Copied to
clipboard

G = C3×D4○D12order 288 = 25·32

Direct product of C3 and D4○D12

direct product, metabelian, supersoluble, monomial

Aliases: C3×D4○D12, C3292+ 1+4, C62.151C23, (S3×D4)⋊5C6, D48(S3×C6), Q89(S3×C6), C4○D128C6, (C3×D4)⋊26D6, (C2×C12)⋊18D6, (C3×Q8)⋊26D6, D1211(C2×C6), (C6×D12)⋊20C2, (C2×D12)⋊13C6, Q83S38C6, (C6×C12)⋊12C22, Dic612(C2×C6), C6.12(C23×C6), C6.80(S3×C23), (C3×C6).49C24, D6.6(C22×C6), (S3×C12)⋊13C22, (C3×D12)⋊37C22, (S3×C6).34C23, C12.26(C22×C6), C32(C3×2+ 1+4), (C3×C12).127C23, C12.177(C22×S3), (C3×Dic6)⋊39C22, (D4×C32)⋊22C22, Dic3.7(C22×C6), (Q8×C32)⋊20C22, (C3×Dic3).35C23, (C2×C4)⋊4(S3×C6), (C3×S3×D4)⋊12C2, C4.33(S3×C2×C6), (C4×S3)⋊2(C2×C6), (C2×C12)⋊5(C2×C6), (C3×C4○D4)⋊8C6, C4○D47(C3×S3), (C3×D4)⋊9(C2×C6), C3⋊D45(C2×C6), C22.3(S3×C2×C6), (C3×C4○D4)⋊12S3, (S3×C2×C6)⋊15C22, (C3×Q8)⋊10(C2×C6), C2.13(S3×C22×C6), (C3×C4○D12)⋊18C2, (C22×S3)⋊4(C2×C6), (C32×C4○D4)⋊6C2, (C2×C6).4(C22×C6), (C3×Q83S3)⋊12C2, (C3×C3⋊D4)⋊18C22, (C2×C6).23(C22×S3), SmallGroup(288,999)

Series: Derived Chief Lower central Upper central

C1C6 — C3×D4○D12
C1C3C6C3×C6S3×C6S3×C2×C6C3×S3×D4 — C3×D4○D12
C3C6 — C3×D4○D12
C1C6C3×C4○D4

Generators and relations for C3×D4○D12
 G = < a,b,c,d,e | a3=b4=c2=e2=1, d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >

Subgroups: 834 in 352 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2 [×9], C3 [×2], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], S3 [×6], C6 [×2], C6 [×16], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×15], Q8, Q8, C23 [×6], C32, Dic3 [×2], C12 [×2], C12 [×6], C12 [×6], D6 [×6], D6 [×6], C2×C6 [×6], C2×C6 [×15], C2×D4 [×9], C4○D4, C4○D4 [×5], C3×S3 [×6], C3×C6, C3×C6 [×3], Dic6, C4×S3 [×6], D12 [×9], C3⋊D4 [×6], C2×C12 [×6], C2×C12 [×9], C3×D4 [×6], C3×D4 [×18], C3×Q8 [×2], C3×Q8 [×2], C22×S3 [×6], C22×C6 [×6], 2+ 1+4, C3×Dic3 [×2], C3×C12, C3×C12 [×3], S3×C6 [×6], S3×C6 [×6], C62 [×3], C2×D12 [×3], C4○D12 [×3], S3×D4 [×6], Q83S3 [×2], C6×D4 [×9], C3×C4○D4 [×2], C3×C4○D4 [×6], C3×Dic6, S3×C12 [×6], C3×D12 [×9], C3×C3⋊D4 [×6], C6×C12 [×3], D4×C32 [×3], Q8×C32, S3×C2×C6 [×6], D4○D12, C3×2+ 1+4, C6×D12 [×3], C3×C4○D12 [×3], C3×S3×D4 [×6], C3×Q83S3 [×2], C32×C4○D4, C3×D4○D12
Quotients: C1, C2 [×15], C3, C22 [×35], S3, C6 [×15], C23 [×15], D6 [×7], C2×C6 [×35], C24, C3×S3, C22×S3 [×7], C22×C6 [×15], 2+ 1+4, S3×C6 [×7], S3×C23, C23×C6, S3×C2×C6 [×7], D4○D12, C3×2+ 1+4, S3×C22×C6, C3×D4○D12

Smallest permutation representation of C3×D4○D12
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 13 7 19)(2 14 8 20)(3 15 9 21)(4 16 10 22)(5 17 11 23)(6 18 12 24)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 37 34 43)(29 38 35 44)(30 39 36 45)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 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)(2 26)(3 25)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)

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

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

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

81 conjugacy classes

class 1 2A2B2C2D2E···2J3A3B3C3D3E4A4B4C4D4E4F6A6B6C···6K6L···6T6U···6AF12A···12N12O···12W12X12Y12Z12AA
order122222···233333444444666···66···66···612···1212···1212121212
size112226···611222222266112···24···46···62···24···46666

81 irreducible representations

dim111111111111222222224444
type++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D6D6D6C3×S3S3×C6S3×C6S3×C62+ 1+4D4○D12C3×2+ 1+4C3×D4○D12
kernelC3×D4○D12C6×D12C3×C4○D12C3×S3×D4C3×Q83S3C32×C4○D4D4○D12C2×D12C4○D12S3×D4Q83S3C3×C4○D4C3×C4○D4C2×C12C3×D4C3×Q8C4○D4C2×C4D4Q8C32C3C3C1
# reps1336212661242133126621224

Matrix representation of C3×D4○D12 in GL4(𝔽13) generated by

3000
0300
0030
0003
,
12200
12100
00122
00121
,
1000
11200
0010
00112
,
7000
0700
0020
0002
,
0020
0002
7000
0700
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[12,12,0,0,2,1,0,0,0,0,12,12,0,0,2,1],[1,1,0,0,0,12,0,0,0,0,1,1,0,0,0,12],[7,0,0,0,0,7,0,0,0,0,2,0,0,0,0,2],[0,0,7,0,0,0,0,7,2,0,0,0,0,2,0,0] >;

C3×D4○D12 in GAP, Magma, Sage, TeX

C_3\times D_4\circ D_{12}
% in TeX

G:=Group("C3xD4oD12");
// GroupNames label

G:=SmallGroup(288,999);
// by ID

G=gap.SmallGroup(288,999);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,555,1571,192,9414]);
// Polycyclic

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

׿
×
𝔽