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G = C3xQ8:3D6  order 288 = 25·32

Direct product of C3 and Q8:3D6

direct product, metabelian, supersoluble, monomial

Aliases: C3xQ8:3D6, D24:6C6, C24:13D6, C8:3(S3xC6), C24:3(C2xC6), D4:S3:3C6, (S3xD4):3C6, Q8:4(S3xC6), C8:S3:1C6, D12:2(C2xC6), (C3xQ8):15D6, D6.7(C3xD4), D4.3(S3xC6), C6.31(C6xD4), (C3xD24):14C2, Q8:2S3:2C6, Q8:3S3:4C6, (C3xSD16):5S3, (C3xSD16):1C6, SD16:1(C3xS3), (C3xD4).26D6, (S3xC6).43D4, C6.191(S3xD4), (C3xC24):10C22, C12.5(C22xC6), Dic3.9(C3xD4), (C3xD12):11C22, C32:20(C8:C22), (C3xC12).76C23, (C3xDic3).46D4, (C32xSD16):1C2, (Q8xC32):6C22, (S3xC12).27C22, C12.156(C22xS3), (D4xC32).13C22, C3:C8:2(C2xC6), (C3xS3xD4):6C2, C4.5(S3xC2xC6), C2.19(C3xS3xD4), C3:3(C3xC8:C22), (C3xQ8):3(C2xC6), (C3xC8:S3):5C2, (C3xD4:S3):12C2, (C3xC3:C8):19C22, (C4xS3).2(C2xC6), (C3xD4).3(C2xC6), (C3xQ8:2S3):9C2, (C3xQ8:3S3):4C2, (C3xC6).219(C2xD4), SmallGroup(288,685)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C3xQ8:3D6
Chief series
C1 — C3 — C6 — C12 — C3xC12 — S3xC12 — C3xS3xD4 — C3xQ8:3D6
Lower central
C3 — C6 — C12 — C3xQ8:3D6
Upper central
C1 — C6 — C12 — C3xSD16

Generators and relations for C3xQ8:3D6
 G = < a,b,c,d,e | a3=b4=d6=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 434 in 146 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2xC4, D4, D4, Q8, C23, C32, Dic3, C12, C12, D6, D6, C2xC6, M4(2), D8, SD16, SD16, C2xD4, C4oD4, C3xS3, C3xC6, C3xC6, C3:C8, C24, C24, C4xS3, C4xS3, D12, D12, C3:D4, C2xC12, C3xD4, C3xD4, C3xQ8, C3xQ8, C22xS3, C22xC6, C8:C22, C3xDic3, C3xC12, C3xC12, S3xC6, S3xC6, C62, C8:S3, D24, D4:S3, Q8:2S3, C3xM4(2), C3xD8, C3xSD16, C3xSD16, S3xD4, Q8:3S3, C6xD4, C3xC4oD4, C3xC3:C8, C3xC24, S3xC12, S3xC12, C3xD12, C3xD12, C3xC3:D4, D4xC32, Q8xC32, S3xC2xC6, Q8:3D6, C3xC8:C22, C3xC8:S3, C3xD24, C3xD4:S3, C3xQ8:2S3, C32xSD16, C3xS3xD4, C3xQ8:3S3, C3xQ8:3D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2xC6, C2xD4, C3xS3, C3xD4, C22xS3, C22xC6, C8:C22, S3xC6, S3xD4, C6xD4, S3xC2xC6, Q8:3D6, C3xC8:C22, C3xS3xD4, C3xQ8:3D6

Smallest permutation representation of C3xQ8:3D6
►On 48 points
Generators in S48
(1 2 3)(4 6 5)(7 9 8)(10 11 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 47 45)(44 48 46)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C6A6B6C6D6E6F6G6H6I6J6K6L6M6N6O6P8A8B12A12B12C···12G12H12I12J12K12L24A···24H24I24J
order12222233333444666666666666666688121212···12121212121224···242424
size114612121122224611222446688812121212412224···4668884···41212

54 irreducible representations

dim1111111111111111222222222222444444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6S3D4D4D6D6D6C3xS3C3xD4C3xD4S3xC6S3xC6S3xC6C8:C22S3xD4Q8:3D6C3xC8:C22C3xS3xD4C3xQ8:3D6
kernelC3xQ8:3D6C3xC8:S3C3xD24C3xD4:S3C3xQ8:2S3C32xSD16C3xS3xD4C3xQ8:3S3Q8:3D6C8:S3D24D4:S3Q8:2S3C3xSD16S3xD4Q8:3S3C3xSD16C3xDic3S3xC6C24C3xD4C3xQ8SD16Dic3D6C8D4Q8C32C6C3C3C2C1
# reps1111111122222222111111222222112224

Matrix representation of C3xQ8:3D6 ►in GL8(F73)

640000000
064000000
006400000
000640000
00001000
00000100
00000010
00000001
,
720000000
072000000
007200000
000720000
000017100
000017200
000000722
000000721
,
01000000
10000000
00010000
00100000
00000010
00000001
000072000
000007200
,
80000000
065000000
006400000
00090000
00001000
000017200
000000171
000000072
,
006400000
00090000
80000000
065000000
00003704429
00003736220
0000044361
00005144037

G:=sub<GL(8,GF(73))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,2,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[8,0,0,0,0,0,0,0,0,65,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,71,72],[0,0,8,0,0,0,0,0,0,0,0,65,0,0,0,0,64,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,37,37,0,51,0,0,0,0,0,36,44,44,0,0,0,0,44,22,36,0,0,0,0,0,29,0,1,37] >;
 

C3xQ8:3D6 in GAP, Magma, Sage, TeX 

C_3\times Q_8\rtimes_3D_6
 
% in TeX
 
G:=Group("C3xQ8:3D6");
 
// GroupNames label
 
G:=SmallGroup(288,685);
 
// by ID
 
G=gap.SmallGroup(288,685);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,1094,303,268,1271,648,102,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^4=d^6=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
 
// generators/relations
 

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