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G = D4.2D4  order 64 = 26

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

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

Aliases: D4.2D4, C42.20C22, C4⋊C8⋊5C2, (C4×D4)⋊4C2, (C2×D8).2C2, (C2×C4).28D4, C4.33(C2×D4), Q8⋊C4⋊6C2, C2.8(C4○D8), C4.4D4⋊3C2, D4⋊C4⋊11C2, (C2×SD16)⋊12C2, C4.43(C4○D4), C4⋊C4.60C22, (C2×C4).91C23, (C2×C8).31C22, C22.87(C2×D4), (C2×Q8).9C22, C2.15(C4⋊D4), C2.11(C8⋊C22), (C2×D4).13C22, SmallGroup(64,144)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — D4.2D4
Chief series
C1 — C2 — C4 — C2×C4 — C2×D4 — C4×D4 — D4.2D4
Lower central
C1 — C2 — C2×C4 — D4.2D4
Upper central
C1 — C22 — C42 — D4.2D4
Jennings
C1 — C2 — C2 — C2×C4 — D4.2D4

Generators and relations for D4.2D4
 G = < a,b,c,d | a4=b2=c4=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 125 in 62 conjugacy classes, 27 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, D4.2D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C4⋊D4, C4○D8, C8⋊C22, D4.2D4

Character table of D4.2D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H8A8B8C8D
 size 1111448222244484444
ρ11111111111111111111    trivial
ρ2111111-11111111-1-1-1-1-1    linear of order 2
ρ31111-1-1-1-111-11-11111-1-1    linear of order 2
ρ41111-1-11-111-11-11-1-1-111    linear of order 2
ρ51111-1-111111-11-11-1-1-1-1    linear of order 2
ρ61111-1-1-11111-11-1-11111    linear of order 2
ρ7111111-1-111-1-1-1-11-1-111    linear of order 2
ρ81111111-111-1-1-1-1-111-1-1    linear of order 2
ρ92222000-2-2-2-202000000    orthogonal lifted from D4
ρ1022220002-2-220-2000000    orthogonal lifted from D4
ρ112-22-2-22002-2000000000    orthogonal lifted from D4
ρ122-22-22-2002-2000000000    orthogonal lifted from D4
ρ132-22-20000-2202i0-2i00000    complex lifted from C4○D4
ρ142-22-20000-220-2i02i00000    complex lifted from C4○D4
ρ152-2-22000-2i002i0000√2-√2-√-2√-2    complex lifted from C4○D8
ρ162-2-220002i00-2i0000-√2√2-√-2√-2    complex lifted from C4○D8
ρ172-2-220002i00-2i0000√2-√2√-2-√-2    complex lifted from C4○D8
ρ182-2-22000-2i002i0000-√2√2√-2-√-2    complex lifted from C4○D8
ρ1944-4-4000000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of D4.2D4
►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 12)(2 11)(3 10)(4 9)(5 14)(6 13)(7 16)(8 15)(17 26)(18 25)(19 28)(20 27)(21 30)(22 29)(23 32)(24 31)

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

D4.2D4 is a maximal subgroup of
 C42.D2p: C42.443D4  C42.211D4  C42.446D4  C42.384D4  C42.225D4  C42.229D4  C42.232D4  C42.265D4 ...
 (Cp×D4).D4: C42.15C23  D8⋊9D4  SD16⋊D4  D8⋊10D4  SD16⋊8D4  D8⋊5D4  SD16⋊2D4  D8⋊12D4 ...
 C4⋊C4.D2p: C42.14C23  C42.18C23  C42.19C23  C42.352C23  C42.358C23  C42.359C23  C42.406C23  C42.407C23 ...
D4.2D4 is a maximal quotient of
 C4.67(C4×D4)  C2.(C4×Q16)  C42.31Q8  (C2×C8).24Q8
 D4.D4p: D4.2D8  D4.1D12  D4.1D20  D4.1D28 ...
 (Cp×D4).D4: D4.2SD16  D4.3SD16  D4.Q16  C42.248C23  C42.250C23  C42.252C23  C42.254C23  C42.100D4 ...
 C42.D2p: C42.119D4  D12.19D4  D12.23D4  D20.19D4  D20.23D4  D28.19D4  D28.23D4 ...
 (C2×C8).D2p: C4⋊C4.106D4  (C2×C8).52D4  D12.D4  D12.12D4  D20.D4  D20.12D4  D28.D4  D28.12D4 ...

Matrix representation of D4.2D4 ►in GL4(𝔽17) generated by

0100
16000
0010
0001
,
14300
3300
00160
00016
,
13000
01300
0001
00160
,
13000
0400
0001
0010
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[14,3,0,0,3,3,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,0,13,0,0,0,0,0,16,0,0,1,0],[13,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0] >;
 

D4.2D4 in GAP, Magma, Sage, TeX 

D_4._2D_4
 
% in TeX
 
G:=Group("D4.2D4");
 
// GroupNames label
 
G:=SmallGroup(64,144);
 
// by ID
 
G=gap.SmallGroup(64,144);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,121,247,362,1444,376,88]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^4=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
 
// generators/relations
 

Export

Character table of D4.2D4 in TeX

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