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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 — C12 — C3×D4⋊Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C6×C12 — C3×C4⋊Dic3 — C3×D4⋊Dic3
 Lower central C3 — C6 — C12 — C3×D4⋊Dic3
 Upper central C1 — C2×C6 — C2×C12 — C6×D4

Generators and relations for C3×D4⋊Dic3
G = < a,b,c,d,e | a3=b4=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 298 in 127 conjugacy classes, 50 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4, C22, C22 [×4], C6 [×6], C6 [×11], C8, C2×C4, C2×C4, D4 [×2], D4, C23, C32, Dic3, C12 [×4], C12 [×3], C2×C6 [×2], C2×C6 [×17], C4⋊C4, C2×C8, C2×D4, C3×C6 [×3], C3×C6 [×2], C3⋊C8, C24, C2×Dic3, C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C3×D4 [×6], C22×C6 [×4], D4⋊C4, C3×Dic3, C3×C12 [×2], C62, C62 [×4], C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C2×C24, C6×D4 [×2], C6×D4, C3×C3⋊C8, C6×Dic3, C6×C12, D4×C32 [×2], D4×C32, C2×C62, D4⋊Dic3, C3×D4⋊C4, C6×C3⋊C8, C3×C4⋊Dic3, D4×C3×C6, C3×D4⋊Dic3
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], Dic3 [×2], C12 [×2], D6, C2×C6, C22⋊C4, D8, SD16, C3×S3, C2×Dic3, C3⋊D4 [×2], C2×C12, C3×D4 [×2], D4⋊C4, C3×Dic3 [×2], S3×C6, D4⋊S3, D4.S3, C6.D4, C3×C22⋊C4, C3×D8, C3×SD16, C6×Dic3, C3×C3⋊D4 [×2], D4⋊Dic3, C3×D4⋊C4, C3×D4⋊S3, C3×D4.S3, C3×C6.D4, C3×D4⋊Dic3

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6O 6P ··· 6AE 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 24A ··· 24H order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 4 4 1 1 2 2 2 2 2 12 12 1 ··· 1 2 ··· 2 4 ··· 4 6 6 6 6 2 2 2 2 4 ··· 4 12 12 12 12 6 ··· 6

72 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 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D4 D4 D6 Dic3 D8 SD16 C3×S3 C3⋊D4 C3×D4 C3⋊D4 C3×D4 S3×C6 C3×Dic3 C3×D8 C3×SD16 C3×C3⋊D4 C3×C3⋊D4 D4⋊S3 D4.S3 C3×D4⋊S3 C3×D4.S3 kernel C3×D4⋊Dic3 C6×C3⋊C8 C3×C4⋊Dic3 D4×C3×C6 D4⋊Dic3 D4×C32 C2×C3⋊C8 C4⋊Dic3 C6×D4 C3×D4 C6×D4 C3×C12 C62 C2×C12 C3×D4 C3×C6 C3×C6 C2×D4 C12 C12 C2×C6 C2×C6 C2×C4 D4 C6 C6 C4 C22 C6 C6 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 1 1 2 2

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
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 65 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 72 0 0 0 0 0 57 16 0 0 16 16
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[65,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,1,0,0,0,0,0,57,16,0,0,16,16] >;

C3×D4⋊Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,266);
// by ID

G=gap.SmallGroup(288,266);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,2524,1271,102,9414]);
// Polycyclic

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

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