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G = D12⋊19D4  order 192 = 26·3

7th semidirect product of D12 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12⋊19D4, C6.1142+ 1+4, C3⋊3D42, C4⋊3(S3×D4), C4⋊C4⋊23D6, D6⋊7(C2×D4), C12⋊6(C2×D4), C3⋊D4⋊1D4, (C2×D4)⋊23D6, C22⋊3(S3×D4), C4⋊D4⋊12S3, C22⋊C4⋊11D6, Dic3⋊4(C2×D4), (C22×C4)⋊20D6, D6⋊D4⋊13C2, C12⋊D4⋊22C2, Dic3⋊D4⋊20C2, C23⋊2D6⋊10C2, C12⋊3D4⋊17C2, D6⋊C4⋊53C22, (C6×D4)⋊13C22, Dic3⋊5D4⋊20C2, C6.68(C22×D4), C2.28(D4○D12), (C22×D12)⋊15C2, (C2×D12)⋊46C22, (C2×C6).153C24, (C2×C12).40C23, Dic3⋊C4⋊52C22, (S3×C23)⋊10C22, (C22×C12)⋊21C22, (C4×Dic3)⋊22C22, C6.D4⋊51C22, (C22×S3).64C23, C23.191(C22×S3), (C22×C6).188C23, C22.174(S3×C23), (C2×Dic3).227C23, (C2×S3×D4)⋊11C2, (C2×C6)⋊3(C2×D4), C2.41(C2×S3×D4), (C4×C3⋊D4)⋊16C2, (S3×C2×C4)⋊14C22, (C3×C4⋊D4)⋊15C2, (C3×C4⋊C4)⋊11C22, (C2×C3⋊D4)⋊15C22, (C3×C22⋊C4)⋊13C22, (C2×C4).176(C22×S3), SmallGroup(192,1168)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — D12⋊19D4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — D12⋊19D4
Lower central
C3 — C2×C6 — D12⋊19D4
Upper central
C1 — C22 — C4⋊D4

Generators and relations for D12⋊19D4
 G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=dad=a7, cbc-1=dbd=a6b, dcd=c-1 >

Subgroups: 1408 in 428 conjugacy classes, 115 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C22×C6, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C4⋊1D4, C22×D4, C4×Dic3, Dic3⋊C4, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C2×D12, S3×D4, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, S3×C23, D42, D6⋊D4, Dic3⋊D4, Dic3⋊5D4, C12⋊D4, C4×C3⋊D4, C23⋊2D6, C12⋊3D4, C3×C4⋊D4, C22×D12, C2×S3×D4, C2×S3×D4, D12⋊19D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2+ 1+4, S3×D4, S3×C23, D42, C2×S3×D4, D4○D12, D12⋊19D4

Smallest permutation representation of D12⋊19D4
►On 48 points
Generators in S48
(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)

(1 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 38)(39 48)(40 47)(41 46)(42 45)(43 44)

(1 16 44 28)(2 23 45 35)(3 18 46 30)(4 13 47 25)(5 20 48 32)(6 15 37 27)(7 22 38 34)(8 17 39 29)(9 24 40 36)(10 19 41 31)(11 14 42 26)(12 21 43 33)

(1 38)(2 45)(3 40)(4 47)(5 42)(6 37)(7 44)(8 39)(9 46)(10 41)(11 48)(12 43)(14 20)(16 22)(18 24)(26 32)(28 34)(30 36)
 
G:=sub<Sym(48)| (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,30)(26,29)(27,28)(31,36)(32,35)(33,34)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44), (1,16,44,28)(2,23,45,35)(3,18,46,30)(4,13,47,25)(5,20,48,32)(6,15,37,27)(7,22,38,34)(8,17,39,29)(9,24,40,36)(10,19,41,31)(11,14,42,26)(12,21,43,33), (1,38)(2,45)(3,40)(4,47)(5,42)(6,37)(7,44)(8,39)(9,46)(10,41)(11,48)(12,43)(14,20)(16,22)(18,24)(26,32)(28,34)(30,36)>;
 
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,30)(26,29)(27,28)(31,36)(32,35)(33,34)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44), (1,16,44,28)(2,23,45,35)(3,18,46,30)(4,13,47,25)(5,20,48,32)(6,15,37,27)(7,22,38,34)(8,17,39,29)(9,24,40,36)(10,19,41,31)(11,14,42,26)(12,21,43,33), (1,38)(2,45)(3,40)(4,47)(5,42)(6,37)(7,44)(8,39)(9,46)(10,41)(11,48)(12,43)(14,20)(16,22)(18,24)(26,32)(28,34)(30,36) );
 
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22),(25,30),(26,29),(27,28),(31,36),(32,35),(33,34),(37,38),(39,48),(40,47),(41,46),(42,45),(43,44)], [(1,16,44,28),(2,23,45,35),(3,18,46,30),(4,13,47,25),(5,20,48,32),(6,15,37,27),(7,22,38,34),(8,17,39,29),(9,24,40,36),(10,19,41,31),(11,14,42,26),(12,21,43,33)], [(1,38),(2,45),(3,40),(4,47),(5,42),(6,37),(7,44),(8,39),(9,46),(10,41),(11,48),(12,43),(14,20),(16,22),(18,24),(26,32),(28,34),(30,36)]])
 

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2M2N2O 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E6F6G12A12B12C12D12E12F
order122222222···22234444444446666666121212121212
size111122446···612122224446612122224488444488

39 irreducible representations

dim1111111111122222224444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2S3D4D4D6D6D6D62+ 1+4S3×D4S3×D4D4○D12
kernelD12⋊19D4D6⋊D4Dic3⋊D4Dic3⋊5D4C12⋊D4C4×C3⋊D4C23⋊2D6C12⋊3D4C3×C4⋊D4C22×D12C2×S3×D4C4⋊D4D12C3⋊D4C22⋊C4C4⋊C4C22×C4C2×D4C6C4C22C2
# reps1221112111314421131222

Matrix representation of D12⋊19D4 ►in GL6(ℤ)

010000
-100000
000100
00-1100
000010
000001
,
010000
100000
00-1100
000100
000010
000001
,
-100000
010000
001000
000100
000001
0000-10
,
-100000
010000
001000
000100
0000-10
000001

G:=sub<GL(6,Integers())| [0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1] >;
 

D12⋊19D4 in GAP, Magma, Sage, TeX 

D_{12}\rtimes_{19}D_4
 
% in TeX
 
G:=Group("D12:19D4");
 
// GroupNames label
 
G:=SmallGroup(192,1168);
 
// by ID
 
G=gap.SmallGroup(192,1168);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,297,192,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^7,c*b*c^-1=d*b*d=a^6*b,d*c*d=c^-1>;
 
// generators/relations
 

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