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G = C3×D4.D6order 288 = 25·32

Direct product of C3 and D4.D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×D4.D6, C24.42D6, Dic126C6, C8.2(S3×C6), (S3×Q8)⋊5C6, C8⋊S32C6, C24.9(C2×C6), D42S3.C6, D4.S34C6, C3⋊Q161C6, D4.4(S3×C6), D6.8(C3×D4), C6.32(C6×D4), SD162(C3×S3), (C3×SD16)⋊6S3, (C3×SD16)⋊2C6, (S3×C6).44D4, (C3×D4).27D6, C6.192(S3×D4), (C3×Q8).48D6, Q8.11(S3×C6), C12.6(C22×C6), Dic6.2(C2×C6), (C3×Dic12)⋊14C2, (C3×C24).32C22, (C3×C12).77C23, Dic3.10(C3×D4), (C3×Dic3).47D4, (C32×SD16)⋊2C2, (S3×C12).28C22, C12.157(C22×S3), C3218(C8.C22), (C3×Dic6).25C22, (D4×C32).14C22, (Q8×C32).11C22, C4.6(S3×C2×C6), (C3×S3×Q8)⋊5C2, C3⋊C8.1(C2×C6), C2.20(C3×S3×D4), (C3×C8⋊S3)⋊6C2, (C4×S3).3(C2×C6), (C3×D4).4(C2×C6), C32(C3×C8.C22), (C3×Q8).6(C2×C6), (C3×D4.S3)⋊10C2, (C3×C3⋊Q16)⋊11C2, (C3×C6).220(C2×D4), (C3×C3⋊C8).20C22, (C3×D42S3).2C2, SmallGroup(288,686)

Series: Derived Chief Lower central Upper central

C1C12 — C3×D4.D6
C1C3C6C12C3×C12S3×C12C3×S3×Q8 — C3×D4.D6
C3C6C12 — C3×D4.D6
C1C6C12C3×SD16

Generators and relations for C3×D4.D6
 G = < a,b,c,d,e | a3=b4=c2=1, d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b-1c, ede-1=d5 >

Subgroups: 306 in 130 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×4], C22 [×2], S3, C6 [×2], C6 [×5], C8, C8, C2×C4 [×3], D4, D4, Q8, Q8 [×3], C32, Dic3, Dic3 [×2], C12 [×2], C12 [×7], D6, C2×C6 [×4], M4(2), SD16, SD16, Q16 [×2], C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6 [×2], Dic6, C4×S3, C4×S3, C2×Dic3, C3⋊D4, C2×C12 [×3], C3×D4 [×2], C3×D4 [×2], C3×Q8 [×2], C3×Q8 [×4], C8.C22, C3×Dic3, C3×Dic3 [×2], C3×C12, C3×C12, S3×C6, C62, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×M4(2), C3×SD16 [×2], C3×SD16 [×2], C3×Q16 [×2], D42S3, S3×Q8, C6×Q8, C3×C4○D4, C3×C3⋊C8, C3×C24, C3×Dic6 [×2], C3×Dic6, S3×C12, S3×C12, C6×Dic3, C3×C3⋊D4, D4×C32, Q8×C32, D4.D6, C3×C8.C22, C3×C8⋊S3, C3×Dic12, C3×D4.S3, C3×C3⋊Q16, C32×SD16, C3×D42S3, C3×S3×Q8, C3×D4.D6
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, D4.D6, C3×C8.C22, C3×S3×D4, C3×D4.D6

Smallest permutation representation of C3×D4.D6
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 27 7 33)(2 34 8 28)(3 29 9 35)(4 36 10 30)(5 31 11 25)(6 26 12 32)(13 44 19 38)(14 39 20 45)(15 46 21 40)(16 41 22 47)(17 48 23 42)(18 43 24 37)
(1 33)(2 8)(3 35)(4 10)(5 25)(6 12)(7 27)(9 29)(11 31)(13 38)(14 20)(15 40)(16 22)(17 42)(18 24)(19 44)(21 46)(23 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 37 7 43)(2 42 8 48)(3 47 9 41)(4 40 10 46)(5 45 11 39)(6 38 12 44)(13 26 19 32)(14 31 20 25)(15 36 21 30)(16 29 22 35)(17 34 23 28)(18 27 24 33)

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

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

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

54 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J6K6L8A8B12A12B12C···12G12H12I12J12K12L12M12N12O12P24A···24H24I24J
order1222333334444466666666666688121212···1212121212121212121224···242424
size1146112222461212112224466888412224···466888121212124···41212

54 irreducible representations

dim1111111111111111222222222222444444
type++++++++++++++-+-
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6S3D4D4D6D6D6C3×S3C3×D4C3×D4S3×C6S3×C6S3×C6C8.C22S3×D4D4.D6C3×C8.C22C3×S3×D4C3×D4.D6
kernelC3×D4.D6C3×C8⋊S3C3×Dic12C3×D4.S3C3×C3⋊Q16C32×SD16C3×D42S3C3×S3×Q8D4.D6C8⋊S3Dic12D4.S3C3⋊Q16C3×SD16D42S3S3×Q8C3×SD16C3×Dic3S3×C6C24C3×D4C3×Q8SD16Dic3D6C8D4Q8C32C6C3C3C2C1
# reps1111111122222222111111222222112224

Matrix representation of C3×D4.D6 in GL4(𝔽73) generated by

8000
0800
0080
0008
,
0100
72000
0001
00720
,
0100
1000
00072
00720
,
195400
545400
004825
002525
,
004825
002525
195400
545400
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,72,0,0,1,0,0,0,0,0,0,72,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,72,0,0,72,0],[19,54,0,0,54,54,0,0,0,0,48,25,0,0,25,25],[0,0,19,54,0,0,54,54,48,25,0,0,25,25,0,0] >;

C3×D4.D6 in GAP, Magma, Sage, TeX

C_3\times D_4.D_6
% in TeX

G:=Group("C3xD4.D6");
// GroupNames label

G:=SmallGroup(288,686);
// by ID

G=gap.SmallGroup(288,686);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,1094,303,1271,648,102,9414]);
// Polycyclic

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

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