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G = D4⋊4D8  order 128 = 27

1st semidirect product of D4 and D8 acting through Inn(D4)

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

Aliases: D4⋊4D8, C42.459C23, C4.392+ 1+4, D42⋊8C2, (C8×D4)⋊8C2, (C4×D8)⋊11C2, C4.44(C2×D8), D4⋊5(C4○D4), C22⋊D8⋊9C2, D4⋊6D4⋊7C2, C8⋊7D4⋊11C2, C4⋊D8⋊13C2, C4⋊C8⋊62C22, C4⋊C4.258D4, (C4×C8)⋊12C22, (C2×D8)⋊9C22, C4⋊Q8⋊20C22, D4○2(D4⋊C4), C22.4(C2×D8), D4⋊Q8⋊12C2, (C2×D4).350D4, C4.4D8⋊15C2, (C4×D4)⋊23C22, C22⋊C4.98D4, C2.19(C22×D8), C2.D8⋊10C22, D4⋊C4⋊6C22, C4⋊C4.398C23, C4⋊D4⋊14C22, C22⋊C8⋊55C22, (C2×C8).181C23, (C2×C4).486C24, (C22×C8)⋊12C22, C22.D8⋊7C2, C23.469(C2×D4), C2.66(D4○SD16), (C2×D4).219C23, C4⋊1D4.82C22, C2.122(D4⋊5D4), C22.746(C22×D4), (C22×C4).1130C23, (C22×D4).405C22, (C2×D4)○(D4⋊C4), (C2×C4⋊C4)⋊55C22, C4.211(C2×C4○D4), (C2×C4).163(C2×D4), (C2×D4⋊C4)⋊25C2, SmallGroup(128,2026)

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

Derived series
C1 — C2×C4 — D4⋊4D8
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — D42 — D4⋊4D8
Lower central
C1 — C2 — C2×C4 — D4⋊4D8
Upper central
C1 — C22 — C4×D4 — D4⋊4D8
Jennings
C1 — C2 — C2 — C2×C4 — D4⋊4D8

Generators and relations for D4⋊4D8
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 608 in 247 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, D8, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, C4⋊C8, C2.D8, C2.D8, C2×C4⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊1D4, C4⋊Q8, C22×C8, C2×D8, C2×D8, C22×D4, C22×D4, C2×C4○D4, C2×D4⋊C4, C8×D4, C4×D8, C22⋊D8, C4⋊D8, C8⋊7D4, D4⋊Q8, C22.D8, C4.4D8, D42, D4⋊6D4, D4⋊4D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C24, C2×D8, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, C22×D8, D4○SD16, D4⋊4D8

Smallest permutation representation of D4⋊4D8
►On 32 points
Generators in S32
(1 14 30 19)(2 15 31 20)(3 16 32 21)(4 9 25 22)(5 10 26 23)(6 11 27 24)(7 12 28 17)(8 13 29 18)

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

(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 29)(2 28)(3 27)(4 26)(5 25)(6 32)(7 31)(8 30)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(16 24)
 
G:=sub<Sym(32)| (1,14,30,19)(2,15,31,20)(3,16,32,21)(4,9,25,22)(5,10,26,23)(6,11,27,24)(7,12,28,17)(8,13,29,18), (1,23)(2,11)(3,17)(4,13)(5,19)(6,15)(7,21)(8,9)(10,30)(12,32)(14,26)(16,28)(18,25)(20,27)(22,29)(24,31), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,32)(7,31)(8,30)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(16,24)>;
 
G:=Group( (1,14,30,19)(2,15,31,20)(3,16,32,21)(4,9,25,22)(5,10,26,23)(6,11,27,24)(7,12,28,17)(8,13,29,18), (1,23)(2,11)(3,17)(4,13)(5,19)(6,15)(7,21)(8,9)(10,30)(12,32)(14,26)(16,28)(18,25)(20,27)(22,29)(24,31), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,32)(7,31)(8,30)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(16,24) );
 
G=PermutationGroup([[(1,14,30,19),(2,15,31,20),(3,16,32,21),(4,9,25,22),(5,10,26,23),(6,11,27,24),(7,12,28,17),(8,13,29,18)], [(1,23),(2,11),(3,17),(4,13),(5,19),(6,15),(7,21),(8,9),(10,30),(12,32),(14,26),(16,28),(18,25),(20,27),(22,29),(24,31)], [(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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,32),(7,31),(8,30),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17),(16,24)]])
 

35 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L4A4B4C4D4E···4I4J4K4L8A8B8C8D8E···8J
order122222222222244444···444488888···8
size111122224488822224···488822224···4

35 irreducible representations

dim1111111111112222244
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D8C4○D42+ 1+4D4○SD16
kernelD4⋊4D8C2×D4⋊C4C8×D4C4×D8C22⋊D8C4⋊D8C8⋊7D4D4⋊Q8C22.D8C4.4D8D42D4⋊6D4C22⋊C4C4⋊C4C2×D4D4D4C4C2
# reps1211212121112118412

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

16000
01600
00162
00161
,
1000
0100
00162
0001
,
01100
31100
00138
00134
,
01100
14000
0049
00413
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,16,0,0,2,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,2,1],[0,3,0,0,11,11,0,0,0,0,13,13,0,0,8,4],[0,14,0,0,11,0,0,0,0,0,4,4,0,0,9,13] >;
 

D4⋊4D8 in GAP, Magma, Sage, TeX 

D_4\rtimes_4D_8
 
% in TeX
 
G:=Group("D4:4D8");
 
// GroupNames label
 
G:=SmallGroup(128,2026);
 
// by ID
 
G=gap.SmallGroup(128,2026);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,346,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
 
// generators/relations
 

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