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## G = C3×C12.53D4order 288 = 25·32

### Direct product of C3 and C12.53D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×C12.53D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C6×C12 — C6×C3⋊C8 — C3×C12.53D4
 Lower central C3 — C6 — C12 — C3×C12.53D4
 Upper central C1 — C12 — C2×C12 — C3×M4(2)

Generators and relations for C3×C12.53D4
G = < a,b,c,d | a3=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b6c3 >

Subgroups: 122 in 70 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4 [×2], C22, C6 [×2], C6 [×4], C8 [×4], C2×C4, C32, C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C2×C8, M4(2), M4(2), C3×C6, C3×C6, C3⋊C8 [×2], C3⋊C8, C24 [×7], C2×C12 [×2], C2×C12, C8.C4, C3×C12 [×2], C62, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2) [×2], C3×M4(2) [×2], C3×C3⋊C8 [×2], C3×C3⋊C8, C3×C24, C6×C12, C12.53D4, C3×C8.C4, C6×C3⋊C8, C3×C4.Dic3, C32×M4(2), C3×C12.53D4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4, Q8, C12 [×2], D6, C2×C6, C4⋊C4, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C8.C4, S3×C6, Dic3⋊C4, C3×C4⋊C4, C3×Dic6, S3×C12, C3×C3⋊D4, C12.53D4, C3×C8.C4, C3×Dic3⋊C4, C3×C12.53D4

Smallest permutation representation of C3×C12.53D4
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 35 4 26 7 29 10 32)(2 28 5 31 8 34 11 25)(3 33 6 36 9 27 12 30)(13 42 22 39 19 48 16 45)(14 47 23 44 20 41 17 38)(15 40 24 37 21 46 18 43)
(1 48 10 45 7 42 4 39)(2 41 11 38 8 47 5 44)(3 46 12 43 9 40 6 37)(13 26 22 35 19 32 16 29)(14 31 23 28 20 25 17 34)(15 36 24 33 21 30 18 27)

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

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

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

72 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C ··· 6G 6H 6I 6J 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E ··· 12L 12M 12N 12O 24A ··· 24P 24Q ··· 24X 24Y 24Z 24AA 24AB order 1 2 2 3 3 3 3 3 4 4 4 6 6 6 ··· 6 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 24 ··· 24 24 ··· 24 24 24 24 24 size 1 1 2 1 1 2 2 2 1 1 2 1 1 2 ··· 2 4 4 4 4 4 6 6 6 6 12 12 1 1 1 1 2 ··· 2 4 4 4 4 ··· 4 6 ··· 6 12 12 12 12

72 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 2 2 2 2 4 4 type + + + + + + - + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D4 Q8 D6 C3×S3 C4×S3 C3⋊D4 C3×D4 Dic6 C3×Q8 C8.C4 S3×C6 S3×C12 C3×C3⋊D4 C3×Dic6 C3×C8.C4 C12.53D4 C3×C12.53D4 kernel C3×C12.53D4 C6×C3⋊C8 C3×C4.Dic3 C32×M4(2) C12.53D4 C3×C3⋊C8 C2×C3⋊C8 C4.Dic3 C3×M4(2) C3⋊C8 C3×M4(2) C3×C12 C62 C2×C12 M4(2) C12 C12 C12 C2×C6 C2×C6 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 1 1 1 2 2 2 2 2 2 4 2 4 4 4 8 2 4

Matrix representation of C3×C12.53D4 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 8 0 0 0 0 8
,
 27 0 0 0 0 27 0 0 0 0 8 0 0 0 0 64
,
 22 45 0 0 0 10 0 0 0 0 0 1 0 0 1 0
,
 63 0 0 0 27 10 0 0 0 0 0 72 0 0 72 0
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[27,0,0,0,0,27,0,0,0,0,8,0,0,0,0,64],[22,0,0,0,45,10,0,0,0,0,0,1,0,0,1,0],[63,27,0,0,0,10,0,0,0,0,0,72,0,0,72,0] >;

C3×C12.53D4 in GAP, Magma, Sage, TeX

C_3\times C_{12}._{53}D_4
% in TeX

G:=Group("C3xC12.53D4");
// GroupNames label

G:=SmallGroup(288,256);
// by ID

G=gap.SmallGroup(288,256);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,168,365,92,136,1271,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^6*c^3>;
// generators/relations

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