Copied to
clipboard

## G = C3×C12.31D6order 432 = 24·33

### Direct product of C3 and C12.31D6

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 2A 2B 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 6A 6B 6C ··· 6H 6I 6J 6K 6L 6M 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12P 12Q ··· 12V 12W 12X 24A ··· 24AF order 1 2 2 3 3 3 ··· 3 3 3 3 4 4 4 6 6 6 ··· 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 12 ··· 12 12 12 24 ··· 24 size 1 1 18 1 1 2 ··· 2 4 4 4 1 1 18 1 1 2 ··· 2 4 4 4 18 18 6 6 6 6 1 1 1 1 2 ··· 2 4 ··· 4 18 18 6 ··· 6

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 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 D6 M4(2) C3×S3 C4×S3 S3×C6 C8⋊S3 C3×M4(2) S3×C12 C3×C8⋊S3 S32 C6.D6 C3×S32 C12.31D6 C3×C6.D6 C3×C12.31D6 kernel C3×C12.31D6 C32×C3⋊C8 C12×C3⋊S3 C12.31D6 C3×C3⋊Dic3 C6×C3⋊S3 C3×C3⋊C8 C4×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C3×C3⋊C8 C3×C12 C33 C3⋊C8 C3×C6 C12 C32 C32 C6 C3 C12 C6 C4 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 2 2 4 4 4 8 4 8 16 1 1 2 2 2 4

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

 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 1 0 0 0 0 72 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 32 36 0 0 0 0 23 41 0 0 0 0 0 0 46 27 0 0 0 0 0 27 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 37 12 0 0 0 0 32 36 0 0 0 0 0 0 72 1 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 72 1

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

׿
×
𝔽