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## G = C3×D4.Dic3order 288 = 25·32

### Direct product of C3 and D4.Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×D4.Dic3
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×C3⋊C8 — C6×C3⋊C8 — C3×D4.Dic3
 Lower central C3 — C6 — C3×D4.Dic3
 Upper central C1 — C12 — C3×C4○D4

Generators and relations for C3×D4.Dic3
G = < a,b,c,d,e | a3=b4=c2=1, d6=b2, e2=b2d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >

Subgroups: 226 in 144 conjugacy classes, 90 normal (24 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], C6 [×2], C6 [×10], C8 [×4], C2×C4 [×3], D4 [×3], Q8, C32, C12 [×2], C12 [×6], C12 [×4], C2×C6 [×6], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C3×C6, C3×C6 [×3], C3⋊C8, C3⋊C8 [×3], C24 [×4], C2×C12 [×6], C2×C12 [×3], C3×D4 [×6], C3×D4 [×3], C3×Q8 [×2], C3×Q8, C8○D4, C3×C12, C3×C12 [×3], C62 [×3], C2×C3⋊C8 [×3], C4.Dic3 [×3], C2×C24 [×3], C3×M4(2) [×3], C3×C4○D4 [×2], C3×C4○D4, C3×C3⋊C8, C3×C3⋊C8 [×3], C6×C12 [×3], D4×C32 [×3], Q8×C32, D4.Dic3, C3×C8○D4, C6×C3⋊C8 [×3], C3×C4.Dic3 [×3], C32×C4○D4, C3×D4.Dic3
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], C23, Dic3 [×4], C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C3×S3, C2×Dic3 [×6], C2×C12 [×6], C22×S3, C22×C6, C8○D4, C3×Dic3 [×4], S3×C6 [×3], C22×Dic3, C22×C12, C6×Dic3 [×6], S3×C2×C6, D4.Dic3, C3×C8○D4, Dic3×C2×C6, C3×D4.Dic3

Smallest permutation representation of C3×D4.Dic3
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(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 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)(25 46)(26 47)(27 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(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 10 29 7 26 4 35)(2 25 11 34 8 31 5 28)(3 30 12 27 9 36 6 33)(13 38 22 47 19 44 16 41)(14 43 23 40 20 37 17 46)(15 48 24 45 21 42 18 39)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C ··· 6K 6L ··· 6T 8A 8B 8C 8D 8E ··· 8J 12A 12B 12C 12D 12E ··· 12P 12Q ··· 12Y 24A ··· 24H 24I ··· 24T order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 8 8 8 8 8 ··· 8 12 12 12 12 12 ··· 12 12 ··· 12 24 ··· 24 24 ··· 24 size 1 1 2 2 2 1 1 2 2 2 1 1 2 2 2 1 1 2 ··· 2 4 ··· 4 3 3 3 3 6 ··· 6 1 1 1 1 2 ··· 2 4 ··· 4 3 ··· 3 6 ··· 6

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 4 4 type + + + + + + - - image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 S3 D6 Dic3 Dic3 C3×S3 C8○D4 S3×C6 C3×Dic3 C3×Dic3 C3×C8○D4 D4.Dic3 C3×D4.Dic3 kernel C3×D4.Dic3 C6×C3⋊C8 C3×C4.Dic3 C32×C4○D4 D4.Dic3 D4×C32 Q8×C32 C2×C3⋊C8 C4.Dic3 C3×C4○D4 C3×D4 C3×Q8 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C4○D4 C32 C2×C4 D4 Q8 C3 C3 C1 # reps 1 3 3 1 2 6 2 6 6 2 12 4 1 3 3 1 2 4 6 6 2 8 2 4

Matrix representation of C3×D4.Dic3 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 0 72 0 0 0 0 46 0 0 0 43 27
,
 1 0 0 0 0 1 0 0 0 0 27 39 0 0 30 46
,
 9 0 0 0 0 65 0 0 0 0 46 0 0 0 0 46
,
 0 1 0 0 72 0 0 0 0 0 22 0 0 0 0 22
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,46,43,0,0,0,27],[1,0,0,0,0,1,0,0,0,0,27,30,0,0,39,46],[9,0,0,0,0,65,0,0,0,0,46,0,0,0,0,46],[0,72,0,0,1,0,0,0,0,0,22,0,0,0,0,22] >;

C3×D4.Dic3 in GAP, Magma, Sage, TeX

C_3\times D_4.{\rm Dic}_3
% in TeX

G:=Group("C3xD4.Dic3");
// GroupNames label

G:=SmallGroup(288,719);
// by ID

G=gap.SmallGroup(288,719);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,555,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=1,d^6=b^2,e^2=b^2*d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^5>;
// generators/relations

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