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## G = C3×Dic3⋊C4order 144 = 24·32

### Direct product of C3 and Dic3⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×Dic3⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — C3×Dic3⋊C4
 Lower central C3 — C6 — C3×Dic3⋊C4
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C3×Dic3⋊C4
G = < a,b,c,d | a3=b6=d4=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 104 in 60 conjugacy classes, 34 normal (30 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×4], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×2], C32, Dic3 [×2], Dic3, C12 [×7], C2×C6 [×2], C2×C6, C4⋊C4, C3×C6 [×3], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×Dic3 [×2], C3×Dic3, C3×C12, C62, Dic3⋊C4, C3×C4⋊C4, C6×Dic3 [×2], C6×C12, C3×Dic3⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4, Q8, C12 [×2], D6, C2×C6, C4⋊C4, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, S3×C6, Dic3⋊C4, C3×C4⋊C4, C3×Dic6, S3×C12, C3×C3⋊D4, C3×Dic3⋊C4

Smallest permutation representation of C3×Dic3⋊C4
On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 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 32 4 35)(2 31 5 34)(3 36 6 33)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 44 16 47)(14 43 17 46)(15 48 18 45)(19 38 22 41)(20 37 23 40)(21 42 24 39)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(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)| (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48), (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), (1,32,4,35)(2,31,5,34)(3,36,6,33)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,38,22,41)(20,37,23,40)(21,42,24,39), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(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( (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48), (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), (1,32,4,35)(2,31,5,34)(3,36,6,33)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,38,22,41)(20,37,23,40)(21,42,24,39), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(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([(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,23,21),(20,24,22),(25,27,29),(26,28,30),(31,33,35),(32,34,36),(37,39,41),(38,40,42),(43,45,47),(44,46,48)], [(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)], [(1,32,4,35),(2,31,5,34),(3,36,6,33),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,44,16,47),(14,43,17,46),(15,48,18,45),(19,38,22,41),(20,37,23,40),(21,42,24,39)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18),(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 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6O 12A ··· 12P 12Q ··· 12X order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 1 1 2 2 2 2 2 6 6 6 6 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Q8 D6 C3×S3 Dic6 C4×S3 C3⋊D4 C3×D4 C3×Q8 S3×C6 C3×Dic6 S3×C12 C3×C3⋊D4 kernel C3×Dic3⋊C4 C6×Dic3 C6×C12 Dic3⋊C4 C3×Dic3 C2×Dic3 C2×C12 Dic3 C2×C12 C3×C6 C3×C6 C2×C6 C2×C4 C6 C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 2 1 2 4 4 2 8 1 1 1 1 2 2 2 2 2 2 2 4 4 4

Matrix representation of C3×Dic3⋊C4 in GL5(𝔽13)

 9 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 3
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 3 0 0 0 0 0 9
,
 1 0 0 0 0 0 6 9 0 0 0 6 7 0 0 0 0 0 0 1 0 0 0 1 0
,
 8 0 0 0 0 0 1 11 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12

G:=sub<GL(5,GF(13))| [9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,3,0,0,0,0,0,9],[1,0,0,0,0,0,6,6,0,0,0,9,7,0,0,0,0,0,0,1,0,0,0,1,0],[8,0,0,0,0,0,1,0,0,0,0,11,12,0,0,0,0,0,12,0,0,0,0,0,12] >;

C3×Dic3⋊C4 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3\rtimes C_4
% in TeX

G:=Group("C3xDic3:C4");
// GroupNames label

G:=SmallGroup(144,77);
// by ID

G=gap.SmallGroup(144,77);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,144,313,79,3461]);
// Polycyclic

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

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