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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 — C12 — C3×D8⋊S3
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×S3×D4 — C3×D8⋊S3
 Lower central C3 — C6 — C12 — C3×D8⋊S3
 Upper central C1 — C6 — C12 — C3×D8

Generators and relations for C3×D8⋊S3
G = < a,b,c,d,e | a3=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >

Subgroups: 426 in 147 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×2], C6 [×2], C6 [×9], C8, C8, C2×C4 [×2], D4 [×2], D4 [×3], Q8, C23, C32, Dic3, Dic3, C12 [×2], C12 [×3], D6, D6 [×3], C2×C6 [×10], M4(2), D8, D8, SD16 [×2], C2×D4, C4○D4, C3×S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8, C24 [×2], C24 [×2], Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C3×D4 [×4], C3×D4 [×5], C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3×Dic3, C3×C12, S3×C6, S3×C6 [×3], C62 [×2], C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×M4(2), C3×D8 [×2], C3×D8 [×2], C3×SD16 [×2], S3×D4, D42S3, C6×D4, C3×C4○D4, C3×C3⋊C8, C3×C24, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4 [×2], D4×C32 [×2], S3×C2×C6, D8⋊S3, C3×C8⋊C22, C3×C8⋊S3, C3×C24⋊C2, C3×D4⋊S3, C3×D4.S3, C32×D8, C3×S3×D4, C3×D42S3, C3×D8⋊S3
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3×D4 [×2], C22×S3, C22×C6, C8⋊C22, S3×C6 [×3], S3×D4, C6×D4, S3×C2×C6, D8⋊S3, C3×C8⋊C22, C3×S3×D4, C3×D8⋊S3

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

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L ··· 6Q 6R 6S 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 12I 24A ··· 24H 24I 24J order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 ··· 6 6 6 8 8 12 12 12 12 12 12 12 12 12 24 ··· 24 24 24 size 1 1 4 4 6 12 1 1 2 2 2 2 6 12 1 1 2 2 2 4 4 4 4 6 6 8 ··· 8 12 12 4 12 2 2 4 4 4 6 6 12 12 4 ··· 4 12 12

54 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 4 4 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 C3×S3 C3×D4 C3×D4 S3×C6 S3×C6 C8⋊C22 S3×D4 D8⋊S3 C3×C8⋊C22 C3×S3×D4 C3×D8⋊S3 kernel C3×D8⋊S3 C3×C8⋊S3 C3×C24⋊C2 C3×D4⋊S3 C3×D4.S3 C32×D8 C3×S3×D4 C3×D4⋊2S3 D8⋊S3 C8⋊S3 C24⋊C2 D4⋊S3 D4.S3 C3×D8 S3×D4 D4⋊2S3 C3×D8 C3×Dic3 S3×C6 C24 C3×D4 D8 Dic3 D6 C8 D4 C32 C6 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 4 1 1 2 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
,
 0 6 2 3 5 5 3 5 1 3 4 0 6 4 0 5
,
 0 5 2 6 2 0 2 1 3 3 0 1 1 6 3 0
,
 2 5 1 1 3 3 3 6 4 3 4 6 2 2 6 3
,
 5 1 2 2 0 0 6 5 4 3 5 6 5 5 1 4
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,5,1,6,6,5,3,4,2,3,4,0,3,5,0,5],[0,2,3,1,5,0,3,6,2,2,0,3,6,1,1,0],[2,3,4,2,5,3,3,2,1,3,4,6,1,6,6,3],[5,0,4,5,1,0,3,5,2,6,5,1,2,5,6,4] >;

C3×D8⋊S3 in GAP, Magma, Sage, TeX

C_3\times D_8\rtimes S_3
% in TeX

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

G:=SmallGroup(288,682);
// by ID

G=gap.SmallGroup(288,682);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,1094,303,1271,648,102,9414]);
// Polycyclic

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

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