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G = C3×Q8⋊3D6  order 288 = 25·32

Direct product of C3 and Q8⋊3D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×Q8⋊3D6, D24⋊6C6, C24⋊13D6, C8⋊3(S3×C6), C24⋊3(C2×C6), D4⋊S3⋊3C6, (S3×D4)⋊3C6, Q8⋊4(S3×C6), C8⋊S3⋊1C6, D12⋊2(C2×C6), (C3×Q8)⋊15D6, D6.7(C3×D4), D4.3(S3×C6), C6.31(C6×D4), (C3×D24)⋊14C2, Q8⋊2S3⋊2C6, Q8⋊3S3⋊4C6, (C3×SD16)⋊5S3, (C3×SD16)⋊1C6, SD16⋊1(C3×S3), (C3×D4).26D6, (S3×C6).43D4, C6.191(S3×D4), (C3×C24)⋊10C22, C12.5(C22×C6), Dic3.9(C3×D4), (C3×D12)⋊11C22, C32⋊20(C8⋊C22), (C3×C12).76C23, (C3×Dic3).46D4, (C32×SD16)⋊1C2, (Q8×C32)⋊6C22, (S3×C12).27C22, C12.156(C22×S3), (D4×C32).13C22, C3⋊C8⋊2(C2×C6), (C3×S3×D4)⋊6C2, C4.5(S3×C2×C6), C2.19(C3×S3×D4), C3⋊3(C3×C8⋊C22), (C3×Q8)⋊3(C2×C6), (C3×C8⋊S3)⋊5C2, (C3×D4⋊S3)⋊12C2, (C3×C3⋊C8)⋊19C22, (C4×S3).2(C2×C6), (C3×D4).3(C2×C6), (C3×Q8⋊2S3)⋊9C2, (C3×Q8⋊3S3)⋊4C2, (C3×C6).219(C2×D4), SmallGroup(288,685)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C3×Q8⋊3D6
Chief series
C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×S3×D4 — C3×Q8⋊3D6
Lower central
C3 — C6 — C12 — C3×Q8⋊3D6
Upper central
C1 — C6 — C12 — C3×SD16

Generators and relations for C3×Q8⋊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, C2×C4, D4, D4, Q8, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, M4(2), D8, SD16, SD16, C2×D4, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C24, C4×S3, C4×S3, D12, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, C8⋊S3, D24, D4⋊S3, Q8⋊2S3, C3×M4(2), C3×D8, C3×SD16, C3×SD16, S3×D4, Q8⋊3S3, C6×D4, C3×C4○D4, C3×C3⋊C8, C3×C24, S3×C12, S3×C12, C3×D12, C3×D12, C3×C3⋊D4, D4×C32, Q8×C32, S3×C2×C6, Q8⋊3D6, C3×C8⋊C22, C3×C8⋊S3, C3×D24, C3×D4⋊S3, C3×Q8⋊2S3, C32×SD16, C3×S3×D4, C3×Q8⋊3S3, C3×Q8⋊3D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, C8⋊C22, S3×C6, S3×D4, C6×D4, S3×C2×C6, Q8⋊3D6, C3×C8⋊C22, C3×S3×D4, C3×Q8⋊3D6

Smallest permutation representation of C3×Q8⋊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+++++++++++++++++
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6S3D4D4D6D6D6C3×S3C3×D4C3×D4S3×C6S3×C6S3×C6C8⋊C22S3×D4Q8⋊3D6C3×C8⋊C22C3×S3×D4C3×Q8⋊3D6
kernelC3×Q8⋊3D6C3×C8⋊S3C3×D24C3×D4⋊S3C3×Q8⋊2S3C32×SD16C3×S3×D4C3×Q8⋊3S3Q8⋊3D6C8⋊S3D24D4⋊S3Q8⋊2S3C3×SD16S3×D4Q8⋊3S3C3×SD16C3×Dic3S3×C6C24C3×D4C3×Q8SD16Dic3D6C8D4Q8C32C6C3C3C2C1
# reps1111111122222222111111222222112224

Matrix representation of C3×Q8⋊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] >;
 

C3×Q8⋊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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