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## G = C3×C12.10D4order 288 = 25·32

### Direct product of C3 and C12.10D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C12.10D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C6×C12 — C3×C4.Dic3 — C3×C12.10D4
 Lower central C3 — C6 — C2×C6 — C3×C12.10D4
 Upper central C1 — C6 — C2×C12 — C6×Q8

Generators and relations for C3×C12.10D4
G = < a,b,c,d | a3=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b5, dcd-1=b9c3 >

Subgroups: 170 in 95 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×2], C6 [×5], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C32, C12 [×4], C12 [×10], C2×C6 [×2], C2×C6, M4(2) [×2], C2×Q8, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×3], C3×Q8 [×8], C4.10D4, C3×C12 [×2], C3×C12 [×2], C62, C4.Dic3 [×2], C3×M4(2) [×2], C6×Q8 [×2], C6×Q8, C3×C3⋊C8 [×2], C6×C12, C6×C12 [×2], Q8×C32 [×2], C12.10D4, C3×C4.10D4, C3×C4.Dic3 [×2], Q8×C3×C6, C3×C12.10D4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], Dic3 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C3×S3, C2×Dic3, C3⋊D4 [×2], C2×C12, C3×D4 [×2], C4.10D4, C3×Dic3 [×2], S3×C6, C6.D4, C3×C22⋊C4, C6×Dic3, C3×C3⋊D4 [×2], C12.10D4, C3×C4.10D4, C3×C6.D4, C3×C12.10D4

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

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

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

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

63 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C ··· 6M 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12Z 24A ··· 24H order 1 2 2 3 3 3 3 3 4 4 4 4 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 2 1 1 2 2 2 2 2 4 4 1 1 2 ··· 2 12 12 12 12 2 2 2 2 4 ··· 4 12 ··· 12

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + - image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Dic3 D6 C3×S3 C3⋊D4 C3×D4 C3×Dic3 S3×C6 C3×C3⋊D4 C4.10D4 C12.10D4 C3×C4.10D4 C3×C12.10D4 kernel C3×C12.10D4 C3×C4.Dic3 Q8×C3×C6 C12.10D4 C6×C12 C4.Dic3 C6×Q8 C2×C12 C6×Q8 C3×C12 C2×C12 C2×C12 C2×Q8 C12 C12 C2×C4 C2×C4 C4 C32 C3 C3 C1 # reps 1 2 1 2 4 4 2 8 1 2 2 1 2 4 4 4 2 8 1 2 2 4

Matrix representation of C3×C12.10D4 in GL4(𝔽7) generated by

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

C3×C12.10D4 in GAP, Magma, Sage, TeX

C_3\times C_{12}._{10}D_4
% in TeX

G:=Group("C3xC12.10D4");
// GroupNames label

G:=SmallGroup(288,270);
// by ID

G=gap.SmallGroup(288,270);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,344,850,136,2524,9414]);
// 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=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^9*c^3>;
// generators/relations

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