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G = C3×Q8.7D6order 288 = 25·32

Direct product of C3 and Q8.7D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×Q8.7D6, C24.65D6, (S3×C8)⋊5C6, D4⋊S34C6, (S3×C24)⋊8C2, C24⋊C26C6, C8.11(S3×C6), C3⋊Q162C6, D4.5(S3×C6), D6.2(C3×D4), C6.33(C6×D4), D42S33C6, C24.11(C2×C6), Q83S35C6, (C3×SD16)⋊7S3, (C3×SD16)⋊4C6, SD163(C3×S3), D12.3(C2×C6), (C3×D4).28D6, (S3×C6).26D4, C6.193(S3×D4), Q8.12(S3×C6), (C3×Q8).49D6, C3220(C4○D8), C12.7(C22×C6), Dic6.3(C2×C6), (C3×C24).38C22, (C3×C12).78C23, (C3×Dic3).50D4, Dic3.13(C3×D4), (C32×SD16)⋊6C2, (S3×C12).50C22, C12.158(C22×S3), (C3×D12).27C22, (C3×Dic6).26C22, (D4×C32).15C22, (Q8×C32).12C22, C4.7(S3×C2×C6), C33(C3×C4○D8), C3⋊C8.6(C2×C6), C2.21(C3×S3×D4), (C3×D4⋊S3)⋊10C2, (C3×D4).5(C2×C6), (C3×C3⋊Q16)⋊9C2, (C3×C24⋊C2)⋊14C2, (C3×Q8).7(C2×C6), (C3×D42S3)⋊6C2, (C4×S3).10(C2×C6), (C3×Q83S3)⋊5C2, (C3×C6).221(C2×D4), (C3×C3⋊C8).34C22, SmallGroup(288,687)

Series: Derived Chief Lower central Upper central

C1C12 — C3×Q8.7D6
C1C3C6C12C3×C12S3×C12C3×D42S3 — C3×Q8.7D6
C3C6C12 — C3×Q8.7D6
C1C6C12C3×SD16

Generators and relations for C3×Q8.7D6
 G = < a,b,c,d,e | a3=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe-1=b-1, dcd-1=b-1c, ece-1=bc, ede-1=b2d-1 >

Subgroups: 338 in 134 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×2], C6 [×2], C6 [×6], C8, C8, C2×C4 [×3], D4, D4 [×3], Q8, Q8, C32, Dic3, Dic3, C12 [×2], C12 [×6], D6, D6, C2×C6 [×5], C2×C8, D8, SD16, SD16, Q16, C4○D4 [×2], C3×S3 [×2], C3×C6, C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12 [×3], C3×D4 [×2], C3×D4 [×4], C3×Q8 [×2], C3×Q8 [×2], C4○D8, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C2×C24, C3×D8, C3×SD16 [×2], C3×SD16 [×2], C3×Q16, D42S3, Q83S3, C3×C4○D4 [×2], C3×C3⋊C8, C3×C24, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×D12, C6×Dic3, C3×C3⋊D4, D4×C32, Q8×C32, Q8.7D6, C3×C4○D8, S3×C24, C3×C24⋊C2, C3×D4⋊S3, C3×C3⋊Q16, C32×SD16, C3×D42S3, C3×Q83S3, C3×Q8.7D6
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, C4○D8, S3×C6 [×3], S3×D4, C6×D4, S3×C2×C6, Q8.7D6, C3×C4○D8, C3×S3×D4, C3×Q8.7D6

Smallest permutation representation of C3×Q8.7D6
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 23 21)(20 24 22)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 39 41)(38 40 42)(43 47 45)(44 48 46)
(1 19 10 22)(2 23 11 20)(3 21 12 24)(4 13 8 16)(5 17 9 14)(6 15 7 18)(25 28 42 39)(26 40 37 29)(27 30 38 41)(31 45 48 34)(32 35 43 46)(33 47 44 36)
(1 43 10 32)(2 47 11 36)(3 45 12 34)(4 27 8 38)(5 25 9 42)(6 29 7 40)(13 41 16 30)(14 28 17 39)(15 37 18 26)(19 35 22 46)(20 44 23 33)(21 31 24 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 14 10 17)(2 16 11 13)(3 18 12 15)(4 23 8 20)(5 19 9 22)(6 21 7 24)(25 32 42 43)(26 48 37 31)(27 36 38 47)(28 46 39 35)(29 34 40 45)(30 44 41 33)

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

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

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

63 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J6K6L6M6N8A8B8C8D12A12B12C12D12E12F12G···12K12L12M12N12O12P24A24B24C24D24E···24J24K24L24M24N
order12222333334444466666666666666888812121212121212···1212121212122424242424···2424242424
size11461211222233412112224466888121222662233334···4888121222224···46666

63 irreducible representations

dim1111111111111111222222222222224444
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6S3D4D4D6D6D6C3×S3C3×D4C3×D4C4○D8S3×C6S3×C6S3×C6C3×C4○D8S3×D4Q8.7D6C3×S3×D4C3×Q8.7D6
kernelC3×Q8.7D6S3×C24C3×C24⋊C2C3×D4⋊S3C3×C3⋊Q16C32×SD16C3×D42S3C3×Q83S3Q8.7D6S3×C8C24⋊C2D4⋊S3C3⋊Q16C3×SD16D42S3Q83S3C3×SD16C3×Dic3S3×C6C24C3×D4C3×Q8SD16Dic3D6C32C8D4Q8C3C6C3C2C1
# reps1111111122222222111111222422281224

Matrix representation of C3×Q8.7D6 in GL4(𝔽73) generated by

1000
0100
0080
0008
,
17100
17200
0010
0001
,
611200
671200
0010
0001
,
1000
17200
00640
0008
,
271900
04600
0008
00640
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[1,1,0,0,71,72,0,0,0,0,1,0,0,0,0,1],[61,67,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[1,1,0,0,0,72,0,0,0,0,64,0,0,0,0,8],[27,0,0,0,19,46,0,0,0,0,0,64,0,0,8,0] >;

C3×Q8.7D6 in GAP, Magma, Sage, TeX

C_3\times Q_8._7D_6
% in TeX

G:=Group("C3xQ8.7D6");
// GroupNames label

G:=SmallGroup(288,687);
// by ID

G=gap.SmallGroup(288,687);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,1094,303,268,1271,648,102,9414]);
// Polycyclic

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

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