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## G = C2×C6×C3⋊D4order 288 = 25·32

### Direct product of C2×C6 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C6×C3⋊D4
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — S3×C22×C6 — C2×C6×C3⋊D4
 Lower central C3 — C6 — C2×C6×C3⋊D4
 Upper central C1 — C22×C6 — C23×C6

Generators and relations for C2×C6×C3⋊D4
G = < a,b,c,d,e | a2=b6=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1130 in 539 conjugacy classes, 210 normal (22 characteristic)
C1, C2, C2 [×6], C2 [×8], C3 [×2], C3, C4 [×4], C22 [×11], C22 [×28], S3 [×4], C6 [×2], C6 [×12], C6 [×27], C2×C4 [×6], D4 [×16], C23, C23 [×6], C23 [×14], C32, Dic3 [×4], C12 [×4], D6 [×4], D6 [×12], C2×C6 [×22], C2×C6 [×75], C22×C4, C2×D4 [×12], C24, C24, C3×S3 [×4], C3×C6, C3×C6 [×6], C3×C6 [×4], C2×Dic3 [×6], C3⋊D4 [×16], C2×C12 [×6], C3×D4 [×16], C22×S3 [×6], C22×S3 [×4], C22×C6 [×2], C22×C6 [×12], C22×C6 [×33], C22×D4, C3×Dic3 [×4], S3×C6 [×4], S3×C6 [×12], C62 [×11], C62 [×12], C22×Dic3, C2×C3⋊D4 [×12], C22×C12, C6×D4 [×12], S3×C23, C23×C6 [×2], C23×C6 [×2], C6×Dic3 [×6], C3×C3⋊D4 [×16], S3×C2×C6 [×6], S3×C2×C6 [×4], C2×C62, C2×C62 [×6], C2×C62 [×4], C22×C3⋊D4, D4×C2×C6, Dic3×C2×C6, C6×C3⋊D4 [×12], S3×C22×C6, C22×C62, C2×C6×C3⋊D4
Quotients: C1, C2 [×15], C3, C22 [×35], S3, C6 [×15], D4 [×4], C23 [×15], D6 [×7], C2×C6 [×35], C2×D4 [×6], C24, C3×S3, C3⋊D4 [×4], C3×D4 [×4], C22×S3 [×7], C22×C6 [×15], C22×D4, S3×C6 [×7], C2×C3⋊D4 [×6], C6×D4 [×6], S3×C23, C23×C6, C3×C3⋊D4 [×4], S3×C2×C6 [×7], C22×C3⋊D4, D4×C2×C6, C6×C3⋊D4 [×6], S3×C22×C6, C2×C6×C3⋊D4

Smallest permutation representation of C2×C6×C3⋊D4
On 48 points
Generators in S48
(1 22)(2 23)(3 24)(4 19)(5 20)(6 21)(7 35)(8 36)(9 31)(10 32)(11 33)(12 34)(13 29)(14 30)(15 25)(16 26)(17 27)(18 28)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)
(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 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 9 16 43)(2 10 17 44)(3 11 18 45)(4 12 13 46)(5 7 14 47)(6 8 15 48)(19 34 29 38)(20 35 30 39)(21 36 25 40)(22 31 26 41)(23 32 27 42)(24 33 28 37)
(1 12)(2 7)(3 8)(4 9)(5 10)(6 11)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)

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

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

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

108 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6N 6O ··· 6BO 6BP ··· 6BW 12A ··· 12H order 1 2 ··· 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 6 6 6 6 1 1 2 2 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D4 D6 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×C3⋊D4 kernel C2×C6×C3⋊D4 Dic3×C2×C6 C6×C3⋊D4 S3×C22×C6 C22×C62 C22×C3⋊D4 C22×Dic3 C2×C3⋊D4 S3×C23 C23×C6 C23×C6 C62 C22×C6 C24 C2×C6 C2×C6 C23 C22 # reps 1 1 12 1 1 2 2 24 2 2 1 4 7 2 8 8 14 16

Matrix representation of C2×C6×C3⋊D4 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 10 0 0 0 0 3 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 3
,
 12 0 0 0 0 1 0 0 0 0 0 12 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[10,0,0,0,0,3,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,3],[12,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C2×C6×C3⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_3\rtimes D_4
% in TeX

G:=Group("C2xC6xC3:D4");
// GroupNames label

G:=SmallGroup(288,1002);
// by ID

G=gap.SmallGroup(288,1002);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,1571,9414]);
// Polycyclic

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

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