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G = C3×Q8.13D6order 288 = 25·32

Direct product of C3 and Q8.13D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×Q8.13D6, C62.67D4, D4⋊S37C6, C4○D124C6, D4.S37C6, D4.8(S3×C6), C3⋊Q167C6, C6.60(C6×D4), Q82S37C6, (C3×D4).48D6, C12.66(C3×D4), (C3×Q8).73D6, Q8.18(S3×C6), D12.12(C2×C6), (C2×C12).333D6, (C3×C12).174D4, C3224(C4○D8), (C3×C12).89C23, C12.18(C22×C6), Dic6.11(C2×C6), C12.149(C3⋊D4), C12.169(C22×S3), (C6×C12).132C22, (C3×D12).41C22, (C3×Dic6).41C22, (D4×C32).24C22, (Q8×C32).25C22, (C2×C3⋊C8)⋊8C6, (C6×C3⋊C8)⋊14C2, C35(C3×C4○D8), C4.18(S3×C2×C6), C4○D46(C3×S3), (C3×C4○D4)⋊6C6, C3⋊C8.10(C2×C6), (C3×D4⋊S3)⋊15C2, (C3×C4○D4)⋊11S3, (C3×C4○D12)⋊8C2, (C2×C4).59(S3×C6), (C3×D4).8(C2×C6), (C2×C6).10(C3×D4), C4.32(C3×C3⋊D4), C2.24(C6×C3⋊D4), (C2×C12).43(C2×C6), (C3×D4.S3)⋊15C2, (C3×C3⋊Q16)⋊15C2, (C3×C6).268(C2×D4), (C32×C4○D4)⋊2C2, C6.161(C2×C3⋊D4), (C3×C3⋊C8).41C22, (C3×Q8).20(C2×C6), C22.1(C3×C3⋊D4), (C3×Q82S3)⋊15C2, (C2×C6).29(C3⋊D4), SmallGroup(288,721)

Series: Derived Chief Lower central Upper central

C1C12 — C3×Q8.13D6
C1C3C6C12C3×C12C3×D12C3×C4○D12 — C3×Q8.13D6
C3C6C12 — C3×Q8.13D6
C1C12C2×C12C3×C4○D4

Generators and relations for C3×Q8.13D6
 G = < a,b,c,d,e | a3=b4=1, c2=d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d5 >

Subgroups: 322 in 144 conjugacy classes, 58 normal (all characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3, C6 [×2], C6 [×9], C8 [×2], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8, C32, Dic3, C12 [×4], C12 [×7], D6, C2×C6 [×2], C2×C6 [×6], C2×C8, D8, SD16 [×2], Q16, C4○D4, C4○D4, C3×S3, C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12 [×2], C2×C12 [×6], C3×D4 [×2], C3×D4 [×7], C3×Q8 [×2], C3×Q8 [×2], C4○D8, C3×Dic3, C3×C12 [×2], C3×C12, S3×C6, C62, C62, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C2×C24, C3×D8, C3×SD16 [×2], C3×Q16, C4○D12, C3×C4○D4 [×2], C3×C4○D4 [×2], C3×C3⋊C8 [×2], C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q8.13D6, C3×C4○D8, C6×C3⋊C8, C3×D4⋊S3, C3×D4.S3, C3×Q82S3, C3×C3⋊Q16, C3×C4○D12, C32×C4○D4, C3×Q8.13D6
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], C3×D4 [×2], C22×S3, C22×C6, C4○D8, S3×C6 [×3], C2×C3⋊D4, C6×D4, C3×C3⋊D4 [×2], S3×C2×C6, Q8.13D6, C3×C4○D8, C6×C3⋊D4, C3×Q8.13D6

Smallest permutation representation of C3×Q8.13D6
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 4 7 10)(2 5 8 11)(3 6 9 12)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)
(1 17 7 23)(2 18 8 24)(3 19 9 13)(4 20 10 14)(5 21 11 15)(6 22 12 16)(25 37 31 43)(26 38 32 44)(27 39 33 45)(28 40 34 46)(29 41 35 47)(30 42 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 27 7 33)(2 32 8 26)(3 25 9 31)(4 30 10 36)(5 35 11 29)(6 28 12 34)(13 46 19 40)(14 39 20 45)(15 44 21 38)(16 37 22 43)(17 42 23 48)(18 47 24 41)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E4A4B4C4D4E6A6B6C···6G6H···6R6S6T8A8B8C8D12A12B12C12D12E···12L12M···12W12X12Y24A···24H
order122223333344444666···66···66688881212121212···1212···12121224···24
size11241211222112412112···24···41212666611112···24···412126···6

72 irreducible representations

dim111111111111111122222222222222222244
type++++++++++++++
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6S3D4D4D6D6D6C3×S3C3⋊D4C3×D4C3⋊D4C3×D4C4○D8S3×C6S3×C6S3×C6C3×C3⋊D4C3×C3⋊D4C3×C4○D8Q8.13D6C3×Q8.13D6
kernelC3×Q8.13D6C6×C3⋊C8C3×D4⋊S3C3×D4.S3C3×Q82S3C3×C3⋊Q16C3×C4○D12C32×C4○D4Q8.13D6C2×C3⋊C8D4⋊S3D4.S3Q82S3C3⋊Q16C4○D12C3×C4○D4C3×C4○D4C3×C12C62C2×C12C3×D4C3×Q8C4○D4C12C12C2×C6C2×C6C32C2×C4D4Q8C4C22C3C3C1
# reps111111112222222211111122222422244824

Matrix representation of C3×Q8.13D6 in GL4(𝔽73) generated by

64000
06400
0010
0001
,
1000
0100
00270
00046
,
1000
0100
0001
00720
,
8000
06400
00270
00027
,
06400
8000
00063
00220
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,27,0,0,0,0,46],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[8,0,0,0,0,64,0,0,0,0,27,0,0,0,0,27],[0,8,0,0,64,0,0,0,0,0,0,22,0,0,63,0] >;

C3×Q8.13D6 in GAP, Magma, Sage, TeX

C_3\times Q_8._{13}D_6
% in TeX

G:=Group("C3xQ8.13D6");
// GroupNames label

G:=SmallGroup(288,721);
// by ID

G=gap.SmallGroup(288,721);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,2524,648,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=1,c^2=d^6=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^5>;
// generators/relations

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