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G = C42:9D6  order 192 = 26·3

7th semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42:9D6, C6.932+ 1+4, C4:C4:54D6, (C4xD12):5C2, (C2xD12):18C4, D12:26(C2xC4), (C4xC12):4C22, C42:C2:6S3, D6:C4:60C22, C2.1(D4oD12), C6.17(C23xC4), (C2xC6).65C24, Dic3:5D4:11C2, D6.4(C22xC4), C4:Dic3:82C22, C22:C4.125D6, (C22xC4).204D6, C12.120(C22xC4), (C2xC12).583C23, C3:2(C22.11C24), (C4xDic3):10C22, (C22xD12).17C2, C22.27(S3xC23), (C2xD12).255C22, C23.26D6:24C2, (S3xC23).35C22, (C22xC6).135C23, C23.163(C22xS3), (C22xS3).162C23, (C22xC12).225C22, (C2xDic3).195C23, C6.D4.94C22, (C2xC4):6(C4xS3), C4.58(S3xC2xC4), (C2xC12):11(C2xC4), (S3xC2xC4):43C22, C22.27(S3xC2xC4), C2.19(S3xC22xC4), (C3xC4:C4):51C22, (S3xC22:C4):25C2, (C22xS3):7(C2xC4), (C3xC42:C2):7C2, (C2xC6).21(C22xC4), (C2xC4).271(C22xS3), (C3xC22:C4).135C22, SmallGroup(192,1080)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C42:9D6
Chief series
C1 — C3 — C6 — C2xC6 — C22xS3 — S3xC23 — C22xD12 — C42:9D6
Lower central
C3 — C6 — C42:9D6
Upper central
C1 — C22 — C42:C2

Generators and relations for C42:9D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, ac=ca, dad=a-1, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 936 in 338 conjugacy classes, 151 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC6, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C24, C4xS3, D12, C2xDic3, C2xC12, C2xC12, C22xS3, C22xS3, C22xC6, C2xC22:C4, C42:C2, C42:C2, C4xD4, C22xD4, C4xDic3, C4:Dic3, D6:C4, C6.D4, C4xC12, C3xC22:C4, C3xC4:C4, S3xC2xC4, C2xD12, C22xC12, S3xC23, C22.11C24, C4xD12, S3xC22:C4, Dic3:5D4, C23.26D6, C3xC42:C2, C22xD12, C42:9D6
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, C22xC4, C24, C4xS3, C22xS3, C23xC4, 2+ 1+4, S3xC2xC4, S3xC23, C22.11C24, S3xC22xC4, D4oD12, C42:9D6

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F···2M 3 4A···4L4M···4T6A6B6C6D6E12A12B12C12D12E···12N
order1222222···234···44···4666661212121212···12
size1111226···622···26···62224422224···4

54 irreducible representations

dim1111111122222244
type++++++++++++++
imageC1C2C2C2C2C2C2C4S3D6D6D6D6C4xS32+ 1+4D4oD12
kernelC42:9D6C4xD12S3xC22:C4Dic3:5D4C23.26D6C3xC42:C2C22xD12C2xD12C42:C2C42C22:C4C4:C4C22xC4C2xC4C6C2
# reps14441111612221824

Matrix representation of C42:9D6 ►in GL6(F13)

1200000
0120000
003600
0071000
000036
0000710
,
800000
080000
0010110
0001011
0000120
0000012
,
12120000
100000
00121200
001000
00121211
0010120
,
12120000
010000
0031000
0071000
00310103
0071063

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,7,0,0,0,0,6,10,0,0,0,0,0,0,3,7,0,0,0,0,6,10],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,11,0,12,0,0,0,0,11,0,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,12,1,0,0,12,0,12,0,0,0,0,0,1,12,0,0,0,0,1,0],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,3,7,3,7,0,0,10,10,10,10,0,0,0,0,10,6,0,0,0,0,3,3] >;
 

C42:9D6 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_9D_6
 
% in TeX
 
G:=Group("C4^2:9D6");
 
// GroupNames label
 
G:=SmallGroup(192,1080);
 
// by ID
 
G=gap.SmallGroup(192,1080);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,570,80,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
 
// generators/relations
 

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