Copied to
clipboard

G = D8:12D4  order 128 = 27

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

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

Aliases: D8:12D4, C42.447C23, C4.1342+ 1+4, C2.64D42, C22:C4o2D8, (C4xD8):24C2, (C8xD4):10C2, C8.79(C2xD4), D4:5D4:6C2, D4:D4:5C2, C8:7D4:21C2, C4:C8:68C22, (C4xC8):18C22, C4:C4.253D4, D4.26(C2xD4), D4.2D4:5C2, (C2xD4).227D4, C2.42(D4oD8), C8.18D4:9C2, C8.12D4:7C2, (C4xD4):22C22, (C22xD8):13C2, C22:2(C4oD8), C4.94(C22xD4), C2.D8:60C22, C22:SD16:32C2, C4:C4.219C23, C4:D4:13C22, C22:C8:61C22, (C2xC8).343C23, (C2xC4).478C24, (C22xC8):18C22, C22:C4.192D4, Q8:C4:8C22, (C2xQ16):47C22, C23.464(C2xD4), D4:C4:82C22, (C2xSD16):48C22, (C2xD8).174C22, (C2xD4).417C23, C4.4D4:15C22, (C2xQ8).199C23, C22:Q8.64C22, C22.738(C22xD4), (C22xC4).1122C23, (C22xD4).402C22, C22:C4o(C2xD8), (C2xC4oD8):9C2, C2.52(C2xC4oD8), (C2xC4).917(C2xD4), (C2xC4oD4):17C22, SmallGroup(128,2012)

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

Derived series
C1 — C2xC4 — D8:12D4
Chief series
C1 — C2 — C22 — C2xC4 — C2xD4 — C22xD4 — C22xD8 — D8:12D4
Lower central
C1 — C2 — C2xC4 — D8:12D4
Upper central
C1 — C22 — C4xD4 — D8:12D4
Jennings
C1 — C2 — C2 — C2xC4 — D8:12D4

Generators and relations for D8:12D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 616 in 263 conjugacy classes, 96 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, D4, D4, Q8, C23, C23, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, D8, D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C4xC8, C22:C8, D4:C4, Q8:C4, C4:C8, C2.D8, C2xC22:C4, C4xD4, C4xD4, C22wrC2, C4:D4, C4:D4, C22:Q8, C22.D4, C4.4D4, C22xC8, C2xD8, C2xD8, C2xD8, C2xSD16, C2xQ16, C4oD8, C22xD4, C2xC4oD4, C8xD4, C4xD8, D4:D4, C22:SD16, D4.2D4, C8:7D4, C8.18D4, C8.12D4, D4:5D4, C22xD8, C2xC4oD8, D8:12D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C4oD8, C22xD4, 2+ 1+4, D42, C2xC4oD8, D4oD8, D8:12D4

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F···2J2K2L4A···4F4G4H4I4J4K4L8A8B8C8D8E···8J
order1222222···2224···444444488888···8
size1111224···4882···244888822224···4

35 irreducible representations

dim1111111111112222244
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D4C4oD82+ 1+4D4oD8
kernelD8:12D4C8xD4C4xD8D4:D4C22:SD16D4.2D4C8:7D4C8.18D4C8.12D4D4:5D4C22xD8C2xC4oD8C22:C4C4:C4D8C2xD4C22C4C2
# reps1112221112112141812

Matrix representation of D8:12D4 ►in GL4(F17) generated by

31400
3300
0010
0001
,
31400
141400
00160
00016
,
0400
13000
00016
0010
,
0400
13000
00016
00160
G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,1,0,0,0,0,1],[3,14,0,0,14,14,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,4,0,0,0,0,0,0,1,0,0,16,0],[0,13,0,0,4,0,0,0,0,0,0,16,0,0,16,0] >;
 

D8:12D4 in GAP, Magma, Sage, TeX 

D_8\rtimes_{12}D_4
 
% in TeX
 
G:=Group("D8:12D4");
 
// GroupNames label
 
G:=SmallGroup(128,2012);
 
// by ID
 
G=gap.SmallGroup(128,2012);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,2019,346,2804,1411,375,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
 
// generators/relations
 

׿
x
:
Z
F
o
wr
Q
<