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## G = C3×D4⋊D6order 288 = 25·32

### Direct product of C3 and D4⋊D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×D4⋊D6
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×D12 — C6×D12 — C3×D4⋊D6
 Lower central C3 — C6 — C12 — C3×D4⋊D6
 Upper central C1 — C6 — C2×C12 — C3×C4○D4

Generators and relations for C3×D4⋊D6
G = < a,b,c,d,e | a3=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ece=b-1c, ede=d-1 >

Subgroups: 418 in 156 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×2], C4, C22, C22 [×5], S3 [×2], C6 [×2], C6 [×10], C8 [×2], C2×C4, C2×C4, D4, D4 [×4], Q8, C23, C32, C12 [×4], C12 [×6], D6 [×4], C2×C6 [×2], C2×C6 [×9], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×2], D12 [×2], D12, C2×C12 [×2], C2×C12 [×5], C3×D4 [×2], C3×D4 [×8], C3×Q8 [×2], C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×C12 [×2], C3×C12, S3×C6 [×4], C62, C62, C4.Dic3, D4⋊S3 [×2], Q82S3 [×2], C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C2×D12, C6×D4, C3×C4○D4 [×2], C3×C4○D4, C3×C3⋊C8 [×2], C3×D12 [×2], C3×D12, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, S3×C2×C6, D4⋊D6, C3×C8⋊C22, C3×C4.Dic3, C3×D4⋊S3 [×2], C3×Q82S3 [×2], C6×D12, C32×C4○D4, C3×D4⋊D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C22×C6, C8⋊C22, S3×C6 [×3], C2×C3⋊D4, C6×D4, C3×C3⋊D4 [×2], S3×C2×C6, D4⋊D6, C3×C8⋊C22, C6×C3⋊D4, C3×D4⋊D6

Smallest permutation representation of C3×D4⋊D6
On 48 points
Generators in S48
(1 2 3)(4 6 5)(7 9 8)(10 12 11)(13 14 15)(16 17 18)(19 21 20)(22 23 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 45 47)(44 46 48)
(1 13 18 24)(2 14 16 22)(3 15 17 23)(4 7 19 10)(5 8 20 11)(6 9 21 12)(25 44 28 47)(26 45 29 48)(27 46 30 43)(31 39 34 42)(32 40 35 37)(33 41 36 38)
(1 39)(2 37)(3 41)(4 45)(5 43)(6 47)(7 26)(8 30)(9 28)(10 29)(11 27)(12 25)(13 31)(14 35)(15 33)(16 40)(17 38)(18 42)(19 48)(20 46)(21 44)(22 32)(23 36)(24 34)
(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 20)(2 19)(3 21)(4 16)(5 18)(6 17)(7 14)(8 13)(9 15)(10 22)(11 24)(12 23)(25 38)(26 37)(27 42)(28 41)(29 40)(30 39)(31 43)(32 48)(33 47)(34 46)(35 45)(36 44)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C ··· 6G 6H ··· 6R 6S 6T 6U 6V 8A 8B 12A ··· 12J 12K ··· 12U 24A 24B 24C 24D order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 6 6 6 ··· 6 6 ··· 6 6 6 6 6 8 8 12 ··· 12 12 ··· 12 24 24 24 24 size 1 1 2 4 12 12 1 1 2 2 2 2 2 4 1 1 2 ··· 2 4 ··· 4 12 12 12 12 12 12 2 ··· 2 4 ··· 4 12 12 12 12

63 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 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 D6 C3×S3 C3⋊D4 C3×D4 C3⋊D4 C3×D4 S3×C6 S3×C6 S3×C6 C3×C3⋊D4 C3×C3⋊D4 C8⋊C22 D4⋊D6 C3×C8⋊C22 C3×D4⋊D6 kernel C3×D4⋊D6 C3×C4.Dic3 C3×D4⋊S3 C3×Q8⋊2S3 C6×D12 C32×C4○D4 D4⋊D6 C4.Dic3 D4⋊S3 Q8⋊2S3 C2×D12 C3×C4○D4 C3×C4○D4 C3×C12 C62 C2×C12 C3×D4 C3×Q8 C4○D4 C12 C12 C2×C6 C2×C6 C2×C4 D4 Q8 C4 C22 C32 C3 C3 C1 # reps 1 1 2 2 1 1 2 2 4 4 2 2 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 1 2 2 4

Matrix representation of C3×D4⋊D6 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 27 0 0 0 38 46 0 0 0 0 46 0 0 0 35 27
,
 35 54 0 0 26 38 0 0 0 0 4 2 0 0 29 69
,
 64 0 0 0 36 9 0 0 0 0 8 0 0 0 41 65
,
 0 0 8 0 0 0 41 65 64 0 0 0 36 9 0 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[27,38,0,0,0,46,0,0,0,0,46,35,0,0,0,27],[35,26,0,0,54,38,0,0,0,0,4,29,0,0,2,69],[64,36,0,0,0,9,0,0,0,0,8,41,0,0,0,65],[0,0,64,36,0,0,0,9,8,41,0,0,0,65,0,0] >;

C3×D4⋊D6 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes D_6
% in TeX

G:=Group("C3xD4:D6");
// GroupNames label

G:=SmallGroup(288,720);
// by ID

G=gap.SmallGroup(288,720);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,555,2524,648,102,9414]);
// Polycyclic

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

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