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## G = C3×C12.46D4order 288 = 25·32

### Direct product of C3 and C12.46D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C12.46D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C6×C12 — C6×D12 — C3×C12.46D4
 Lower central C3 — C6 — C2×C6 — C3×C12.46D4
 Upper central C1 — C6 — C2×C12 — C3×M4(2)

Generators and relations for C3×C12.46D4
G = < a,b,c,d,e | a3=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b5, bd=db, ebe=bc, cd=dc, ce=ec, ede=d-1 >

Subgroups: 314 in 102 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, D4, C23, C32, C12, C12, D6, C2×C6, C2×C6, M4(2), M4(2), C2×D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4.D4, C3×C12, S3×C6, C62, C4.Dic3, C3×M4(2), C3×M4(2), C2×D12, C6×D4, C3×C3⋊C8, C3×C24, C3×D12, C6×C12, S3×C2×C6, C12.46D4, C3×C4.D4, C3×C4.Dic3, C32×M4(2), C6×D12, C3×C12.46D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C4.D4, S3×C6, D6⋊C4, C3×C22⋊C4, S3×C12, C3×D12, C3×C3⋊D4, C12.46D4, C3×C4.D4, C3×D6⋊C4, C3×C12.46D4

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

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C ··· 6G 6H 6I 6J 6K 6L 6M 6N 8A 8B 8C 8D 12A ··· 12J 12K 12L 12M 24A ··· 24P 24Q 24R 24S 24T order 1 2 2 2 2 3 3 3 3 3 4 4 6 6 6 ··· 6 6 6 6 6 6 6 6 8 8 8 8 12 ··· 12 12 12 12 24 ··· 24 24 24 24 24 size 1 1 2 12 12 1 1 2 2 2 2 2 1 1 2 ··· 2 4 4 4 12 12 12 12 4 4 12 12 2 ··· 2 4 4 4 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.D4 C12.46D4 C3×C4.D4 C3×C12.46D4 kernel C3×C12.46D4 C3×C4.Dic3 C32×M4(2) C6×D12 C12.46D4 S3×C2×C6 C4.Dic3 C3×M4(2) C2×D12 C22×S3 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×C12.46D4 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 0 1 0 0 27 0 0 0 0 0 0 1 0 0 46 0
,
 1 0 0 0 0 72 0 0 0 0 1 0 0 0 0 72
,
 64 0 0 0 0 64 0 0 0 0 8 0 0 0 0 8
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,27,0,0,1,0,0,0,0,0,0,46,0,0,1,0],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72],[64,0,0,0,0,64,0,0,0,0,8,0,0,0,0,8],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;

C3×C12.46D4 in GAP, Magma, Sage, TeX

C_3\times C_{12}._{46}D_4
% in TeX

G:=Group("C3xC12.46D4");
// GroupNames label

G:=SmallGroup(288,257);
// by ID

G=gap.SmallGroup(288,257);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,1683,136,1271,9414]);
// 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^5,b*d=d*b,e*b*e=b*c,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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