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G = D8⋊C4order 64 = 26

3rd semidirect product of D8 and C4 acting via C4/C2=C2

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

Aliases: D83C4, C42.11C22, C83(C2×C4), (C4×D4)⋊2C2, D42(C2×C4), C8⋊C42C2, C4.Q83C2, (C2×D8).6C2, C2.17(C4×D4), C4.6(C4○D4), (C2×C4).103D4, D4⋊C416C2, C4⋊C4.55C22, C2.5(C8⋊C22), (C2×C4).78C23, (C2×C8).15C22, C4.14(C22×C4), C22.56(C2×D4), (C2×D4).53C22, 2-Sylow(PSigmaL(2,81)), Aut(SD32), SmallGroup(64,123)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — D8⋊C4
C1C2C22C2×C4C42C4×D4 — D8⋊C4
C1C2C4 — D8⋊C4
C1C22C42 — D8⋊C4
C1C2C2C2×C4 — D8⋊C4

Generators and relations for D8⋊C4
 G = < a,b,c | a8=b2=c4=1, bab=a-1, cac-1=a5, cbc-1=a4b >

Subgroups: 125 in 66 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×8], C8 [×2], C8, C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×4], D4 [×2], C23 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4 [×2], C8⋊C4, D4⋊C4 [×2], C4.Q8, C4×D4 [×2], C2×D8, D8⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, C4×D4, C8⋊C22 [×2], D8⋊C4

Character table of D8⋊C4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D
 size 1111444422222244444444
ρ11111111111111111111111    trivial
ρ21111-1-1-1-1111111-1-1-1-11111    linear of order 2
ρ31111-11-111111111-11-1-1-1-1-1    linear of order 2
ρ411111-11-1111111-11-11-1-1-1-1    linear of order 2
ρ51111-11-111-1-1-1-11-11-111-11-1    linear of order 2
ρ611111-11-11-1-1-1-111-11-11-11-1    linear of order 2
ρ7111111111-1-1-1-11-1-1-1-1-11-11    linear of order 2
ρ81111-1-1-1-11-1-1-1-111111-11-11    linear of order 2
ρ91-11-111-1-1-1-ii-ii1-i-iiii-1-i1    linear of order 4
ρ101-11-1-1-111-1-ii-ii1ii-i-ii-1-i1    linear of order 4
ρ111-11-1-111-1-1-ii-ii1-iii-i-i1i-1    linear of order 4
ρ121-11-11-1-11-1-ii-ii1i-i-ii-i1i-1    linear of order 4
ρ131-11-1-111-1-1i-ii-i1i-i-iii1-i-1    linear of order 4
ρ141-11-11-1-11-1i-ii-i1-iii-ii1-i-1    linear of order 4
ρ151-11-111-1-1-1i-ii-i1ii-i-i-i-1i1    linear of order 4
ρ161-11-1-1-111-1i-ii-i1-i-iii-i-1i1    linear of order 4
ρ1722220000-22-2-22-200000000    orthogonal lifted from D4
ρ1822220000-2-222-2-200000000    orthogonal lifted from D4
ρ192-22-200002-2i-2i2i2i-200000000    complex lifted from C4○D4
ρ202-22-2000022i2i-2i-2i-200000000    complex lifted from C4○D4
ρ2144-4-4000000000000000000    orthogonal lifted from C8⋊C22
ρ224-4-44000000000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of D8⋊C4
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 29)(2 28)(3 27)(4 26)(5 25)(6 32)(7 31)(8 30)(9 19)(10 18)(11 17)(12 24)(13 23)(14 22)(15 21)(16 20)
(1 22 25 14)(2 19 26 11)(3 24 27 16)(4 21 28 13)(5 18 29 10)(6 23 30 15)(7 20 31 12)(8 17 32 9)

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

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

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

D8⋊C4 is a maximal subgroup of
C42.352C23  C42.353C23  C42.356C23  C42.359C23  C42.385C23  C42.388C23  C42.391C23  D89D4  D810D4  C42.41C23  C42.42C23  C42.53C23  C42.54C23  C42.59C23  C42.61C23  C42.507C23  C42.511C23  C42.514C23  C42.517C23  C42.74C23  C42.531C23
 D8p⋊C4: D16⋊C4  D24⋊C4  D249C4  D409C4  D4015C4  D40⋊C4  D56⋊C4  D569C4 ...
 C42.D2p: C42.383D4  C4×C8⋊C22  C42.227D4  C42.229D4  C42.232D4  C42.233D4  C42.257D4  C42.260D4 ...
 D4p⋊(C2×C4): C42.275C23  C42.277C23  C42.281C23  D4⋊S3⋊C4  D4⋊D56C4  D4⋊D7⋊C4 ...
 (C2p×D8).C2: D84Q8  D85Q8  D8⋊Dic3  D8⋊Dic5  D8⋊Dic7 ...
D8⋊C4 is a maximal quotient of
D85C8  D42M4(2)  C83M4(2)  C8⋊C42  C2.D85C4  C4.(C4×Q8)
 D8p⋊C4: D16⋊C4  D24⋊C4  D249C4  D409C4  D4015C4  D40⋊C4  D56⋊C4  D569C4 ...
 C42.D2p: D4⋊C42  Q32⋊C4  C42.48D6  C42.48D10  C42.48D14 ...
 (Cp×D8)⋊C4: (C2×C4)⋊9D8  (C2×D8)⋊10C4  D8⋊Dic3  D8⋊Dic5  D8⋊Dic7 ...
 C2p.(C4×D4): D4⋊(C4⋊C4)  C4.67(C4×D4)  C2.(C82D4)  D4⋊S3⋊C4  D4⋊D56C4  D4⋊D7⋊C4 ...

Matrix representation of D8⋊C4 in GL6(𝔽17)

120000
16160000
000789
005740
0089010
00401210
,
1600000
110000
001000
0011600
0000160
0000161
,
1300000
0130000
000010
000001
0016000
0001600

G:=sub<GL(6,GF(17))| [1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,5,8,4,0,0,7,7,9,0,0,0,8,4,0,12,0,0,9,0,10,10],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,16,16,0,0,0,0,0,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

D8⋊C4 in GAP, Magma, Sage, TeX

D_8\rtimes C_4
% in TeX

G:=Group("D8:C4");
// GroupNames label

G:=SmallGroup(64,123);
// by ID

G=gap.SmallGroup(64,123);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,230,963,489,117]);
// Polycyclic

G:=Group<a,b,c|a^8=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^4*b>;
// generators/relations

Export

Character table of D8⋊C4 in TeX

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