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## G = C3×C12.55D4order 288 = 25·32

### Direct product of C3 and C12.55D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C12.55D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C6×C12 — C6×C3⋊C8 — C3×C12.55D4
 Lower central C3 — C6 — C3×C12.55D4
 Upper central C1 — C2×C12 — C22×C12

Generators and relations for C3×C12.55D4
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=b3c3 >

Subgroups: 218 in 127 conjugacy classes, 58 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4, C22, C22 [×2], C22 [×2], C6 [×6], C6 [×11], C8 [×2], C2×C4 [×2], C2×C4 [×2], C23, C32, C12 [×4], C12 [×6], C2×C6 [×2], C2×C6 [×4], C2×C6 [×11], C2×C8 [×2], C22×C4, C3×C6 [×3], C3×C6 [×2], C3⋊C8 [×2], C24 [×2], C2×C12 [×4], C2×C12 [×10], C22×C6 [×2], C22×C6, C22⋊C8, C3×C12 [×2], C3×C12, C62, C62 [×2], C62 [×2], C2×C3⋊C8 [×2], C2×C24 [×2], C22×C12 [×2], C22×C12, C3×C3⋊C8 [×2], C6×C12 [×2], C6×C12 [×2], C2×C62, C12.55D4, C3×C22⋊C8, C6×C3⋊C8 [×2], C2×C6×C12, C3×C12.55D4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, D4 [×2], Dic3 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C2×C8, M4(2), C3×S3, C3⋊C8 [×2], C24 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12, C3×D4 [×2], C22⋊C8, C3×Dic3 [×2], S3×C6, C2×C3⋊C8, C4.Dic3, C6.D4, C3×C22⋊C4, C2×C24, C3×M4(2), C3×C3⋊C8 [×2], C6×Dic3, C3×C3⋊D4 [×2], C12.55D4, C3×C22⋊C8, C6×C3⋊C8, C3×C4.Dic3, C3×C6.D4, C3×C12.55D4

Smallest permutation representation of C3×C12.55D4
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 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)
(1 28 19 40 7 34 13 46)(2 33 20 45 8 27 14 39)(3 26 21 38 9 32 15 44)(4 31 22 43 10 25 16 37)(5 36 23 48 11 30 17 42)(6 29 24 41 12 35 18 47)
(1 43 10 40 7 37 4 46)(2 48 11 45 8 42 5 39)(3 41 12 38 9 47 6 44)(13 31 22 28 19 25 16 34)(14 36 23 33 20 30 17 27)(15 29 24 26 21 35 18 32)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6AE 8A ··· 8H 12A ··· 12H 12I ··· 12AJ 24A ··· 24P order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 1 1 2 2 2 1 1 1 1 2 2 1 ··· 1 2 ··· 2 6 ··· 6 1 ··· 1 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 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 2 2 type + + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 S3 D4 Dic3 D6 Dic3 M4(2) C3×S3 C3⋊D4 C3×D4 C3⋊C8 C3×Dic3 S3×C6 C3×Dic3 C4.Dic3 C3×M4(2) C3×C3⋊D4 C3×C3⋊C8 C3×C4.Dic3 kernel C3×C12.55D4 C6×C3⋊C8 C2×C6×C12 C12.55D4 C6×C12 C2×C62 C2×C3⋊C8 C22×C12 C62 C2×C12 C22×C6 C2×C6 C22×C12 C3×C12 C2×C12 C2×C12 C22×C6 C3×C6 C22×C4 C12 C12 C2×C6 C2×C4 C2×C4 C23 C6 C6 C4 C22 C2 # reps 1 2 1 2 2 2 4 2 8 4 4 16 1 2 1 1 1 2 2 4 4 4 2 2 2 4 4 8 8 8

Matrix representation of C3×C12.55D4 in GL3(𝔽73) generated by

 1 0 0 0 64 0 0 0 64
,
 27 0 0 0 3 0 0 21 24
,
 63 0 0 0 63 63 0 0 10
,
 10 0 0 0 63 63 0 20 10
G:=sub<GL(3,GF(73))| [1,0,0,0,64,0,0,0,64],[27,0,0,0,3,21,0,0,24],[63,0,0,0,63,0,0,63,10],[10,0,0,0,63,20,0,63,10] >;

C3×C12.55D4 in GAP, Magma, Sage, TeX

C_3\times C_{12}._{55}D_4
% in TeX

G:=Group("C3xC12.55D4");
// GroupNames label

G:=SmallGroup(288,264);
// by ID

G=gap.SmallGroup(288,264);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,136,9414]);
// 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^3*c^3>;
// generators/relations

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