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G = C42:3Dic3order 192 = 26·3

1st semidirect product of C42 and Dic3 acting via Dic3/C3=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42:3Dic3, C12.19C42, (C4xC12):1C4, (C2xC12).6Q8, (C4xDic3):3C4, (C2xC4).13D12, C12.28(C4:C4), (C2xC4).2Dic6, C4.33(D6:C4), C3:1(C4.9C42), (C2xC12).104D4, C4.24(C4xDic3), (C22xC6).42D4, (C22xC4).73D6, C12.9(C22:C4), C4.8(Dic3:C4), C42:C2.1S3, C4.12(C4:Dic3), C22.17(D6:C4), C23.23(C3:D4), C6.7(C2.C42), C2.8(C6.C42), C22.3(Dic3:C4), C23.26D6.7C2, (C22xC12).120C22, C22.10(C6.D4), (C2xC3:C8):1C4, (C2xC6).3(C4:C4), (C2xC12).56(C2xC4), (C2xC4).139(C4xS3), (C2xC4).20(C3:D4), (C2xC4).72(C2xDic3), (C2xC4.Dic3).7C2, (C2xC6).89(C22:C4), (C3xC42:C2).1C2, SmallGroup(192,90)

Series: Derived Chief Lower central Upper central

C1C12 — C42:3Dic3
C1C3C6C2xC6C22xC6C22xC12C23.26D6 — C42:3Dic3
C3C12 — C42:3Dic3
C1C4C42:C2

Generators and relations for C42:3Dic3
 G = < a,b,c,d | a4=b4=c6=1, d2=c3, ab=ba, cac-1=ab2, dad-1=ab-1, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 216 in 94 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, C23, Dic3, C12, C12, C2xC6, C2xC6, C42, C42, C22:C4, C4:C4, C2xC8, M4(2), C22xC4, C3:C8, C2xDic3, C2xC12, C2xC12, C22xC6, C42:C2, C42:C2, C2xM4(2), C2xC3:C8, C4.Dic3, C4xDic3, C4:Dic3, C6.D4, C4xC12, C3xC22:C4, C3xC4:C4, C22xC12, C4.9C42, C2xC4.Dic3, C23.26D6, C3xC42:C2, C42:3Dic3
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Q8, Dic3, D6, C42, C22:C4, C4:C4, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2.C42, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C4.9C42, C6.C42, C42:3Dic3

Smallest permutation representation of C42:3Dic3
On 48 points
Generators in S48
(1 36 22 42)(2 34 23 40)(3 32 24 38)(4 26 18 47)(5 30 16 45)(6 28 17 43)(7 29 14 44)(8 27 15 48)(9 25 13 46)(10 33 21 39)(11 31 19 37)(12 35 20 41)
(1 6 10 9)(2 4 11 7)(3 5 12 8)(13 22 17 21)(14 23 18 19)(15 24 16 20)(25 36 28 33)(26 31 29 34)(27 32 30 35)(37 44 40 47)(38 45 41 48)(39 46 42 43)
(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 17)(2 16)(3 18)(4 20)(5 19)(6 21)(7 24)(8 23)(9 22)(10 13)(11 15)(12 14)(25 43 28 46)(26 48 29 45)(27 47 30 44)(31 38 34 41)(32 37 35 40)(33 42 36 39)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I4J4K4L4M6A6B6C6D6E8A8B8C8D12A12B12C12D12E···12N
order12222344444444444446666688881212121212···12
size11222211222444412121212222441212121222224···4

42 irreducible representations

dim11111112222222222244
type++++++-+-+-+
imageC1C2C2C2C4C4C4S3D4Q8D4Dic3D6Dic6C4xS3D12C3:D4C3:D4C4.9C42C42:3Dic3
kernelC42:3Dic3C2xC4.Dic3C23.26D6C3xC42:C2C2xC3:C8C4xDic3C4xC12C42:C2C2xC12C2xC12C22xC6C42C22xC4C2xC4C2xC4C2xC4C2xC4C23C3C1
# reps11114441211212422224

Matrix representation of C42:3Dic3 in GL4(F73) generated by

0010
0001
306000
134300
,
46000
04600
00460
00046
,
07200
1100
0001
007272
,
134300
306000
005966
00714
G:=sub<GL(4,GF(73))| [0,0,30,13,0,0,60,43,1,0,0,0,0,1,0,0],[46,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[0,1,0,0,72,1,0,0,0,0,0,72,0,0,1,72],[13,30,0,0,43,60,0,0,0,0,59,7,0,0,66,14] >;

C42:3Dic3 in GAP, Magma, Sage, TeX

C_4^2\rtimes_3{\rm Dic}_3
% in TeX

G:=Group("C4^2:3Dic3");
// GroupNames label

G:=SmallGroup(192,90);
// by ID

G=gap.SmallGroup(192,90);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,1123,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c^3,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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