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G = D42Q8order 64 = 26

2nd semidirect product of D4 and Q8 acting via Q8/C4=C2

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

Aliases: D42Q8, C4.10SD16, C42.24C22, C4⋊Q84C2, C4⋊C810C2, C4.Q87C2, (C4×D4).8C2, (C2×C4).34D4, C4.13(C2×Q8), D4⋊C4.5C2, C4.25(C4○D4), C4⋊C4.13C22, (C2×C8).34C22, C2.11(C2×SD16), C22.97(C2×D4), C2.14(C8⋊C22), (C2×C4).101C23, (C2×D4).58C22, C2.14(C22⋊Q8), SmallGroup(64,157)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D42Q8
C1C2C4C2×C4C2×D4C4×D4 — D42Q8
C1C2C2×C4 — D42Q8
C1C22C42 — D42Q8
C1C2C2C2×C4 — D42Q8

Generators and relations for D42Q8
 G = < a,b,c,d | a4=b2=c4=1, d2=c2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab, dcd-1=c-1 >

4C2
4C2
2C22
2C4
2C22
4C4
4C22
4C4
4C22
4C4
2C23
2C8
2C2×C4
2C2×C4
2C2×C4
2D4
2C8
4Q8
4Q8
4C2×C4
4C2×C4
2C22×C4
2C22⋊C4
2C2×Q8
2C4⋊C4

Character table of D42Q8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 1111442222444884444
ρ11111111111111111111    trivial
ρ21111-1-1-111-11-11-111-1-11    linear of order 2
ρ3111111-111-1-1-1-1-11-111-1    linear of order 2
ρ41111-1-11111-11-111-1-1-1-1    linear of order 2
ρ51111-1-11111-11-1-1-11111    linear of order 2
ρ6111111-111-1-1-1-11-11-1-11    linear of order 2
ρ71111-1-1-111-11-111-1-111-1    linear of order 2
ρ81111111111111-1-1-1-1-1-1    linear of order 2
ρ92222002-2-220-20000000    orthogonal lifted from D4
ρ10222200-2-2-2-2020000000    orthogonal lifted from D4
ρ112-22-2-220-220000000000    symplectic lifted from Q8, Schur index 2
ρ122-22-22-20-220000000000    symplectic lifted from Q8, Schur index 2
ρ132-22-20002-202i0-2i000000    complex lifted from C4○D4
ρ142-22-20002-20-2i02i000000    complex lifted from C4○D4
ρ152-2-2200-200200000--2-2--2-2    complex lifted from SD16
ρ162-2-2200200-200000--2--2-2-2    complex lifted from SD16
ρ172-2-2200200-200000-2-2--2--2    complex lifted from SD16
ρ182-2-2200-200200000-2--2-2--2    complex lifted from SD16
ρ1944-4-4000000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of D42Q8
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 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 16)(12 15)(17 21)(18 24)(19 23)(20 22)(25 29)(26 32)(27 31)(28 30)
(1 16 5 10)(2 13 6 11)(3 14 7 12)(4 15 8 9)(17 26 23 32)(18 27 24 29)(19 28 21 30)(20 25 22 31)
(1 23 5 17)(2 22 6 20)(3 21 7 19)(4 24 8 18)(9 29 15 27)(10 32 16 26)(11 31 13 25)(12 30 14 28)

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

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

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

D42Q8 is a maximal subgroup of
D42SD16  Q82SD16  D4⋊Q16  C42.195C23  D4.SD16  D4.3Q16  C42.199C23  C811SD16  D4.2SD16  D4.Q16  C83SD16  C42.248C23  C42.254C23  C42.447D4  C42.219D4  C42.448D4  C42.20C23  C42.23C23  C42.222D4  C42.450D4  C42.227D4  C42.234D4  C42.352C23  C42.359C23  C42.279D4  C42.285D4  C42.294D4  C42.295D4  C42.300D4  C42.301D4  C4.2- 1+4  C42.30C23  D47SD16  C42.468C23  C42.41C23  C42.46C23  C42.474C23  C42.480C23  D49SD16  C42.490C23  C42.495C23  C42.498C23  C42.501C23  Q88SD16  C42.507C23  C42.510C23  C42.512C23  C42.514C23  Q8×SD16
 D4p⋊Q8: D86Q8  D84Q8  D123Q8  D12⋊Q8  D125Q8  D203Q8  D20⋊Q8  D205Q8 ...
 C4p.SD16: D4.1Q16  C8.8SD16  C12.38SD16  C20.38SD16  C28.38SD16 ...
 C4p⋊Q8⋊C2: C42.287D4  C42.290D4  C42.423C23  C42.426C23  C42.59C23  C42.64C23  SD16⋊Q8  SD163Q8 ...
D42Q8 is a maximal quotient of
C42.98D4  D4⋊(C4⋊C4)  C4.Q810C4  C42.30Q8  C42.121D4  (C2×C8)⋊Q8  C4⋊C4.106D4  C4.(C4⋊Q8)  (C2×C8).169D4  (C2×C4).23Q16
 D4p⋊Q8: D83Q8  D123Q8  D12⋊Q8  D125Q8  D203Q8  D20⋊Q8  D205Q8  D283Q8 ...
 C4p.SD16: D8.2Q8  C12.38SD16  C20.38SD16  C28.38SD16 ...
 (Cp×D4)⋊Q8: (C2×D4)⋊Q8  D4⋊Dic6  D4⋊Dic10  D4⋊Dic14 ...

Matrix representation of D42Q8 in GL4(𝔽17) generated by

0100
16000
0010
0001
,
01600
16000
00160
00016
,
01600
1000
0001
00160
,
12500
5500
0004
0040
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,16,0,0,0,0,0,16,0,0,0,0,16],[0,1,0,0,16,0,0,0,0,0,0,16,0,0,1,0],[12,5,0,0,5,5,0,0,0,0,0,4,0,0,4,0] >;

D42Q8 in GAP, Magma, Sage, TeX

D_4\rtimes_2Q_8
% in TeX

G:=Group("D4:2Q8");
// GroupNames label

G:=SmallGroup(64,157);
// by ID

G=gap.SmallGroup(64,157);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,240,121,55,362,1444,376,88]);
// Polycyclic

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

Export

Subgroup lattice of D42Q8 in TeX
Character table of D42Q8 in TeX

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