Copied to
clipboard

## G = C3×Dic32order 432 = 24·33

### Direct product of C3, Dic3 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×Dic32
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C3×C62 — Dic3×C3×C6 — C3×Dic32
 Lower central C32 — C3×Dic32
 Upper central C1 — C2×C6

Generators and relations for C3×Dic32
G = < a,b,c,d,e | a3=b6=d6=1, c2=b3, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 448 in 178 conjugacy classes, 72 normal (16 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, C32, C32, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C62, C62, C62, C4×Dic3, C4×C12, C32×C6, C32×C6, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C6×C12, C32×Dic3, C3×C3⋊Dic3, C3×C62, Dic32, Dic3×C12, Dic3×C3×C6, C6×C3⋊Dic3, C3×Dic32
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, C42, C3×S3, C4×S3, C2×Dic3, C2×C12, C3×Dic3, S32, S3×C6, C4×Dic3, C4×C12, S3×Dic3, C6.D6, S3×C12, C6×Dic3, C3×S32, Dic32, Dic3×C12, C3×S3×Dic3, C3×C6.D6, C3×Dic32

Smallest permutation representation of C3×Dic32
On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 45 47)(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 30 4 27)(2 29 5 26)(3 28 6 25)(7 36 10 33)(8 35 11 32)(9 34 12 31)(13 20 16 23)(14 19 17 22)(15 24 18 21)(37 44 40 47)(38 43 41 46)(39 48 42 45)
(1 14 5 18 3 16)(2 15 6 13 4 17)(7 44 11 48 9 46)(8 45 12 43 10 47)(19 26 21 28 23 30)(20 27 22 29 24 25)(31 41 33 37 35 39)(32 42 34 38 36 40)
(1 42 18 36)(2 37 13 31)(3 38 14 32)(4 39 15 33)(5 40 16 34)(6 41 17 35)(7 27 48 24)(8 28 43 19)(9 29 44 20)(10 30 45 21)(11 25 46 22)(12 26 47 23)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 3K 4A ··· 4H 4I 4J 4K 4L 6A ··· 6F 6G ··· 6X 6Y ··· 6AG 12A ··· 12P 12Q ··· 12AN 12AO ··· 12AV order 1 2 2 2 3 3 3 ··· 3 3 3 3 4 ··· 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 1 1 2 ··· 2 4 4 4 3 ··· 3 9 9 9 9 1 ··· 1 2 ··· 2 4 ··· 4 3 ··· 3 6 ··· 6 9 ··· 9

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + - + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 Dic3 D6 C3×S3 C4×S3 C3×Dic3 S3×C6 S3×C12 S32 S3×Dic3 C6.D6 C3×S32 C3×S3×Dic3 C3×C6.D6 kernel C3×Dic32 Dic3×C3×C6 C6×C3⋊Dic3 Dic32 C32×Dic3 C3×C3⋊Dic3 C6×Dic3 C2×C3⋊Dic3 C3×Dic3 C3⋊Dic3 C6×Dic3 C3×Dic3 C62 C2×Dic3 C3×C6 Dic3 C2×C6 C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 2 8 4 4 2 16 8 2 4 2 4 8 8 4 16 1 2 1 2 4 2

Matrix representation of C3×Dic32 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 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 12 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 5 0
,
 1 1 0 0 0 0 12 0 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 5 0 0 0 0 0 8 8 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,8,0,0,0,0,0,8,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

C3×Dic32 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3^2
% in TeX

G:=Group("C3xDic3^2");
// GroupNames label

G:=SmallGroup(432,425);
// by ID

G=gap.SmallGroup(432,425);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,92,2028,14118]);
// Polycyclic

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

׿
×
𝔽