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## G = C3×C23.11D6order 288 = 25·32

### Direct product of C3 and C23.11D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C23.11D6
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C3×D6⋊C4 — C3×C23.11D6
 Lower central C3 — C2×C6 — C3×C23.11D6
 Upper central C1 — C2×C6 — C3×C22⋊C4

Generators and relations for C3×C23.11D6
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e6=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >

Subgroups: 418 in 169 conjugacy classes, 62 normal (58 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C2×D4, C2×Q8, C3×S3, C3×C6, C3×C6, Dic6, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C22×C6, C4.4D4, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C62, C62, C4×Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C6×D4, C6×Q8, C3×Dic6, C6×Dic3, C3×C3⋊D4, C6×C12, S3×C2×C6, C2×C62, C23.11D6, C3×C4.4D4, Dic3×C12, C3×D6⋊C4, C3×C6.D4, C32×C22⋊C4, C6×Dic6, C6×C3⋊D4, C3×C23.11D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C4○D4, C3×S3, C3×D4, C22×S3, C22×C6, C4.4D4, S3×C6, C4○D12, S3×D4, D42S3, C6×D4, C3×C4○D4, S3×C2×C6, C23.11D6, C3×C4.4D4, C3×C4○D12, C3×S3×D4, C3×D42S3, C3×C23.11D6

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6O 6P ··· 6W 6X 6Y 12A 12B 12C 12D 12E ··· 12R 12S ··· 12Z 12AA 12AB order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 12 12 size 1 1 1 1 4 12 1 1 2 2 2 2 2 4 6 6 6 6 12 1 ··· 1 2 ··· 2 4 ··· 4 12 12 2 2 2 2 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 4 4 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 S3 D4 D6 D6 C4○D4 C3×S3 C3×D4 S3×C6 S3×C6 C4○D12 C3×C4○D4 C3×C4○D12 S3×D4 D4⋊2S3 C3×S3×D4 C3×D4⋊2S3 kernel C3×C23.11D6 Dic3×C12 C3×D6⋊C4 C3×C6.D4 C32×C22⋊C4 C6×Dic6 C6×C3⋊D4 C23.11D6 C4×Dic3 D6⋊C4 C6.D4 C3×C22⋊C4 C2×Dic6 C2×C3⋊D4 C3×C22⋊C4 C3×Dic3 C2×C12 C22×C6 C3×C6 C22⋊C4 Dic3 C2×C4 C23 C6 C6 C2 C6 C6 C2 C2 # reps 1 1 2 1 1 1 1 2 2 4 2 2 2 2 1 2 2 1 4 2 4 4 2 4 8 8 1 1 2 2

Matrix representation of C3×C23.11D6 in GL6(𝔽13)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 5 12 0 0 0 0 0 0 1 0 0 0 0 0 10 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 9 0 0 0 0 0 0 3 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 1 5 0 0 0 0 0 12
,
 0 3 0 0 0 0 9 0 0 0 0 0 0 0 8 2 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 0 0 2 5

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,5,0,0,0,0,0,12,0,0,0,0,0,0,1,10,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[9,0,0,0,0,0,0,3,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,5,12],[0,9,0,0,0,0,3,0,0,0,0,0,0,0,8,0,0,0,0,0,2,5,0,0,0,0,0,0,8,2,0,0,0,0,0,5] >;

C3×C23.11D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{11}D_6
% in TeX

G:=Group("C3xC2^3.11D6");
// GroupNames label

G:=SmallGroup(288,656);
// by ID

G=gap.SmallGroup(288,656);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,176,1598,555,9414]);
// Polycyclic

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

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