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G = C3×C12.31D6  order 432 = 24·33

Direct product of C3 and C12.31D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×C12.31D6, C33⋊6M4(2), C12.104S32, C6.2(S3×C12), C12.47(S3×C6), (C3×C12).181D6, C3⋊Dic3.5C12, C32⋊5(C3×M4(2)), C32⋊10(C8⋊S3), C6.24(C6.D6), (C32×C12).63C22, (C3×C3⋊C8)⋊8C6, C3⋊C8⋊5(C3×S3), (C3×C3⋊C8)⋊12S3, C4.16(C3×S32), C3⋊1(C3×C8⋊S3), (C4×C3⋊S3).7C6, (C6×C3⋊S3).4C4, (C2×C3⋊S3).5C12, (C32×C3⋊C8)⋊15C2, (C3×C6).59(C4×S3), (C12×C3⋊S3).16C2, (C3×C12).64(C2×C6), (C3×C6).20(C2×C12), C2.3(C3×C6.D6), (C3×C3⋊Dic3).11C4, (C32×C6).25(C2×C4), SmallGroup(432,417)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C3×C12.31D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C32×C3⋊C8 — C3×C12.31D6
Lower central
C32 — C3×C6 — C3×C12.31D6
Upper central
C1 — C12

Generators and relations for C3×C12.31D6
 G = < a,b,c,d | a3=b12=d2=1, c6=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd=b5, dcd=c5 >

Subgroups: 384 in 118 conjugacy classes, 40 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, C2×C4, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, M4(2), C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C8⋊S3, C3×M4(2), C3×C3⋊S3, C32×C6, C3×C3⋊C8, C3×C3⋊C8, C3×C24, S3×C12, C4×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C12.31D6, C3×C8⋊S3, C32×C3⋊C8, C12×C3⋊S3, C3×C12.31D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, D6, C2×C6, M4(2), C3×S3, C4×S3, C2×C12, S32, S3×C6, C8⋊S3, C3×M4(2), C6.D6, S3×C12, C3×S32, C12.31D6, C3×C8⋊S3, C3×C6.D6, C3×C12.31D6

Smallest permutation representation of C3×C12.31D6
►On 48 points
Generators in S48
(1 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)

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

90 conjugacy classes

class 1 2A2B3A3B3C···3H3I3J3K4A4B4C6A6B6C···6H6I6J6K6L6M8A8B8C8D12A12B12C12D12E···12P12Q···12V12W12X24A···24AF
order122333···3333444666···66666688881212121212···1212···12121224···24
size1118112···24441118112···24441818666611112···24···418186···6

90 irreducible representations

dim11111111112222222222444444
type+++++++
imageC1C2C2C3C4C4C6C6C12C12S3D6M4(2)C3×S3C4×S3S3×C6C8⋊S3C3×M4(2)S3×C12C3×C8⋊S3S32C6.D6C3×S32C12.31D6C3×C6.D6C3×C12.31D6
kernelC3×C12.31D6C32×C3⋊C8C12×C3⋊S3C12.31D6C3×C3⋊Dic3C6×C3⋊S3C3×C3⋊C8C4×C3⋊S3C3⋊Dic3C2×C3⋊S3C3×C3⋊C8C3×C12C33C3⋊C8C3×C6C12C32C32C6C3C12C6C4C3C2C1
# reps121222424422244484816112224

Matrix representation of C3×C12.31D6 ►in GL6(𝔽73)

800000
080000
0064000
0006400
000010
000001
,
2700000
0270000
000100
0072100
000010
000001
,
32360000
23410000
00462700
0002700
0000721
0000720
,
37120000
32360000
0072100
000100
0000720
0000721

G:=sub<GL(6,GF(73))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,23,0,0,0,0,36,41,0,0,0,0,0,0,46,0,0,0,0,0,27,27,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[37,32,0,0,0,0,12,36,0,0,0,0,0,0,72,0,0,0,0,0,1,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;
 

C3×C12.31D6 in GAP, Magma, Sage, TeX 

C_3\times C_{12}._{31}D_6
 
% in TeX
 
G:=Group("C3xC12.31D6");
 
// GroupNames label
 
G:=SmallGroup(432,417);
 
// by ID
 
G=gap.SmallGroup(432,417);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,92,80,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^12=d^2=1,c^6=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^5,d*c*d=c^5>;
 
// generators/relations
 

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