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## G = C3×C12.48D4order 288 = 25·32

### Direct product of C3 and C12.48D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C12.48D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — C6×Dic6 — C3×C12.48D4
 Lower central C3 — C2×C6 — C3×C12.48D4
 Upper central C1 — C2×C6 — C22×C12

Generators and relations for C3×C12.48D4
G = < a,b,c,d | a3=b12=c4=1, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b6c-1 >

Subgroups: 346 in 175 conjugacy classes, 74 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×6], C6 [×11], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C32, Dic3 [×4], C12 [×4], C12 [×10], C2×C6 [×2], C2×C6 [×4], C2×C6 [×11], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3×C6 [×3], C3×C6 [×2], Dic6 [×2], C2×Dic3 [×4], C2×C12 [×4], C2×C12 [×14], C3×Q8 [×2], C22×C6 [×2], C22×C6, C22⋊Q8, C3×Dic3 [×4], C3×C12 [×2], C3×C12, C62, C62 [×2], C62 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×3], C2×Dic6, C22×C12 [×2], C22×C12, C6×Q8, C3×Dic6 [×2], C6×Dic3 [×4], C6×C12 [×2], C6×C12 [×2], C2×C62, C12.48D4, C3×C22⋊Q8, C3×Dic3⋊C4 [×2], C3×C4⋊Dic3, C3×C6.D4 [×2], C6×Dic6, C2×C6×C12, C3×C12.48D4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], Q8 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C2×Q8, C4○D4, C3×S3, Dic6 [×2], C3⋊D4 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6, C22⋊Q8, S3×C6 [×3], C2×Dic6, C4○D12, C2×C3⋊D4, C6×D4, C6×Q8, C3×C4○D4, C3×Dic6 [×2], C3×C3⋊D4 [×2], S3×C2×C6, C12.48D4, C3×C22⋊Q8, C6×Dic6, C3×C4○D12, C6×C3⋊D4, C3×C12.48D4

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

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6AE 12A ··· 12AF 12AG ··· 12AN order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 1 1 2 2 2 2 2 2 2 12 12 12 12 1 ··· 1 2 ··· 2 2 ··· 2 12 ··· 12

90 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 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 Q8 D6 D6 C4○D4 C3×S3 C3⋊D4 C3×D4 Dic6 C3×Q8 S3×C6 S3×C6 C4○D12 C3×C4○D4 C3×C3⋊D4 C3×Dic6 C3×C4○D12 kernel C3×C12.48D4 C3×Dic3⋊C4 C3×C4⋊Dic3 C3×C6.D4 C6×Dic6 C2×C6×C12 C12.48D4 Dic3⋊C4 C4⋊Dic3 C6.D4 C2×Dic6 C22×C12 C22×C12 C3×C12 C62 C2×C12 C22×C6 C3×C6 C22×C4 C12 C12 C2×C6 C2×C6 C2×C4 C23 C6 C6 C4 C22 C2 # reps 1 2 1 2 1 1 2 4 2 4 2 2 1 2 2 2 1 2 2 4 4 4 4 4 2 4 4 8 8 8

Matrix representation of C3×C12.48D4 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 9 0 0 0 0 9
,
 6 0 0 0 4 11 0 0 0 0 11 0 0 0 0 6
,
 1 11 0 0 0 12 0 0 0 0 0 12 0 0 1 0
,
 12 2 0 0 12 1 0 0 0 0 0 1 0 0 12 0
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,9,0,0,0,0,9],[6,4,0,0,0,11,0,0,0,0,11,0,0,0,0,6],[1,0,0,0,11,12,0,0,0,0,0,1,0,0,12,0],[12,12,0,0,2,1,0,0,0,0,0,12,0,0,1,0] >;

C3×C12.48D4 in GAP, Magma, Sage, TeX

C_3\times C_{12}._{48}D_4
% in TeX

G:=Group("C3xC12.48D4");
// GroupNames label

G:=SmallGroup(288,695);
// by ID

G=gap.SmallGroup(288,695);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,701,344,590,9414]);
// Polycyclic

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

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