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G = Q8.D4order 64 = 26

2nd non-split extension by Q8 of D4 acting via D4/C4=C2

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

Aliases: Q8.2D4, C42.21C22, C4⋊C86C2, (C4×Q8)⋊4C2, (C2×Q16)⋊3C2, C4.34(C2×D4), (C2×C4).29D4, C2.9(C4○D8), Q8⋊C412C2, D4⋊C4.2C2, C4.44(C4○D4), C4⋊C4.61C22, (C2×C4).92C23, (C2×C8).32C22, (C2×SD16).4C2, C4.4D4.4C2, C22.88(C2×D4), C2.16(C4⋊D4), (C2×D4).14C22, (C2×Q8).10C22, C2.11(C8.C22), SmallGroup(64,145)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q8.D4
C1C2C4C2×C4C2×Q8C4×Q8 — Q8.D4
C1C2C2×C4 — Q8.D4
C1C22C42 — Q8.D4
C1C2C2C2×C4 — Q8.D4

Generators and relations for Q8.D4
 G = < a,b,c,d | a4=c4=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c-1 >

8C2
2C4
2C4
2C4
2C4
4C4
4C22
4C22
4C22
4C4
2C2×C4
2C8
2C2×C4
2Q8
2Q8
2D4
2D4
2C8
2Q8
2C23
2C2×C4
2SD16
2Q16
2C22⋊C4
2C22⋊C4
2C4⋊C4
2Q16
2SD16
2C42

Character table of Q8.D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J8A8B8C8D
 size 1111822224444484444
ρ11111111111111111111    trivial
ρ21111111-1-111-1-1-1-1-1-111    linear of order 2
ρ31111111-1-1-1-111-1-111-1-1    linear of order 2
ρ4111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ51111-1111111111-1-1-1-1-1    linear of order 2
ρ61111-111-1-111-1-1-1111-1-1    linear of order 2
ρ71111-111-1-1-1-111-11-1-111    linear of order 2
ρ81111-11111-1-1-1-11-11111    linear of order 2
ρ92-22-202-200002-2000000    orthogonal lifted from D4
ρ102-22-202-20000-22000000    orthogonal lifted from D4
ρ1122220-2-2-2-20000200000    orthogonal lifted from D4
ρ1222220-2-2220000-200000    orthogonal lifted from D4
ρ132-22-20-22002i-2i00000000    complex lifted from C4○D4
ρ142-22-20-2200-2i2i00000000    complex lifted from C4○D4
ρ152-2-22000-2i2i000000--2-22-2    complex lifted from C4○D8
ρ162-2-22000-2i2i000000-2--2-22    complex lifted from C4○D8
ρ172-2-220002i-2i000000--2-2-22    complex lifted from C4○D8
ρ182-2-220002i-2i000000-2--22-2    complex lifted from C4○D8
ρ1944-4-4000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of Q8.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 11 3 9)(2 10 4 12)(5 13 7 15)(6 16 8 14)(17 25 19 27)(18 28 20 26)(21 29 23 31)(22 32 24 30)
(1 21 5 19)(2 22 6 20)(3 23 7 17)(4 24 8 18)(9 31 15 25)(10 32 16 26)(11 29 13 27)(12 30 14 28)
(1 17 3 19)(2 20 4 18)(5 23 7 21)(6 22 8 24)(9 28 11 26)(10 27 12 25)(13 32 15 30)(14 31 16 29)

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

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

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

Q8.D4 is a maximal subgroup of
Q8.2D12
 C42.D2p: C42.443D4  C42.212D4  C42.446D4  C42.384D4  C42.226D4  C42.229D4  C42.235D4  C42.268D4 ...
 (Cp×Q8).D4: C42.17C23  C42.505C23  C42.506C23  C42.510C23  C42.512C23  C42.516C23  C42.518C23  C42.527C23 ...
 C4⋊C4.D2p: C42.16C23  C42.18C23  C42.19C23  C42.355C23  C42.358C23  C42.359C23  C42.408C23  C42.409C23 ...
Q8.D4 is a maximal quotient of
D4⋊C4⋊C4  C4.68(C4×D4)  C42.31Q8  (C2×C8).24Q8
 (Cp×Q8).D4: Q8.2SD16  Q8.3SD16  Q8.2D8  Q8.2Q16  C42.249C23  C42.251C23  C42.253C23  C42.255C23 ...
 C42.D2p: C42.119D4  C42.36D6  C42.61D6  C42.36D10  C42.61D10  C42.36D14  C42.61D14 ...
 (C2×C8).D2p: (C2×Q8).8Q8  (C2×C4).23D8  Dic6.D4  Dic6.11D4  Dic10.D4  Dic10.11D4  Dic14.D4  Dic14.11D4 ...

Matrix representation of Q8.D4 in GL4(𝔽17) generated by

1000
0100
00115
00116
,
16000
01600
00107
0057
,
161600
2100
0040
0004
,
161600
0100
0040
00413
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[16,0,0,0,0,16,0,0,0,0,10,5,0,0,7,7],[16,2,0,0,16,1,0,0,0,0,4,0,0,0,0,4],[16,0,0,0,16,1,0,0,0,0,4,4,0,0,0,13] >;

Q8.D4 in GAP, Magma, Sage, TeX

Q_8.D_4
% in TeX

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

G:=SmallGroup(64,145);
// by ID

G=gap.SmallGroup(64,145);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,247,362,158,1444,376,88]);
// Polycyclic

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

Export

Subgroup lattice of Q8.D4 in TeX
Character table of Q8.D4 in TeX

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