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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⋊C4○2D8, (C4×D8)⋊24C2, (C8×D4)⋊10C2, C8.79(C2×D4), D4⋊5D4⋊6C2, D4⋊D4⋊5C2, C8⋊7D4⋊21C2, C4⋊C8⋊68C22, (C4×C8)⋊18C22, C4⋊C4.253D4, D4.26(C2×D4), D4.2D4⋊5C2, (C2×D4).227D4, C2.42(D4○D8), C8.18D4⋊9C2, C8.12D4⋊7C2, (C4×D4)⋊22C22, (C22×D8)⋊13C2, C22⋊2(C4○D8), C4.94(C22×D4), C2.D8⋊60C22, C22⋊SD16⋊32C2, C4⋊C4.219C23, C4⋊D4⋊13C22, C22⋊C8⋊61C22, (C2×C8).343C23, (C2×C4).478C24, (C22×C8)⋊18C22, C22⋊C4.192D4, Q8⋊C4⋊8C22, (C2×Q16)⋊47C22, C23.464(C2×D4), D4⋊C4⋊82C22, (C2×SD16)⋊48C22, (C2×D8).174C22, (C2×D4).417C23, C4.4D4⋊15C22, (C2×Q8).199C23, C22⋊Q8.64C22, C22.738(C22×D4), (C22×C4).1122C23, (C22×D4).402C22, C22⋊C4○(C2×D8), (C2×C4○D8)⋊9C2, C2.52(C2×C4○D8), (C2×C4).917(C2×D4), (C2×C4○D4)⋊17C22, SmallGroup(128,2012)

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

Derived series
C1 — C2×C4 — D8⋊12D4
Chief series
C1 — C2 — C22 — C2×C4 — C2×D4 — C22×D4 — C22×D8 — D8⋊12D4
Lower central
C1 — C2 — C2×C4 — D8⋊12D4
Upper central
C1 — C22 — C4×D4 — D8⋊12D4
Jennings
C1 — C2 — C2 — C2×C4 — 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, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C2×C22⋊C4, C4×D4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×C8, C2×D8, C2×D8, C2×D8, C2×SD16, C2×Q16, C4○D8, C22×D4, C2×C4○D4, C8×D4, C4×D8, D4⋊D4, C22⋊SD16, D4.2D4, C8⋊7D4, C8.18D4, C8.12D4, D4⋊5D4, C22×D8, C2×C4○D8, D8⋊12D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4○D8, C22×D4, 2+ 1+4, D42, C2×C4○D8, D4○D8, 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++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D4C4○D82+ 1+4D4○D8
kernelD8⋊12D4C8×D4C4×D8D4⋊D4C22⋊SD16D4.2D4C8⋊7D4C8.18D4C8.12D4D4⋊5D4C22×D8C2×C4○D8C22⋊C4C4⋊C4D8C2×D4C22C4C2
# 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
 

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