23  C4.2+ 1+4  C4.142+ 1+4  C4.182+ 1+4  C42.406C23  C42.410C23  C42.423C23  C42.425C23  C4.2- 1+4  C42.26C23  C42.29C23  SD167D4  SD161D4  D4×D8  Q1613D4  D47SD16  C42.462C23  C42.470C23  C42.44C23  C42.49C23  C42.53C23  C42.54C23  C42.471C23  C42.474C23  C42.482C23  D46Q16  C42.488C23  C42.490C23  C42.60C23  C42.61C23  C42.496C23  C42.498C23  C42.502C23  Q88SD16  C42.507C23  C42.511C23  C42.516C23  D86Q8  Q8×Q16  SD16⋊Q8  D85Q8  Q85D8  C42.530C23  C42.74C23  C42.533C23
 D8⋊D2p: D89D4  D85D4  D812D4  D813D6  D815D6  D85D6  D813D10  D815D10 ...
 M4(2)⋊D2p: M4(2)⋊16D4  M4(2)⋊11D4  D4.12D12  D4.12D20  D4.12D28 ...
 C4○D4⋊D2p: C4○D4⋊D4  (C2×D4)⋊21D4  C42.18C23  D12.32C23  D20.32C23  D28.32C23 ...

Matrix representation of D4○D8 in GL4(𝔽7) generated by

0626
5021
1155
5212
,
0400
2000
6643
2523
,
2051
1221
1665
5515
,
1063
0545
0441
0644
G:=sub<GL(4,GF(7))| [0,5,1,5,6,0,1,2,2,2,5,1,6,1,5,2],[0,2,6,2,4,0,6,5,0,0,4,2,0,0,3,3],[2,1,1,5,0,2,6,5,5,2,6,1,1,1,5,5],[1,0,0,0,0,5,4,6,6,4,4,4,3,5,1,4] >;

D4○D8 in GAP, Magma, Sage, TeX

D_4\circ D_8
% in TeX

G:=Group("D4oD8");
// GroupNames label

G:=SmallGroup(64,257);
// by ID

G=gap.SmallGroup(64,257);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,-2,217,255,1444,730,88]);
// Polycyclic

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

Export

Character table of D4○D8 in TeX

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