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G = C3×C23.14D6order 288 = 25·32

Direct product of C3 and C23.14D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C23.14D6
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C6×C3⋊D4 — C3×C23.14D6
 Lower central C3 — C2×C6 — C3×C23.14D6
 Upper central C1 — C2×C6 — C6×D4

Generators and relations for C3×C23.14D6
G = < a,b,c,d,e,f | a3=b2=c2=d2=e6=1, f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de-1 >

Subgroups: 522 in 215 conjugacy classes, 70 normal (58 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C3×S3, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C4⋊D4, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C62, C62, C62, Dic3⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C6×Dic3, C6×Dic3, C3×C3⋊D4, C6×C12, D4×C32, S3×C2×C6, C2×C62, C23.14D6, C3×C4⋊D4, C3×Dic3⋊C4, C3×D6⋊C4, C3×C6.D4, Dic3×C2×C6, C6×C3⋊D4, D4×C3×C6, C3×C23.14D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C4○D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C4⋊D4, S3×C6, S3×D4, D42S3, C2×C3⋊D4, C6×D4, C3×C4○D4, C3×C3⋊D4, S3×C2×C6, C23.14D6, C3×C4⋊D4, C3×S3×D4, C3×D42S3, C6×C3⋊D4, C3×C23.14D6

Smallest permutation representation of C3×C23.14D6
On 48 points
Generators in S48
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(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)(2 21)(3 26)(4 23)(5 28)(6 19)(7 34)(8 42)(9 36)(10 38)(11 32)(12 40)(13 20)(14 25)(15 22)(16 27)(17 24)(18 29)(31 48)(33 44)(35 46)(37 47)(39 43)(41 45)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 26)(20 27)(21 28)(22 29)(23 30)(24 25)(31 41)(32 42)(33 37)(34 38)(35 39)(36 40)(43 46)(44 47)(45 48)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 45)(8 46)(9 47)(10 48)(11 43)(12 44)(19 29)(20 30)(21 25)(22 26)(23 27)(24 28)(31 38)(32 39)(33 40)(34 41)(35 42)(36 37)
(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 40 13 33)(2 32 14 39)(3 38 15 31)(4 36 16 37)(5 42 17 35)(6 34 18 41)(7 26 45 22)(8 21 46 25)(9 30 47 20)(10 19 48 29)(11 28 43 24)(12 23 44 27)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6S 6T ··· 6AG 6AH 6AI 12A ··· 12H 12I ··· 12P 12Q 12R order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 6 12 ··· 12 12 ··· 12 12 12 size 1 1 1 1 2 2 4 12 1 1 2 2 2 4 6 6 6 6 12 1 ··· 1 2 ··· 2 4 ··· 4 12 12 4 ··· 4 6 ··· 6 12 12

72 irreducible representations

 dim 1 1 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 4 4 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 C4○D4 C3×S3 C3×D4 C3⋊D4 C3×D4 S3×C6 S3×C6 C3×C4○D4 C3×C3⋊D4 S3×D4 D4⋊2S3 C3×S3×D4 C3×D4⋊2S3 kernel C3×C23.14D6 C3×Dic3⋊C4 C3×D6⋊C4 C3×C6.D4 Dic3×C2×C6 C6×C3⋊D4 D4×C3×C6 C23.14D6 Dic3⋊C4 D6⋊C4 C6.D4 C22×Dic3 C2×C3⋊D4 C6×D4 C6×D4 C3×Dic3 C62 C2×C12 C22×C6 C3×C6 C2×D4 Dic3 C2×C6 C2×C6 C2×C4 C23 C6 C22 C6 C6 C2 C2 # reps 1 1 1 1 1 2 1 2 2 2 2 2 4 2 1 2 2 1 2 2 2 4 4 4 2 4 4 8 1 1 2 2

Matrix representation of C3×C23.14D6 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 2 12 0 0 0 0 0 1 0 0 1 0
,
 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 4 0 0 0 7 10 0 0 0 0 12 0 0 0 0 1
,
 11 2 0 0 5 2 0 0 0 0 0 8 0 0 8 0
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[1,2,0,0,0,12,0,0,0,0,0,1,0,0,1,0],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[4,7,0,0,0,10,0,0,0,0,12,0,0,0,0,1],[11,5,0,0,2,2,0,0,0,0,0,8,0,0,8,0] >;

C3×C23.14D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{14}D_6
% in TeX

G:=Group("C3xC2^3.14D6");
// GroupNames label

G:=SmallGroup(288,710);
// by ID

G=gap.SmallGroup(288,710);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,590,555,9414]);
// Polycyclic

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

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