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G = C3×Q83D6order 288 = 25·32

Direct product of C3 and Q83D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×Q83D6, D246C6, C2413D6, C83(S3×C6), C243(C2×C6), D4⋊S33C6, (S3×D4)⋊3C6, Q84(S3×C6), C8⋊S31C6, D122(C2×C6), (C3×Q8)⋊15D6, D6.7(C3×D4), D4.3(S3×C6), C6.31(C6×D4), (C3×D24)⋊14C2, Q82S32C6, Q83S34C6, (C3×SD16)⋊5S3, (C3×SD16)⋊1C6, SD161(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, C3220(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⋊C82(C2×C6), (C3×S3×D4)⋊6C2, C4.5(S3×C2×C6), C2.19(C3×S3×D4), C33(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×Q82S3)⋊9C2, (C3×Q83S3)⋊4C2, (C3×C6).219(C2×D4), SmallGroup(288,685)

Series: Derived Chief Lower central Upper central

C1C12 — C3×Q83D6
C1C3C6C12C3×C12S3×C12C3×S3×D4 — C3×Q83D6
C3C6C12 — C3×Q83D6
C1C6C12C3×SD16

Generators and relations for C3×Q83D6
 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 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×3], C6 [×2], C6 [×7], C8, C8, C2×C4 [×2], D4, D4 [×4], Q8, C23, C32, Dic3, C12 [×2], C12 [×5], D6, D6 [×4], C2×C6 [×8], M4(2), D8 [×2], SD16, SD16, C2×D4, C4○D4, C3×S3 [×3], C3×C6, C3×C6, C3⋊C8, C24 [×2], C24 [×2], C4×S3, C4×S3, D12 [×2], D12, C3⋊D4, C2×C12 [×2], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6 [×4], C62, C8⋊S3, D24, D4⋊S3, Q82S3, C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C3×SD16 [×2], S3×D4, Q83S3, C6×D4, C3×C4○D4, C3×C3⋊C8, C3×C24, S3×C12, S3×C12, C3×D12 [×2], C3×D12, C3×C3⋊D4, D4×C32, Q8×C32, S3×C2×C6, Q83D6, C3×C8⋊C22, C3×C8⋊S3, C3×D24, C3×D4⋊S3, C3×Q82S3, C32×SD16, C3×S3×D4, C3×Q83S3, C3×Q83D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3×D4 [×2], C22×S3, C22×C6, C8⋊C22, S3×C6 [×3], S3×D4, C6×D4, S3×C2×C6, Q83D6, C3×C8⋊C22, C3×S3×D4, C3×Q83D6

Smallest permutation representation of C3×Q83D6
On 48 points
Generators in S48
(1 2 3)(4 6 5)(7 9 8)(10 11 12)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 31 12 34)(2 35 10 32)(3 33 11 36)(4 25 9 28)(5 29 7 26)(6 27 8 30)(13 16 20 23)(14 24 21 17)(15 18 22 19)(37 43 46 40)(38 41 47 44)(39 45 48 42)
(1 43 12 40)(2 47 10 38)(3 45 11 42)(4 17 9 24)(5 15 7 22)(6 13 8 20)(14 25 21 28)(16 27 23 30)(18 29 19 26)(31 37 34 46)(32 44 35 41)(33 39 36 48)
(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 5)(2 4)(3 6)(7 12)(8 11)(9 10)(13 39)(14 38)(15 37)(16 42)(17 41)(18 40)(19 43)(20 48)(21 47)(22 46)(23 45)(24 44)(25 32)(26 31)(27 36)(28 35)(29 34)(30 33)

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

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

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

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×D4Q83D6C3×C8⋊C22C3×S3×D4C3×Q83D6
kernelC3×Q83D6C3×C8⋊S3C3×D24C3×D4⋊S3C3×Q82S3C32×SD16C3×S3×D4C3×Q83S3Q83D6C8⋊S3D24D4⋊S3Q82S3C3×SD16S3×D4Q83S3C3×SD16C3×Dic3S3×C6C24C3×D4C3×Q8SD16Dic3D6C8D4Q8C32C6C3C3C2C1
# reps1111111122222222111111222222112224

Matrix representation of C3×Q83D6 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×Q83D6 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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