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## G = C3×C12⋊7D4order 288 = 25·32

### Direct product of C3 and C12⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C12⋊7D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C6×D12 — C3×C12⋊7D4
 Lower central C3 — C2×C6 — C3×C12⋊7D4
 Upper central C1 — C2×C6 — C22×C12

Generators and relations for C3×C127D4
G = < a,b,c,d | a3=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 538 in 215 conjugacy classes, 74 normal (42 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], S3 [×2], C6 [×6], C6 [×13], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C32, Dic3 [×2], C12 [×4], C12 [×8], D6 [×6], C2×C6 [×2], C2×C6 [×4], C2×C6 [×17], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×2], C3×C6 [×3], C3×C6 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×4], C2×C12 [×12], C3×D4 [×6], C22×S3 [×2], C22×C6 [×2], C22×C6 [×3], C4⋊D4, C3×Dic3 [×2], C3×C12 [×2], C3×C12, S3×C6 [×6], C62, C62 [×2], C62 [×2], C4⋊Dic3, D6⋊C4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C2×D12, C2×C3⋊D4 [×2], C22×C12 [×2], C22×C12, C6×D4 [×3], C3×D12 [×2], C6×Dic3 [×2], C3×C3⋊D4 [×4], C6×C12 [×2], C6×C12 [×2], S3×C2×C6 [×2], C2×C62, C127D4, C3×C4⋊D4, C3×C4⋊Dic3, C3×D6⋊C4 [×2], C6×D12, C6×C3⋊D4 [×2], C2×C6×C12, C3×C127D4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×4], C23, D6 [×3], C2×C6 [×7], C2×D4 [×2], C4○D4, C3×S3, D12 [×2], C3⋊D4 [×2], C3×D4 [×4], C22×S3, C22×C6, C4⋊D4, S3×C6 [×3], C2×D12, C4○D12, C2×C3⋊D4, C6×D4 [×2], C3×C4○D4, C3×D12 [×2], C3×C3⋊D4 [×2], S3×C2×C6, C127D4, C3×C4⋊D4, C6×D12, C3×C4○D12, C6×C3⋊D4, C3×C127D4

Smallest permutation representation of C3×C127D4
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 33 29)(26 34 30)(27 35 31)(28 36 32)(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 16 27 40)(2 15 28 39)(3 14 29 38)(4 13 30 37)(5 24 31 48)(6 23 32 47)(7 22 33 46)(8 21 34 45)(9 20 35 44)(10 19 36 43)(11 18 25 42)(12 17 26 41)
(1 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 48)(12 47)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6AE 6AF 6AG 6AH 6AI 12A ··· 12AF 12AG 12AH 12AI 12AJ order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 12 12 12 12 size 1 1 1 1 2 2 12 12 1 1 2 2 2 2 2 2 2 12 12 1 ··· 1 2 ··· 2 12 12 12 12 2 ··· 2 12 12 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 D4 D6 D6 C4○D4 C3×S3 C3⋊D4 C3×D4 D12 C3×D4 S3×C6 S3×C6 C4○D12 C3×C4○D4 C3×C3⋊D4 C3×D12 C3×C4○D12 kernel C3×C12⋊7D4 C3×C4⋊Dic3 C3×D6⋊C4 C6×D12 C6×C3⋊D4 C2×C6×C12 C12⋊7D4 C4⋊Dic3 D6⋊C4 C2×D12 C2×C3⋊D4 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 1 2 1 2 1 2 2 4 2 4 2 1 2 2 2 1 2 2 4 4 4 4 4 2 4 4 8 8 8

Matrix representation of C3×C127D4 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 4 4 0 0 0 10 0 0 0 0 7 0 0 0 11 2
,
 12 10 0 0 5 1 0 0 0 0 7 11 0 0 11 6
,
 1 0 0 0 8 12 0 0 0 0 6 2 0 0 2 7
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[4,0,0,0,4,10,0,0,0,0,7,11,0,0,0,2],[12,5,0,0,10,1,0,0,0,0,7,11,0,0,11,6],[1,8,0,0,0,12,0,0,0,0,6,2,0,0,2,7] >;

C3×C127D4 in GAP, Magma, Sage, TeX

C_3\times C_{12}\rtimes_7D_4
% in TeX

G:=Group("C3xC12:7D4");
// GroupNames label

G:=SmallGroup(288,701);
// by ID

G=gap.SmallGroup(288,701);
# by ID

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

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

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