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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 [×3], C2 [×2], C3 [×2], C3, C4 [×6], C22, C22 [×6], S3, C6 [×6], C6 [×8], C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×2], C23, C23, C32, Dic3 [×2], Dic3 [×2], C12 [×10], D6 [×3], C2×C6 [×2], C2×C6 [×14], C42, C22⋊C4, C22⋊C4 [×3], C2×D4, C2×Q8, C3×S3, C3×C6 [×3], C3×C6, Dic6 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×4], C2×C12 [×5], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6 [×2], C22×C6 [×2], C4.4D4, C3×Dic3 [×2], C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×3], C62, C62 [×3], C4×Dic3, D6⋊C4 [×2], C6.D4, C4×C12, C3×C22⋊C4 [×2], C3×C22⋊C4 [×4], C2×Dic6, C2×C3⋊D4, C6×D4, C6×Q8, C3×Dic6 [×2], C6×Dic3 [×3], C3×C3⋊D4 [×2], C6×C12 [×2], S3×C2×C6, C2×C62, C23.11D6, C3×C4.4D4, Dic3×C12, C3×D6⋊C4 [×2], C3×C6.D4, C32×C22⋊C4, C6×Dic6, C6×C3⋊D4, C3×C23.11D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C4○D4 [×2], C3×S3, C3×D4 [×2], C22×S3, C22×C6, C4.4D4, S3×C6 [×3], C4○D12, S3×D4, D42S3, C6×D4, C3×C4○D4 [×2], 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 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(2 32)(4 34)(6 36)(8 26)(10 28)(12 30)(13 43)(14 20)(15 45)(16 22)(17 47)(18 24)(19 37)(21 39)(23 41)(38 44)(40 46)(42 48)
(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 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(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 24 25 42)(2 41 26 23)(3 22 27 40)(4 39 28 21)(5 20 29 38)(6 37 30 19)(7 18 31 48)(8 47 32 17)(9 16 33 46)(10 45 34 15)(11 14 35 44)(12 43 36 13)

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

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

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

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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