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G = C3xDic6:S3order 432 = 24·33

Direct product of C3 and Dic6:S3

direct product, metabelian, supersoluble, monomial

Aliases: C3xDic6:S3, C33:9SD16, C12.97S32, C12.27(S3xC6), C32:4C8:9C6, (C3xDic6):1S3, (C3xDic6):1C6, Dic6:2(C3xS3), D12.2(C3xS3), (C3xD12).5C6, (C3xD12).9S3, (C3xC12).110D6, C32:6(C3xSD16), (C32xC6).20D4, (C32xDic6):1C2, (C32xD12).1C2, C6.28(D6:S3), C32:12(D4.S3), (C32xC12).3C22, C32:12(Q8:2S3), C4.9(C3xS32), C3:2(C3xD4.S3), C6.8(C3xC3:D4), C3:2(C3xQ8:2S3), (C3xC6).19(C3xD4), (C3xC12).37(C2xC6), C2.4(C3xD6:S3), (C3xC32:4C8):15C2, (C3xC6).83(C3:D4), SmallGroup(432,420)

Series: Derived Chief Lower central Upper central

C1C3xC12 — C3xDic6:S3
C1C3C32C3xC6C3xC12C32xC12C32xD12 — C3xDic6:S3
C32C3xC6C3xC12 — C3xDic6:S3
C1C6C12

Generators and relations for C3xDic6:S3
 G = < a,b,c,d,e | a3=b12=d3=e2=1, c2=b6, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=b7, cd=dc, ece=b3c, ede=d-1 >

Subgroups: 400 in 122 conjugacy classes, 36 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2xC6, SD16, C3xS3, C3xC6, C3xC6, C3:C8, C24, Dic6, D12, C3xD4, C3xQ8, C33, C3xDic3, C3xC12, C3xC12, S3xC6, C62, D4.S3, Q8:2S3, C3xSD16, S3xC32, C32xC6, C3xC3:C8, C32:4C8, C3xDic6, C3xDic6, C3xD12, C3xD12, D4xC32, Q8xC32, C32xDic3, C32xC12, S3xC3xC6, Dic6:S3, C3xD4.S3, C3xQ8:2S3, C3xC32:4C8, C32xDic6, C32xD12, C3xDic6:S3
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2xC6, SD16, C3xS3, C3:D4, C3xD4, S32, S3xC6, D4.S3, Q8:2S3, C3xSD16, D6:S3, C3xC3:D4, C3xS32, Dic6:S3, C3xD4.S3, C3xQ8:2S3, C3xD6:S3, C3xDic6:S3

Smallest permutation representation of C3xDic6:S3
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(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 45 41)(38 46 42)(39 47 43)(40 48 44)
(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 7 20)(2 13 8 19)(3 24 9 18)(4 23 10 17)(5 22 11 16)(6 21 12 15)(25 37 31 43)(26 48 32 42)(27 47 33 41)(28 46 34 40)(29 45 35 39)(30 44 36 38)
(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 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 28)(2 35)(3 30)(4 25)(5 32)(6 27)(7 34)(8 29)(9 36)(10 31)(11 26)(12 33)(13 48)(14 43)(15 38)(16 45)(17 40)(18 47)(19 42)(20 37)(21 44)(22 39)(23 46)(24 41)

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

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

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

63 conjugacy classes

class 1 2A2B3A3B3C···3H3I3J3K4A4B6A6B6C···6H6I6J6K6L···6S8A8B12A12B12C···12N12O···12V24A24B24C24D
order122333···333344666···66666···688121212···1212···1224242424
size1112112···2444212112···244412···121818224···412···1218181818

63 irreducible representations

dim111111112222222222224444444444
type+++++++++-+-
imageC1C2C2C2C3C6C6C6S3S3D4D6SD16C3xS3C3xS3C3:D4C3xD4S3xC6C3xSD16C3xC3:D4S32D4.S3Q8:2S3D6:S3C3xS32Dic6:S3C3xD4.S3C3xQ8:2S3C3xD6:S3C3xDic6:S3
kernelC3xDic6:S3C3xC32:4C8C32xDic6C32xD12Dic6:S3C32:4C8C3xDic6C3xD12C3xDic6C3xD12C32xC6C3xC12C33Dic6D12C3xC6C3xC6C12C32C6C12C32C32C6C4C3C3C3C2C1
# reps111122221112222424481111222224

Matrix representation of C3xDic6:S3 in GL6(F73)

100000
010000
008000
000800
000010
000001
,
1710000
1720000
001000
000100
0000072
0000172
,
19220000
30540000
0072000
0007200
000001
000010
,
100000
010000
0007200
0017200
000010
000001
,
59190000
32140000
000100
001000
000010
000001

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,71,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[19,30,0,0,0,0,22,54,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[59,32,0,0,0,0,19,14,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3xDic6:S3 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_6\rtimes S_3
% in TeX

G:=Group("C3xDic6:S3");
// GroupNames label

G:=SmallGroup(432,420);
// by ID

G=gap.SmallGroup(432,420);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,176,1011,514,80,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^12=d^3=e^2=1,c^2=b^6,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,e*b*e=b^7,c*d=d*c,e*c*e=b^3*c,e*d*e=d^-1>;
// generators/relations

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