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## G = C3×C4⋊Dic3order 144 = 24·32

### Direct product of C3 and C4⋊Dic3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C4⋊Dic3
G = < a,b,c,d | a3=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 104 in 60 conjugacy classes, 38 normal (26 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×2], C32, Dic3 [×2], C12 [×4], C12 [×4], C2×C6 [×2], C2×C6, C4⋊C4, C3×C6 [×3], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×Dic3 [×2], C3×C12 [×2], C62, C4⋊Dic3, C3×C4⋊C4, C6×Dic3 [×2], C6×C12, C3×C4⋊Dic3
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4, Q8, Dic3 [×2], C12 [×2], D6, C2×C6, C4⋊C4, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×Dic3 [×2], S3×C6, C4⋊Dic3, C3×C4⋊C4, C3×Dic6, C3×D12, C6×Dic3, C3×C4⋊Dic3

Smallest permutation representation of C3×C4⋊Dic3
On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 30 17 22)(2 25 18 23)(3 26 13 24)(4 27 14 19)(5 28 15 20)(6 29 16 21)(7 32 44 40)(8 33 45 41)(9 34 46 42)(10 35 47 37)(11 36 48 38)(12 31 43 39)
(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 35 4 32)(2 34 5 31)(3 33 6 36)(7 30 10 27)(8 29 11 26)(9 28 12 25)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)

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

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

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

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 Dic3 D6 C3×S3 Dic6 D12 C3×D4 C3×Q8 C3×Dic3 S3×C6 C3×Dic6 C3×D12 kernel C3×C4⋊Dic3 C6×Dic3 C6×C12 C4⋊Dic3 C3×C12 C2×Dic3 C2×C12 C12 C2×C12 C3×C6 C3×C6 C12 C2×C6 C2×C4 C6 C6 C6 C6 C4 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 1 1 1 2 1 2 2 2 2 2 4 2 4 4

Matrix representation of C3×C4⋊Dic3 in GL4(𝔽13) generated by

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

C3×C4⋊Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC4:Dic3");
// GroupNames label

G:=SmallGroup(144,78);
// by ID

G=gap.SmallGroup(144,78);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,313,151,3461]);
// Polycyclic

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

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