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G = C42:12D6  order 192 = 26·3

10th semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42:12D6, C6.962+ 1+4, C4:C4:44D6, (C4xD12):8C2, (C4xC12):6C22, D6:Q8:4C2, C42:2S3:1C2, C42:7S3:6C2, C42:3S3:3C2, D6:C4:40C22, D6.D4:4C2, D6:D4.1C2, C2.8(D4oD12), (C2xC6).71C24, C22:C4.95D6, C42:C2:11S3, D6.15(C4oD4), C4:Dic3:56C22, (C2xDic6):5C22, (C22xC4).208D6, Dic3:4D4:43C2, (C2xC12).146C23, Dic3:C4:32C22, Dic3.D4:4C2, (C4xDic3):50C22, (C2xD12).23C22, C3:2(C22.45C24), C22.18(C4oD12), C23.28D6:27C2, (C22xS3).21C23, (S3xC23).37C22, C23.169(C22xS3), (C22xC6).141C23, C22.100(S3xC23), C6.D4.4C22, (C22xC12).435C22, (C2xDic3).198C23, (C22xDic3).88C22, C4:C4:S3:4C2, (C2xD6:C4):40C2, (S3xC2xC4):45C22, C6.28(C2xC4oD4), C2.10(S3xC4oD4), (S3xC22:C4):26C2, (C3xC4:C4):54C22, C2.30(C2xC4oD12), (C2xC6).41(C4oD4), (C2xC3:D4).9C22, (C3xC42:C2):13C2, (C2xC4).274(C22xS3), (C3xC22:C4).138C22, SmallGroup(192,1086)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC6 — C42:12D6
Chief series
C1 — C3 — C6 — C2xC6 — C22xS3 — S3xC23 — S3xC22:C4 — C42:12D6
Lower central
C3 — C2xC6 — C42:12D6
Upper central
C1 — C22 — C42:C2

Generators and relations for C42:12D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 696 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, C24, Dic6, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C22xS3, C22xS3, C22xC6, C2xC22:C4, C42:C2, C42:C2, C4xD4, C22wrC2, C22:Q8, C22.D4, C4.4D4, C42:2C2, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C4xC12, C3xC22:C4, C3xC4:C4, C2xDic6, S3xC2xC4, C2xD12, C22xDic3, C2xC3:D4, C22xC12, S3xC23, C22.45C24, C42:2S3, C4xD12, C42:7S3, C42:3S3, Dic3.D4, S3xC22:C4, Dic3:4D4, D6:D4, D6.D4, D6:Q8, C4:C4:S3, C2xD6:C4, C23.28D6, C3xC42:C2, C42:12D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C24, C22xS3, C2xC4oD4, 2+ 1+4, C4oD12, S3xC23, C22.45C24, C2xC4oD12, S3xC4oD4, D4oD12, C42:12D6

Smallest permutation representation of C42:12D6
►On 48 points
Generators in S48
(1 31 7 25)(2 35 8 29)(3 33 9 27)(4 34 10 28)(5 32 11 26)(6 36 12 30)(13 43 19 37)(14 47 20 41)(15 45 21 39)(16 46 22 40)(17 44 23 38)(18 48 24 42)

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

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

45 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A···4F4G4H4I4J4K4L4M4N4O6A6B6C6D6E12A12B12C12D12E···12N
order122222222234···4444444444666661212121212···12
size11112266121222···244466121212122224422224···4

45 irreducible representations

dim11111111111111122222222444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2S3D6D6D6D6C4oD4C4oD4C4oD122+ 1+4S3xC4oD4D4oD12
kernelC42:12D6C42:2S3C4xD12C42:7S3C42:3S3Dic3.D4S3xC22:C4Dic3:4D4D6:D4D6.D4D6:Q8C4:C4:S3C2xD6:C4C23.28D6C3xC42:C2C42:C2C42C22:C4C4:C4C22xC4D6C2xC6C22C6C2C2
# reps11111111121111112221448122

Matrix representation of C42:12D6 ►in GL6(F13)

800000
080000
0012000
0001200
000001
000010
,
010000
100000
0012000
0001200
000080
000008
,
100000
010000
00121200
001000
000010
0000012
,
100000
0120000
00121200
000100
0000120
000001

G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1] >;
 

C42:12D6 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{12}D_6
 
% in TeX
 
G:=Group("C4^2:12D6");
 
// GroupNames label
 
G:=SmallGroup(192,1086);
 
// by ID
 
G=gap.SmallGroup(192,1086);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,100,675,136,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
 
// generators/relations
 

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