Copied to
clipboard

## G = C3×S3×SD16order 288 = 25·32

### Direct product of C3, S3 and SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×S3×SD16
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×S3×D4 — C3×S3×SD16
 Lower central C3 — C6 — C12 — C3×S3×SD16
 Upper central C1 — C6 — C12 — C3×SD16

Generators and relations for C3×S3×SD16
G = < a,b,c,d,e | a3=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 402 in 146 conjugacy classes, 58 normal (54 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×3], C22 [×5], S3 [×2], S3, C6 [×2], C6 [×7], C8, C8, C2×C4 [×2], D4, D4 [×2], Q8, Q8 [×2], C23, C32, Dic3, Dic3, C12 [×2], C12 [×6], D6, D6 [×3], C2×C6 [×7], C2×C8, SD16, SD16 [×3], C2×D4, C2×Q8, C3×S3 [×2], C3×S3, C3×C6, C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12 [×2], C3×D4 [×2], C3×D4 [×3], C3×Q8 [×2], C3×Q8 [×3], C22×S3, C22×C6, C2×SD16, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6 [×3], C62, S3×C8, C24⋊C2, D4.S3, Q82S3, C2×C24, C3×SD16 [×2], C3×SD16 [×4], S3×D4, S3×Q8, C6×D4, C6×Q8, C3×C3⋊C8, C3×C24, C3×Dic6, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×C3⋊D4, D4×C32, Q8×C32, S3×C2×C6, S3×SD16, C6×SD16, S3×C24, C3×C24⋊C2, C3×D4.S3, C3×Q82S3, C32×SD16, C3×S3×D4, C3×S3×Q8, C3×S3×SD16
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], SD16 [×2], C2×D4, C3×S3, C3×D4 [×2], C22×S3, C22×C6, C2×SD16, S3×C6 [×3], C3×SD16 [×2], S3×D4, C6×D4, S3×C2×C6, S3×SD16, C6×SD16, C3×S3×D4, C3×S3×SD16

Smallest permutation representation of C3×S3×SD16
On 48 points
Generators in S48
(1 25 40)(2 26 33)(3 27 34)(4 28 35)(5 29 36)(6 30 37)(7 31 38)(8 32 39)(9 42 17)(10 43 18)(11 44 19)(12 45 20)(13 46 21)(14 47 22)(15 48 23)(16 41 24)
(1 25 40)(2 26 33)(3 27 34)(4 28 35)(5 29 36)(6 30 37)(7 31 38)(8 32 39)(9 17 42)(10 18 43)(11 19 44)(12 20 45)(13 21 46)(14 22 47)(15 23 48)(16 24 41)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)
(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)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)(26 28)(27 31)(30 32)(33 35)(34 38)(37 39)(41 47)(43 45)(44 48)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O 6P 8A 8B 8C 8D 12A 12B 12C ··· 12G 12H 12I 12J 12K 12L 12M 12N 24A 24B 24C 24D 24E ··· 24J 24K 24L 24M 24N order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 ··· 12 12 12 12 12 12 12 12 24 24 24 24 24 ··· 24 24 24 24 24 size 1 1 3 3 4 12 1 1 2 2 2 2 4 6 12 1 1 2 2 2 3 3 3 3 4 4 8 8 8 12 12 2 2 6 6 2 2 4 ··· 4 6 6 8 8 8 12 12 2 2 2 2 4 ··· 4 6 6 6 6

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 D6 SD16 C3×S3 C3×D4 C3×D4 S3×C6 S3×C6 S3×C6 C3×SD16 S3×D4 S3×SD16 C3×S3×D4 C3×S3×SD16 kernel C3×S3×SD16 S3×C24 C3×C24⋊C2 C3×D4.S3 C3×Q8⋊2S3 C32×SD16 C3×S3×D4 C3×S3×Q8 S3×SD16 S3×C8 C24⋊C2 D4.S3 Q8⋊2S3 C3×SD16 S3×D4 S3×Q8 C3×SD16 C3×Dic3 S3×C6 C24 C3×D4 C3×Q8 C3×S3 SD16 Dic3 D6 C8 D4 Q8 S3 C6 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 4 2 2 2 2 2 2 8 1 2 2 4

Matrix representation of C3×S3×SD16 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 64 0 0 0 65 8 0 0 0 0 1 0 0 0 0 1
,
 1 7 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 0 72 0 0 0 0 0 61 0 0 6 12
,
 72 0 0 0 0 72 0 0 0 0 1 0 0 0 72 72
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[64,65,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,7,72,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,0,6,0,0,61,12],[72,0,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;

C3×S3×SD16 in GAP, Magma, Sage, TeX

C_3\times S_3\times {\rm SD}_{16}
% in TeX

G:=Group("C3xS3xSD16");
// GroupNames label

G:=SmallGroup(288,684);
// by ID

G=gap.SmallGroup(288,684);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,303,268,1271,648,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=e^2=1,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=d^3>;
// generators/relations

׿
×
𝔽