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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, (C8xD4):8C2, (C4xD8):11C2, C4.44(C2xD8), D4:5(C4oD4), C22:D8:9C2, D4:6D4:7C2, C8:7D4:11C2, C4:D8:13C2, C4:C8:62C22, C4:C4.258D4, (C4xC8):12C22, (C2xD8):9C22, C4:Q8:20C22, D4o2(D4:C4), C22.4(C2xD8), D4:Q8:12C2, (C2xD4).350D4, C4.4D8:15C2, (C4xD4):23C22, C22:C4.98D4, C2.19(C22xD8), C2.D8:10C22, D4:C4:6C22, C4:C4.398C23, C4:D4:14C22, C22:C8:55C22, (C2xC8).181C23, (C2xC4).486C24, (C22xC8):12C22, C22.D8:7C2, C23.469(C2xD4), C2.66(D4oSD16), (C2xD4).219C23, C4:1D4.82C22, C2.122(D4:5D4), C22.746(C22xD4), (C22xC4).1130C23, (C22xD4).405C22, (C2xD4)o(D4:C4), (C2xC4:C4):55C22, C4.211(C2xC4oD4), (C2xC4).163(C2xD4), (C2xD4:C4):25C2, SmallGroup(128,2026)

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

Derived series
C1 — C2xC4 — D4:4D8
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C22xD4 — D42 — D4:4D8
Lower central
C1 — C2 — C2xC4 — D4:4D8
Upper central
C1 — C22 — C4xD4 — D4:4D8
Jennings
C1 — C2 — C2 — C2xC4 — 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, C2xC4, C2xC4, C2xC4, D4, D4, Q8, C23, C23, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, C2xC8, D8, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C4xC8, C22:C8, D4:C4, D4:C4, C4:C8, C2.D8, C2.D8, C2xC4:C4, C4xD4, C22wrC2, C4:D4, C4:D4, C22:Q8, C22.D4, C4:1D4, C4:Q8, C22xC8, C2xD8, C2xD8, C22xD4, C22xD4, C2xC4oD4, C2xD4:C4, C8xD4, C4xD8, C22:D8, C4:D8, C8:7D4, D4:Q8, C22.D8, C4.4D8, D42, D4:6D4, D4:4D8
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C4oD4, C24, C2xD8, C22xD4, C2xC4oD4, 2+ 1+4, D4:5D4, C22xD8, D4oSD16, 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+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D8C4oD42+ 1+4D4oSD16
kernelD4:4D8C2xD4:C4C8xD4C4xD8C22:D8C4:D8C8:7D4D4:Q8C22.D8C4.4D8D42D4:6D4C22:C4C4:C4C2xD4D4D4C4C2
# 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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