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G = C3×Q83Dic3order 288 = 25·32

Direct product of C3 and Q83Dic3

direct product, metabelian, supersoluble, monomial

Aliases: C3×Q83Dic3, C62.36D4, C3212C4≀C2, (C3×D4)⋊2C12, (C3×Q8)⋊4C12, C12.9(C2×C12), (C3×D4)⋊5Dic3, (C4×Dic3)⋊2C6, Q84(C3×Dic3), D42(C3×Dic3), (C3×Q8)⋊7Dic3, C12.65(C3×D4), (D4×C32)⋊5C4, C4.Dic34C6, C4.3(C6×Dic3), (Q8×C32)⋊5C4, (Dic3×C12)⋊7C2, (C2×C12).319D6, (C3×C12).167D4, (C6×C12).51C22, C12.38(C2×Dic3), C12.148(C3⋊D4), C6.36(C6.D4), C33(C3×C4≀C2), (C2×C6).4(C3×D4), (C2×C4).40(S3×C6), (C3×C4○D4).9C6, C4○D4.5(C3×S3), C4.31(C3×C3⋊D4), (C2×C12).21(C2×C6), (C3×C12).45(C2×C4), (C3×C4○D4).20S3, C6.18(C3×C22⋊C4), C22.3(C3×C3⋊D4), (C2×C6).44(C3⋊D4), (C3×C4.Dic3)⋊20C2, C2.8(C3×C6.D4), (C32×C4○D4).1C2, (C3×C6).69(C22⋊C4), SmallGroup(288,271)

Series: Derived Chief Lower central Upper central

C1C12 — C3×Q83Dic3
C1C3C6C12C2×C12C6×C12C3×C4.Dic3 — C3×Q83Dic3
C3C6C12 — C3×Q83Dic3
C1C12C2×C12C3×C4○D4

Generators and relations for C3×Q83Dic3
 G = < a,b,c,d,e | a3=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >

Subgroups: 226 in 108 conjugacy classes, 42 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22, C6 [×2], C6 [×8], C8, C2×C4, C2×C4 [×2], D4, D4, Q8, C32, Dic3 [×2], C12 [×4], C12 [×8], C2×C6 [×2], C2×C6 [×5], C42, M4(2), C4○D4, C3×C6, C3×C6 [×2], C3⋊C8, C24, C2×Dic3, C2×C12 [×2], C2×C12 [×6], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8, C4≀C2, C3×Dic3 [×2], C3×C12 [×2], C3×C12, C62, C62, C4.Dic3, C4×Dic3, C4×C12, C3×M4(2), C3×C4○D4 [×2], C3×C4○D4, C3×C3⋊C8, C6×Dic3, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q83Dic3, C3×C4≀C2, C3×C4.Dic3, Dic3×C12, C32×C4○D4, C3×Q83Dic3
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], Dic3 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C3×S3, C2×Dic3, C3⋊D4 [×2], C2×C12, C3×D4 [×2], C4≀C2, C3×Dic3 [×2], S3×C6, C6.D4, C3×C22⋊C4, C6×Dic3, C3×C3⋊D4 [×2], Q83Dic3, C3×C4≀C2, C3×C6.D4, C3×Q83Dic3

Smallest permutation representation of C3×Q83Dic3
On 48 points
Generators in S48
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 11 36 13)(2 12 31 14)(3 7 32 15)(4 8 33 16)(5 9 34 17)(6 10 35 18)(19 26 38 47)(20 27 39 48)(21 28 40 43)(22 29 41 44)(23 30 42 45)(24 25 37 46)
(1 16 36 8)(2 9 31 17)(3 18 32 10)(4 11 33 13)(5 14 34 12)(6 7 35 15)(19 41 38 22)(20 23 39 42)(21 37 40 24)(25 43 46 28)(26 29 47 44)(27 45 48 30)
(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 26 4 29)(2 25 5 28)(3 30 6 27)(7 42 10 39)(8 41 11 38)(9 40 12 37)(13 19 16 22)(14 24 17 21)(15 23 18 20)(31 46 34 43)(32 45 35 48)(33 44 36 47)

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

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

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

72 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E4F4G4H6A6B6C···6G6H···6R8A8B12A12B12C12D12E···12L12M···12W12X···12AE24A24B24C24D
order12223333344444444666···66···6881212121212···1212···1212···1224242424
size11241122211246666112···24···4121211112···24···46···612121212

72 irreducible representations

dim11111111111122222222222222222244
type++++++++--
imageC1C2C2C2C3C4C4C6C6C6C12C12S3D4D4D6Dic3Dic3C3×S3C3⋊D4C3×D4C3⋊D4C3×D4C4≀C2S3×C6C3×Dic3C3×Dic3C3×C3⋊D4C3×C3⋊D4C3×C4≀C2Q83Dic3C3×Q83Dic3
kernelC3×Q83Dic3C3×C4.Dic3Dic3×C12C32×C4○D4Q83Dic3D4×C32Q8×C32C4.Dic3C4×Dic3C3×C4○D4C3×D4C3×Q8C3×C4○D4C3×C12C62C2×C12C3×D4C3×Q8C4○D4C12C12C2×C6C2×C6C32C2×C4D4Q8C4C22C3C3C1
# reps11112222224411111122222422244824

Matrix representation of C3×Q83Dic3 in GL4(𝔽73) generated by

1000
0100
00640
00064
,
0100
72000
00720
00072
,
46000
02700
0010
00072
,
04600
27000
00650
0009
,
145900
141400
0001
00720
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[0,72,0,0,1,0,0,0,0,0,72,0,0,0,0,72],[46,0,0,0,0,27,0,0,0,0,1,0,0,0,0,72],[0,27,0,0,46,0,0,0,0,0,65,0,0,0,0,9],[14,14,0,0,59,14,0,0,0,0,0,72,0,0,1,0] >;

C3×Q83Dic3 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes_3{\rm Dic}_3
% in TeX

G:=Group("C3xQ8:3Dic3");
// GroupNames label

G:=SmallGroup(288,271);
// by ID

G=gap.SmallGroup(288,271);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,136,2524,1271,102,9414]);
// Polycyclic

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

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