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G = C3×D8.S3order 288 = 25·32

Direct product of C3 and D8.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C3×D8.S3
 Chief series C1 — C3 — C6 — C12 — C24 — C3×C24 — C3×Dic12 — C3×D8.S3
 Lower central C3 — C6 — C12 — C24 — C3×D8.S3
 Upper central C1 — C6 — C12 — C24 — C3×D8

Generators and relations for C3×D8.S3
G = < a,b,c,d,e | a3=b8=c2=d3=1, e2=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=b5c, ede-1=d-1 >

Subgroups: 186 in 63 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, C6 [×2], C6 [×5], C8, D4, Q8, C32, Dic3, C12 [×2], C12 [×2], C2×C6 [×4], C16, D8, Q16, C3×C6, C3×C6, C24 [×2], C24, Dic6, C3×D4 [×4], C3×Q8, SD32, C3×Dic3, C3×C12, C62, C3⋊C16, C48, Dic12, C3×D8 [×2], C3×D8, C3×Q16, C3×C24, C3×Dic6, D4×C32, D8.S3, C3×SD32, C3×C3⋊C16, C3×Dic12, C32×D8, C3×D8.S3
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, D8, C3×S3, C3⋊D4, C3×D4, SD32, S3×C6, D4⋊S3, C3×D8, C3×C3⋊D4, D8.S3, C3×SD32, C3×D4⋊S3, C3×D8.S3

Smallest permutation representation of C3×D8.S3
On 48 points
Generators in S48
(1 34 25)(2 35 26)(3 36 27)(4 37 28)(5 38 29)(6 39 30)(7 40 31)(8 33 32)(9 48 22)(10 41 23)(11 42 24)(12 43 17)(13 44 18)(14 45 19)(15 46 20)(16 47 21)
(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 8)(2 7)(3 6)(4 5)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)(25 32)(26 31)(27 30)(28 29)(33 34)(35 40)(36 39)(37 38)(42 48)(43 47)(44 46)
(1 25 34)(2 26 35)(3 27 36)(4 28 37)(5 29 38)(6 30 39)(7 31 40)(8 32 33)(9 48 22)(10 41 23)(11 42 24)(12 43 17)(13 44 18)(14 45 19)(15 46 20)(16 47 21)
(1 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(33 47 37 43)(34 46 38 42)(35 45 39 41)(36 44 40 48)

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

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

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

54 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F ··· 6M 8A 8B 12A 12B 12C 12D 12E 12F 12G 16A 16B 16C 16D 24A 24B 24C 24D 24E ··· 24J 48A ··· 48H order 1 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 ··· 6 8 8 12 12 12 12 12 12 12 16 16 16 16 24 24 24 24 24 ··· 24 48 ··· 48 size 1 1 8 1 1 2 2 2 2 24 1 1 2 2 2 8 ··· 8 2 2 2 2 4 4 4 24 24 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D8 C3×S3 C3⋊D4 C3×D4 SD32 S3×C6 C3×D8 C3×C3⋊D4 C3×SD32 D4⋊S3 D8.S3 C3×D4⋊S3 C3×D8.S3 kernel C3×D8.S3 C3×C3⋊C16 C3×Dic12 C32×D8 D8.S3 C3⋊C16 Dic12 C3×D8 C3×D8 C3×C12 C24 C3×C6 D8 C12 C12 C32 C8 C6 C4 C3 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 4 2 4 4 8 1 2 2 4

Matrix representation of C3×D8.S3 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 3 3 0 2 1 3 1 1 2 2 6 3 3 4 2 3
,
 0 6 4 3 4 4 4 4 3 0 0 4 0 6 6 3
,
 4 4 4 6 1 5 6 3 5 5 6 1 3 4 5 4
,
 6 3 3 1 6 3 6 6 5 4 6 4 0 3 2 6
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,1,2,3,3,3,2,4,0,1,6,2,2,1,3,3],[0,4,3,0,6,4,0,6,4,4,0,6,3,4,4,3],[4,1,5,3,4,5,5,4,4,6,6,5,6,3,1,4],[6,6,5,0,3,3,4,3,3,6,6,2,1,6,4,6] >;

C3×D8.S3 in GAP, Magma, Sage, TeX

C_3\times D_8.S_3
% in TeX

G:=Group("C3xD8.S3");
// GroupNames label

G:=SmallGroup(288,261);
// by ID

G=gap.SmallGroup(288,261);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,336,197,1011,514,192,2524,1271,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^3=1,e^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^5*c,e*d*e^-1=d^-1>;
// generators/relations

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