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## G = C3×D6⋊C4order 144 = 24·32

### Direct product of C3 and D6⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×D6⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C3×D6⋊C4
 Lower central C3 — C6 — C3×D6⋊C4
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C3×D6⋊C4
G = < a,b,c,d | a3=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 168 in 76 conjugacy classes, 34 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C2×C4, C2×C4, C23, C32, Dic3, C12 [×5], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C22⋊C4, C3×S3 [×2], C3×C6 [×3], C2×Dic3, C2×C12 [×2], C2×C12 [×2], C22×S3, C22×C6, C3×Dic3, C3×C12, S3×C6 [×2], S3×C6 [×2], C62, D6⋊C4, C3×C22⋊C4, C6×Dic3, C6×C12, S3×C2×C6, C3×D6⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], S3×C6, D6⋊C4, C3×C22⋊C4, S3×C12, C3×D12, C3×C3⋊D4, C3×D6⋊C4

Smallest permutation representation of C3×D6⋊C4
On 48 points
Generators in S48
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(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)(2 29)(3 28)(4 27)(5 26)(6 25)(7 34)(8 33)(9 32)(10 31)(11 36)(12 35)(13 40)(14 39)(15 38)(16 37)(17 42)(18 41)(19 46)(20 45)(21 44)(22 43)(23 48)(24 47)
(1 23 11 17)(2 24 12 18)(3 19 7 13)(4 20 8 14)(5 21 9 15)(6 22 10 16)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 43 34 37)(29 44 35 38)(30 45 36 39)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6O 6P 6Q 6R 6S 12A ··· 12P 12Q 12R 12S 12T order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 12 12 12 12 size 1 1 1 1 6 6 1 1 2 2 2 2 2 6 6 1 ··· 1 2 ··· 2 6 6 6 6 2 ··· 2 6 6 6 6

54 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 type + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D4 D6 C3×S3 C4×S3 D12 C3⋊D4 C3×D4 S3×C6 S3×C12 C3×D12 C3×C3⋊D4 kernel C3×D6⋊C4 C6×Dic3 C6×C12 S3×C2×C6 D6⋊C4 S3×C6 C2×Dic3 C2×C12 C22×S3 D6 C2×C12 C3×C6 C2×C6 C2×C4 C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 2 1 2 2 2 2 4 2 4 4 4

Matrix representation of C3×D6⋊C4 in GL3(𝔽13) generated by

 9 0 0 0 3 0 0 0 3
,
 1 0 0 0 4 0 0 0 10
,
 12 0 0 0 0 10 0 4 0
,
 5 0 0 0 1 0 0 0 12
G:=sub<GL(3,GF(13))| [9,0,0,0,3,0,0,0,3],[1,0,0,0,4,0,0,0,10],[12,0,0,0,0,4,0,10,0],[5,0,0,0,1,0,0,0,12] >;

C3×D6⋊C4 in GAP, Magma, Sage, TeX

C_3\times D_6\rtimes C_4
% in TeX

G:=Group("C3xD6:C4");
// GroupNames label

G:=SmallGroup(144,79);
// by ID

G=gap.SmallGroup(144,79);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,313,79,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

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