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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(S3xD4), C4:C4:23D6, D6:7(C2xD4), C12:6(C2xD4), C3:D4:1D4, (C2xD4):23D6, C22:3(S3xD4), C4:D4:12S3, C22:C4:11D6, Dic3:4(C2xD4), (C22xC4):20D6, D6:D4:13C2, C12:D4:22C2, Dic3:D4:20C2, C23:2D6:10C2, C12:3D4:17C2, D6:C4:53C22, (C6xD4):13C22, Dic3:5D4:20C2, C6.68(C22xD4), C2.28(D4oD12), (C22xD12):15C2, (C2xD12):46C22, (C2xC6).153C24, (C2xC12).40C23, Dic3:C4:52C22, (S3xC23):10C22, (C22xC12):21C22, (C4xDic3):22C22, C6.D4:51C22, (C22xS3).64C23, C23.191(C22xS3), (C22xC6).188C23, C22.174(S3xC23), (C2xDic3).227C23, (C2xS3xD4):11C2, (C2xC6):3(C2xD4), C2.41(C2xS3xD4), (C4xC3:D4):16C2, (S3xC2xC4):14C22, (C3xC4:D4):15C2, (C3xC4:C4):11C22, (C2xC3:D4):15C22, (C3xC22:C4):13C22, (C2xC4).176(C22xS3), SmallGroup(192,1168)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC6 — D12:19D4
Chief series
C1 — C3 — C6 — C2xC6 — C22xS3 — S3xC23 — C2xS3xD4 — D12:19D4
Lower central
C3 — C2xC6 — 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, C2xC4, C2xC4, C2xC4, D4, C23, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C24, C4xS3, D12, D12, C2xDic3, C3:D4, C3:D4, C2xC12, C2xC12, C2xC12, C3xD4, C22xS3, C22xS3, C22xS3, C22xC6, C22xC6, C4xD4, C22wrC2, C4:D4, C4:D4, C4:1D4, C22xD4, C4xDic3, Dic3:C4, D6:C4, D6:C4, C6.D4, C3xC22:C4, C3xC4:C4, S3xC2xC4, S3xC2xC4, C2xD12, C2xD12, C2xD12, S3xD4, C2xC3:D4, C2xC3:D4, C22xC12, C6xD4, C6xD4, S3xC23, D42, D6:D4, Dic3:D4, Dic3:5D4, C12:D4, C4xC3:D4, C23:2D6, C12:3D4, C3xC4:D4, C22xD12, C2xS3xD4, C2xS3xD4, D12:19D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C24, C22xS3, C22xD4, 2+ 1+4, S3xD4, S3xC23, D42, C2xS3xD4, D4oD12, 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+4S3xD4S3xD4D4oD12
kernelD12:19D4D6:D4Dic3:D4Dic3:5D4C12:D4C4xC3:D4C23:2D6C12:3D4C3xC4:D4C22xD12C2xS3xD4C4:D4D12C3:D4C22:C4C4:C4C22xC4C2xD4C6C4C22C2
# 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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