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G = D84D4order 128 = 27

3rd semidirect product of D8 and D4 acting via D4/C4=C2

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

Aliases: D84D4, C42.439C23, C4.1262+ 1+4, D425C2, C82(C2×D4), C2.56D42, D44(C2×D4), (C4×D8)⋊31C2, C86D43C2, C85D48C2, D46D44C2, C4⋊D835C2, C8⋊D430C2, C4⋊C828C22, C42(C8⋊C22), (C4×C8)⋊29C22, C4⋊C4.359D4, C4⋊Q816C22, D4⋊D437C2, (C2×D4).309D4, (C4×D4)⋊19C22, (C2×D8)⋊29C22, C22⋊C4.42D4, C4.86(C22×D4), C2.D871C22, D4.D419C2, C22⋊SD1618C2, C4⋊C4.211C23, C22⋊C824C22, (C2×C4).470C24, (C2×C8).283C23, C23.312(C2×D4), C22⋊Q811C22, D4⋊C481C22, C2.60(D4○SD16), Q8⋊C440C22, (C2×SD16)⋊46C22, (C2×D4).210C23, C4⋊D4.61C22, C41D4.76C22, (C2×Q8).194C23, (C2×M4(2))⋊22C22, (C22×C4).320C23, C22.730(C22×D4), (C22×D4).399C22, (C2×C8⋊C22)⋊30C2, (C2×C4).154(C2×D4), C2.72(C2×C8⋊C22), (C2×C4○D4)⋊15C22, SmallGroup(128,2004)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D84D4
C1C2C22C2×C4C2×D4C22×D4C2×C8⋊C22 — D84D4
C1C2C2×C4 — D84D4
C1C22C4×D4 — D84D4
C1C2C2C2×C4 — D84D4

Generators and relations for D84D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a-1, dad=a3, cbc-1=a6b, dbd=a2b, dcd=c-1 >

Subgroups: 648 in 273 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×28], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×4], D4 [×23], Q8 [×4], C23 [×2], C23 [×14], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], D8 [×4], D8 [×4], SD16 [×10], C22×C4 [×2], C22×C4 [×4], C2×D4 [×3], C2×D4 [×2], C2×D4 [×18], C2×Q8 [×2], C4○D4 [×6], C24 [×2], C4×C8, C22⋊C8 [×2], D4⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8, C2.D8, C2×C4⋊C4, C4×D4 [×3], C22≀C2 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×D8, C2×D8 [×2], C2×SD16 [×6], C8⋊C22 [×8], C22×D4 [×2], C22×D4, C2×C4○D4 [×2], C86D4, C4×D8, D4⋊D4 [×2], C22⋊SD16 [×2], C4⋊D8, D4.D4, C8⋊D4 [×2], C85D4, D42, D46D4, C2×C8⋊C22 [×2], D84D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C8⋊C22 [×2], C22×D4 [×2], 2+ 1+4, D42, C2×C8⋊C22, D4○SD16, D84D4

Character table of D84D4

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F
 size 11114444448822224448888444488
ρ111111111111111111111111111111    trivial
ρ2111111-1-11-11-1-11-111-1-11-11-1-1-111-11    linear of order 2
ρ311111-111111-1-11-11-1-11-11-1-111-1-1-11    linear of order 2
ρ411111-1-1-11-1111111-11-1-1-1-11-1-1-1-111    linear of order 2
ρ51111-1111-1-1-11-11-111-1-1-11-11-1-111-11    linear of order 2
ρ61111-11-1-1-11-1-11111111-1-1-1-1111111    linear of order 2
ρ71111-1-111-1-1-1-11111-11-1111-1-1-1-1-111    linear of order 2
ρ81111-1-1-1-1-11-11-11-11-1-111-11111-1-1-11    linear of order 2
ρ91111-11-1-1-1-11-1-11-111-1-111-1111-1-11-1    linear of order 2
ρ101111-1111-111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ111111-1-1-1-1-1-1111111-11-1-111-11111-1-1    linear of order 2
ρ121111-1-111-111-1-11-11-1-11-1-111-1-1111-1    linear of order 2
ρ13111111-1-111-1-11111111-1111-1-1-1-1-1-1    linear of order 2
ρ14111111111-1-11-11-111-1-1-1-11-111-1-11-1    linear of order 2
ρ1511111-1-1-111-11-11-11-1-1111-1-1-1-1111-1    linear of order 2
ρ1611111-1111-1-1-11111-11-11-1-111111-1-1    linear of order 2
ρ172-22-2202-2-20000-2020000000-220000    orthogonal lifted from D4
ρ182-22-2-20-2220000-2020000000-220000    orthogonal lifted from D4
ρ1922220-2000-2002-22-22-220000000000    orthogonal lifted from D4
ρ2022220-2000200-2-2-2-222-20000000000    orthogonal lifted from D4
ρ212-22-2-202-220000-20200000002-20000    orthogonal lifted from D4
ρ22222202000-200-2-2-2-2-2220000000000    orthogonal lifted from D4
ρ232-22-220-22-20000-20200000002-20000    orthogonal lifted from D4
ρ242222020002002-22-2-2-2-20000000000    orthogonal lifted from D4
ρ254-4-4400000000-40400000000000000    orthogonal lifted from C8⋊C22
ρ264-4-440000000040-400000000000000    orthogonal lifted from C8⋊C22
ρ274-44-400000000040-40000000000000    orthogonal lifted from 2+ 1+4
ρ2844-4-40000000000000000000002-2-2-200    complex lifted from D4○SD16
ρ2944-4-4000000000000000000000-2-22-200    complex lifted from D4○SD16

Smallest permutation representation of D84D4
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 20)(2 19)(3 18)(4 17)(5 24)(6 23)(7 22)(8 21)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)
(1 30 21 13)(2 29 22 12)(3 28 23 11)(4 27 24 10)(5 26 17 9)(6 25 18 16)(7 32 19 15)(8 31 20 14)
(1 13)(2 16)(3 11)(4 14)(5 9)(6 12)(7 15)(8 10)(17 26)(18 29)(19 32)(20 27)(21 30)(22 25)(23 28)(24 31)

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

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

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

Matrix representation of D84D4 in GL6(ℤ)

100000
010000
000010
00000-1
000-100
00-1000
,
100000
010000
000010
000001
001000
000100
,
010000
-100000
00-1000
000100
000001
000010
,
010000
100000
00-1000
000100
00000-1
0000-10

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

D84D4 in GAP, Magma, Sage, TeX

D_8\rtimes_4D_4
% in TeX

G:=Group("D8:4D4");
// GroupNames label

G:=SmallGroup(128,2004);
// by ID

G=gap.SmallGroup(128,2004);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,723,346,2804,1411,375,172]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^3,c*b*c^-1=a^6*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of D84D4 in TeX

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