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G = C8⋊D4order 64 = 26

1st semidirect product of C8 and D4 acting via D4/C2=C22

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

Aliases: C81D4, C23.16D4, C2.D812C2, (C2×C4).30D4, C4.56(C2×D4), C22⋊Q84C2, (C2×SD16)⋊1C2, C4⋊D4.4C2, C4⋊C4.8C22, D4⋊C417C2, Q8⋊C417C2, C4.10(C4○D4), (C2×M4(2))⋊1C2, (C2×C4).96C23, (C2×C8).52C22, C22.92(C2×D4), C2.20(C4⋊D4), C2.12(C8⋊C22), (C2×D4).17C22, (C2×Q8).13C22, C2.12(C8.C22), (C22×C4).48C22, SmallGroup(64,149)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8⋊D4
C1C2C22C2×C4C22×C4C2×M4(2) — C8⋊D4
C1C2C2×C4 — C8⋊D4
C1C22C22×C4 — C8⋊D4
C1C2C2C2×C4 — C8⋊D4

Generators and relations for C8⋊D4
 G = < a,b,c | a8=b4=c2=1, bab-1=a-1, cac=a3, cbc=b-1 >

Subgroups: 117 in 60 conjugacy classes, 27 normal (25 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C8 [×2], C8, C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], M4(2) [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C8⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4, C4⋊D4, C8⋊C22, C8.C22, C8⋊D4

Character table of C8⋊D4

 class 12A2B2C2D2E4A4B4C4D4E4F8A8B8C8D
 size 1111482248884444
ρ11111111111111111    trivial
ρ21111-1111-11-1-1-1-111    linear of order 2
ρ31111-1-111-1-111-1-111    linear of order 2
ρ411111-1111-1-1-11111    linear of order 2
ρ51111-1-111-11-1111-1-1    linear of order 2
ρ611111-111111-1-1-1-1-1    linear of order 2
ρ7111111111-1-11-1-1-1-1    linear of order 2
ρ81111-1111-1-11-111-1-1    linear of order 2
ρ9222220-2-2-20000000    orthogonal lifted from D4
ρ102-22-200-220000002-2    orthogonal lifted from D4
ρ112-22-200-22000000-22    orthogonal lifted from D4
ρ122222-20-2-220000000    orthogonal lifted from D4
ρ132-22-2002-200002i-2i00    complex lifted from C4○D4
ρ142-22-2002-20000-2i2i00    complex lifted from C4○D4
ρ154-4-44000000000000    orthogonal lifted from C8⋊C22
ρ1644-4-4000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C8⋊D4
On 32 points
Generators in S32
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(1 23 11 27)(2 22 12 26)(3 21 13 25)(4 20 14 32)(5 19 15 31)(6 18 16 30)(7 17 9 29)(8 24 10 28)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 25)(18 28)(19 31)(20 26)(21 29)(22 32)(23 27)(24 30)

G:=sub<Sym(32)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,23,11,27)(2,22,12,26)(3,21,13,25)(4,20,14,32)(5,19,15,31)(6,18,16,30)(7,17,9,29)(8,24,10,28), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,25)(18,28)(19,31)(20,26)(21,29)(22,32)(23,27)(24,30)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,23,11,27)(2,22,12,26)(3,21,13,25)(4,20,14,32)(5,19,15,31)(6,18,16,30)(7,17,9,29)(8,24,10,28), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,25)(18,28)(19,31)(20,26)(21,29)(22,32)(23,27)(24,30) );

G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,23,11,27),(2,22,12,26),(3,21,13,25),(4,20,14,32),(5,19,15,31),(6,18,16,30),(7,17,9,29),(8,24,10,28)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(17,25),(18,28),(19,31),(20,26),(21,29),(22,32),(23,27),(24,30)])

C8⋊D4 is a maximal subgroup of
C42.256D4  C42.260D4  C24.126D4  C24.129D4  C4.152+ 1+4  C4.162+ 1+4  C42.300D4  C42.302D4  C42.304D4
 C4⋊C4.D2p: C42.386C23  C42.387C23  C42.390C23  C42.391C23  C4.2- 1+4  C42.25C23  C42.27C23  C42.29C23 ...
 C8pD4⋊C2: C24.110D4  M4(2)⋊14D4  (C2×C8)⋊14D4  M4(2)⋊16D4  C42.255D4  C42.259D4  C24.127D4  C24.130D4 ...
 C4p.(C2×D4): M4(2)⋊15D4  (C2×C8)⋊11D4  M4(2)⋊17D4  C242D4  C248D4  C402D4  C408D4  C562D4 ...
C8⋊D4 is a maximal quotient of
C24.76D4  C233SD16
 C8⋊D4p: C8⋊D8  C83D12  C83D20  C83D28 ...
 C23.D4p: C24.83D4  C242D4  C402D4  C562D4 ...
 C4⋊C4.D2p: C8⋊SD16  C83SD16  C8⋊Q16  C42.248C23  C42.249C23  C42.250C23  C42.251C23  C24.86D4 ...
 (C2×C8).D2p: C24.67D4  C24.75D4  C2.(C8⋊D4)  C2.(C82D4)  C4.(C4×Q8)  C8⋊(C22⋊C4)  (C2×C4)⋊5SD16  C24.89D4 ...

Matrix representation of C8⋊D4 in GL6(ℤ)

100000
010000
000010
00-1-1-1-2
000100
001001
,
0-10000
100000
001000
000-100
001112
00-100-1
,
100000
0-10000
001000
000-100
00-1-1-1-2
000101

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,1,0,0,0,-1,1,0,0,0,1,-1,0,0,0,0,0,-2,0,1],[0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,1,-1,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,2,-1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,-1,0,0,0,0,-1,-1,1,0,0,0,0,-1,0,0,0,0,0,-2,1] >;

C8⋊D4 in GAP, Magma, Sage, TeX

C_8\rtimes D_4
% in TeX

G:=Group("C8:D4");
// GroupNames label

G:=SmallGroup(64,149);
// by ID

G=gap.SmallGroup(64,149);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,332,963,117]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations

Export

Character table of C8⋊D4 in TeX

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