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## G = C3×C12.47D4order 288 = 25·32

### Direct product of C3 and C12.47D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C12.47D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C6×C12 — C6×Dic6 — C3×C12.47D4
 Lower central C3 — C6 — C2×C6 — C3×C12.47D4
 Upper central C1 — C6 — C2×C12 — C3×M4(2)

Generators and relations for C3×C12.47D4
G = < a,b,c,d | a3=b12=1, c4=d2=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b9c3 >

Subgroups: 186 in 86 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2, C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×2], C6 [×4], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C32, Dic3 [×2], C12 [×4], C12 [×4], C2×C6 [×2], C2×C6, M4(2), M4(2), C2×Q8, C3×C6, C3×C6, C3⋊C8, C24 [×5], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×Q8 [×2], C4.10D4, C3×Dic3 [×2], C3×C12 [×2], C62, C4.Dic3, C3×M4(2) [×2], C3×M4(2) [×2], C2×Dic6, C6×Q8, C3×C3⋊C8, C3×C24, C3×Dic6 [×2], C6×Dic3 [×2], C6×C12, C12.47D4, C3×C4.10D4, C3×C4.Dic3, C32×M4(2), C6×Dic6, C3×C12.47D4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], C4.10D4, S3×C6, D6⋊C4, C3×C22⋊C4, S3×C12, C3×D12, C3×C3⋊D4, C12.47D4, C3×C4.10D4, C3×D6⋊C4, C3×C12.47D4

Smallest permutation representation of C3×C12.47D4
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(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 40 10 43 7 46 4 37)(2 39 11 42 8 45 5 48)(3 38 12 41 9 44 6 47)(13 27 16 36 19 33 22 30)(14 26 17 35 20 32 23 29)(15 25 18 34 21 31 24 28)
(1 30 7 36)(2 29 8 35)(3 28 9 34)(4 27 10 33)(5 26 11 32)(6 25 12 31)(13 40 19 46)(14 39 20 45)(15 38 21 44)(16 37 22 43)(17 48 23 42)(18 47 24 41)

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

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

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

63 conjugacy classes

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

63 irreducible representations

 dim 1 1 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 C4 C6 C6 C6 C12 S3 D4 D6 C3×S3 D12 C3⋊D4 C3×D4 C4×S3 S3×C6 C3×D12 C3×C3⋊D4 S3×C12 C4.10D4 C12.47D4 C3×C4.10D4 C3×C12.47D4 kernel C3×C12.47D4 C3×C4.Dic3 C32×M4(2) C6×Dic6 C12.47D4 C6×Dic3 C4.Dic3 C3×M4(2) C2×Dic6 C2×Dic3 C3×M4(2) C3×C12 C2×C12 M4(2) C12 C12 C12 C2×C6 C2×C4 C4 C4 C22 C32 C3 C3 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 2 1 2 2 2 4 2 2 4 4 4 1 2 2 4

Matrix representation of C3×C12.47D4 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 3 0 17 0 0 3 0 0 0 0 49 0 0 0 0 49
,
 27 45 0 26 0 0 52 46 43 23 46 0 72 46 21 0
,
 72 0 42 0 0 0 0 1 66 0 1 0 0 72 0 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[3,0,0,0,0,3,0,0,17,0,49,0,0,0,0,49],[27,0,43,72,45,0,23,46,0,52,46,21,26,46,0,0],[72,0,66,0,0,0,0,72,42,0,1,0,0,1,0,0] >;

C3×C12.47D4 in GAP, Magma, Sage, TeX

C_3\times C_{12}._{47}D_4
% in TeX

G:=Group("C3xC12.47D4");
// GroupNames label

G:=SmallGroup(288,258);
// by ID

G=gap.SmallGroup(288,258);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,336,365,92,1683,136,1271,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=1,c^4=d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^9*c^3>;
// generators/relations

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