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G = D8.4D4  order 128 = 27

1st non-split extension by D8 of D4 acting via D4/C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D8.4D4, C4⋊2SD32, C42.138D4, C4⋊C16⋊11C2, (C4×D8).5C2, C8.72(C2×D4), (C2×C8).67D4, C4⋊Q16⋊6C2, (C2×C4).149D8, C2.8(C2×SD32), C8.90(C4○D4), (C4×C8).63C22, (C2×SD32).3C2, C2.Q32⋊10C2, C2.20(C4⋊D8), C4.51(C4⋊D4), C4.16(C8⋊C22), (C2×C8).516C23, (C2×C16).40C22, C2.9(Q32⋊C2), C22.102(C2×D8), (C2×Q16).2C22, (C2×D8).110C22, C2.D8.158C22, (C2×C4).784(C2×D4), SmallGroup(128,940)

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

Derived series
C1 — C2×C8 — D8.4D4
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C2×D8 — C4×D8 — D8.4D4
Lower central
C1 — C2 — C4 — C2×C8 — D8.4D4
Upper central
C1 — C22 — C42 — C4×C8 — D8.4D4
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D8.4D4

Generators and relations for D8.4D4
 G = < a,b,c,d | a8=b2=c4=1, d2=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 220 in 83 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, D8, D8, Q16, C22×C4, C2×D4, C2×Q8, C4×C8, D4⋊C4, C2.D8, C2×C16, SD32, C4×D4, C4⋊Q8, C2×D8, C2×Q16, C2×Q16, C2.Q32, C4⋊C16, C4×D8, C4⋊Q16, C2×SD32, D8.4D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, SD32, C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, C2×SD32, Q32⋊C2, D8.4D4

Character table of D8.4D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111882222488161622224444444444
ρ111111111111111111111111111111    trivial
ρ21111111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-111111-1-1-1-111111111111111    linear of order 2
ρ41111-1-111111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-11-1-11-1111-11111-1-1111-11-1-1-1    linear of order 2
ρ61111-1-11-1-11-111-111111-1-1-1-1-11-1111    linear of order 2
ρ71111111-1-11-1-1-1-111111-1-1111-11-1-1-1    linear of order 2
ρ81111111-1-11-1-1-11-11111-1-1-1-1-11-1111    linear of order 2
ρ92222002-2-22-20000-2-2-2-22200000000    orthogonal lifted from D4
ρ102-2-222-2200-200000-222-20000000000    orthogonal lifted from D4
ρ11222200222220000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ122-2-22-22200-200000-222-20000000000    orthogonal lifted from D4
ρ13222200-2-2-2-220000000000√2-√2√2√2-√2-√2-√2√2    orthogonal lifted from D8
ρ14222200-222-2-20000000000-√2√2-√2√2√2-√2-√2√2    orthogonal lifted from D8
ρ15222200-222-2-20000000000√2-√2√2-√2-√2√2√2-√2    orthogonal lifted from D8
ρ16222200-2-2-2-220000000000-√2√2-√2-√2√2√2√2-√2    orthogonal lifted from D8
ρ172-2-2200200-20-2i2i002-2-220000000000    complex lifted from C4○D4
ρ182-2-2200200-202i-2i002-2-220000000000    complex lifted from C4○D4
ρ192-22-2000-22000000√2-√2√2-√2-√2√2ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ165+ζ163ζ1613+ζ1611ζ1615+ζ169ζ167+ζ16ζ1613+ζ1611    complex lifted from SD32
ρ202-22-2000-22000000√2-√2√2-√2-√2√2ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ1613+ζ1611ζ165+ζ163ζ167+ζ16ζ1615+ζ169ζ165+ζ163    complex lifted from SD32
ρ212-22-2000-22000000-√2√2-√2√2√2-√2ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ1615+ζ169ζ167+ζ16ζ1613+ζ1611ζ165+ζ163ζ167+ζ16    complex lifted from SD32
ρ222-22-2000-22000000-√2√2-√2√2√2-√2ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ167+ζ16ζ1615+ζ169ζ165+ζ163ζ1613+ζ1611ζ1615+ζ169    complex lifted from SD32
ρ232-22-20002-2000000√2-√2√2-√2√2-√2ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ165+ζ163ζ1615+ζ169ζ167+ζ16ζ1613+ζ1611    complex lifted from SD32
ρ242-22-20002-2000000√2-√2√2-√2√2-√2ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ1613+ζ1611ζ167+ζ16ζ1615+ζ169ζ165+ζ163    complex lifted from SD32
ρ252-22-20002-2000000-√2√2-√2√2-√2√2ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ167+ζ16ζ165+ζ163ζ1613+ζ1611ζ1615+ζ169    complex lifted from SD32
ρ262-22-20002-2000000-√2√2-√2√2-√2√2ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ1615+ζ169ζ1613+ζ1611ζ165+ζ163ζ167+ζ16    complex lifted from SD32
ρ274-4-4400-40040000000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-400000000000-2√2-2√22√22√20000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2944-4-4000000000002√22√2-2√2-2√20000000000    symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of D8.4D4
►On 64 points
Generators in S64
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(1 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 39)(34 38)(35 37)(41 47)(42 46)(43 45)(49 55)(50 54)(51 53)(57 63)(58 62)(59 61)

(1 27 11 19)(2 28 12 20)(3 29 13 21)(4 30 14 22)(5 31 15 23)(6 32 16 24)(7 25 9 17)(8 26 10 18)(33 49 41 57)(34 50 42 58)(35 51 43 59)(36 52 44 60)(37 53 45 61)(38 54 46 62)(39 55 47 63)(40 56 48 64)

(1 57 5 61)(2 64 6 60)(3 63 7 59)(4 62 8 58)(9 51 13 55)(10 50 14 54)(11 49 15 53)(12 56 16 52)(17 35 21 39)(18 34 22 38)(19 33 23 37)(20 40 24 36)(25 43 29 47)(26 42 30 46)(27 41 31 45)(28 48 32 44)
 
G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,39)(34,38)(35,37)(41,47)(42,46)(43,45)(49,55)(50,54)(51,53)(57,63)(58,62)(59,61), (1,27,11,19)(2,28,12,20)(3,29,13,21)(4,30,14,22)(5,31,15,23)(6,32,16,24)(7,25,9,17)(8,26,10,18)(33,49,41,57)(34,50,42,58)(35,51,43,59)(36,52,44,60)(37,53,45,61)(38,54,46,62)(39,55,47,63)(40,56,48,64), (1,57,5,61)(2,64,6,60)(3,63,7,59)(4,62,8,58)(9,51,13,55)(10,50,14,54)(11,49,15,53)(12,56,16,52)(17,35,21,39)(18,34,22,38)(19,33,23,37)(20,40,24,36)(25,43,29,47)(26,42,30,46)(27,41,31,45)(28,48,32,44)>;
 
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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,39)(34,38)(35,37)(41,47)(42,46)(43,45)(49,55)(50,54)(51,53)(57,63)(58,62)(59,61), (1,27,11,19)(2,28,12,20)(3,29,13,21)(4,30,14,22)(5,31,15,23)(6,32,16,24)(7,25,9,17)(8,26,10,18)(33,49,41,57)(34,50,42,58)(35,51,43,59)(36,52,44,60)(37,53,45,61)(38,54,46,62)(39,55,47,63)(40,56,48,64), (1,57,5,61)(2,64,6,60)(3,63,7,59)(4,62,8,58)(9,51,13,55)(10,50,14,54)(11,49,15,53)(12,56,16,52)(17,35,21,39)(18,34,22,38)(19,33,23,37)(20,40,24,36)(25,43,29,47)(26,42,30,46)(27,41,31,45)(28,48,32,44) );
 
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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,16),(14,15),(17,20),(18,19),(21,24),(22,23),(25,28),(26,27),(29,32),(30,31),(33,39),(34,38),(35,37),(41,47),(42,46),(43,45),(49,55),(50,54),(51,53),(57,63),(58,62),(59,61)], [(1,27,11,19),(2,28,12,20),(3,29,13,21),(4,30,14,22),(5,31,15,23),(6,32,16,24),(7,25,9,17),(8,26,10,18),(33,49,41,57),(34,50,42,58),(35,51,43,59),(36,52,44,60),(37,53,45,61),(38,54,46,62),(39,55,47,63),(40,56,48,64)], [(1,57,5,61),(2,64,6,60),(3,63,7,59),(4,62,8,58),(9,51,13,55),(10,50,14,54),(11,49,15,53),(12,56,16,52),(17,35,21,39),(18,34,22,38),(19,33,23,37),(20,40,24,36),(25,43,29,47),(26,42,30,46),(27,41,31,45),(28,48,32,44)]])
 

Matrix representation of D8.4D4 ►in GL4(𝔽17) generated by

31400
3300
00160
00016
,
31400
141400
00160
0001
,
16000
01600
00130
0004
,
7100
11000
0008
00150
G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,16,0,0,0,0,16],[3,14,0,0,14,14,0,0,0,0,16,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,4],[7,1,0,0,1,10,0,0,0,0,0,15,0,0,8,0] >;
 

D8.4D4 in GAP, Magma, Sage, TeX 

D_8._4D_4
 
% in TeX
 
G:=Group("D8.4D4");
 
// GroupNames label
 
G:=SmallGroup(128,940);
 
// by ID
 
G=gap.SmallGroup(128,940);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,64,422,1684,438,242,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Character table of D8.4D4 in TeX

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