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## G = C3×C23.8D6order 288 = 25·32

### Direct product of C3 and C23.8D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C23.8D6
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — Dic3×C12 — C3×C23.8D6
 Lower central C3 — C2×C6 — C3×C23.8D6
 Upper central C1 — C2×C6 — C3×C22⋊C4

Generators and relations for C3×C23.8D6
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=e5 >

Subgroups: 290 in 137 conjugacy classes, 58 normal (all characteristic)
C1, C2 [×3], C2, C3 [×2], C3, C4 [×6], C22, C22 [×3], C6 [×6], C6 [×7], C2×C4 [×2], C2×C4 [×4], C23, C32, Dic3 [×4], C12 [×10], C2×C6 [×2], C2×C6 [×11], C42, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×3], C3×C6 [×3], C3×C6, C2×Dic3 [×4], C2×C12 [×4], C2×C12 [×6], C22×C6 [×2], C22×C6, C422C2, C3×Dic3 [×4], C3×C12 [×2], C62, C62 [×3], C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C6×Dic3 [×4], C6×C12 [×2], C2×C62, C23.8D6, C3×C422C2, Dic3×C12, C3×Dic3⋊C4 [×2], C3×C4⋊Dic3, C3×C6.D4 [×2], C32×C22⋊C4, C3×C23.8D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], C23, D6 [×3], C2×C6 [×7], C4○D4 [×3], C3×S3, C22×S3, C22×C6, C422C2, S3×C6 [×3], C4○D12, D42S3 [×2], C3×C4○D4 [×3], S3×C2×C6, C23.8D6, C3×C422C2, C3×C4○D12, C3×D42S3 [×2], C3×C23.8D6

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A ··· 6F 6G ··· 6O 6P ··· 6W 12A 12B 12C 12D 12E ··· 12R 12S ··· 12Z 12AA 12AB 12AC 12AD order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 4 1 1 2 2 2 2 2 4 6 6 6 6 12 12 1 ··· 1 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4 6 ··· 6 12 12 12 12

72 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 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 C4○D12 C3×C4○D4 C3×C4○D12 D4⋊2S3 C3×D4⋊2S3 kernel C3×C23.8D6 Dic3×C12 C3×Dic3⋊C4 C3×C4⋊Dic3 C3×C6.D4 C32×C22⋊C4 C23.8D6 C4×Dic3 Dic3⋊C4 C4⋊Dic3 C6.D4 C3×C22⋊C4 C3×C22⋊C4 C2×C12 C22×C6 C3×C6 C22⋊C4 C2×C4 C23 C6 C6 C2 C6 C2 # reps 1 1 2 1 2 1 2 2 4 2 4 2 1 2 1 6 2 4 2 4 12 8 2 4

Matrix representation of C3×C23.8D6 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 12 0 0 0 0 1 0 0 0 3 12
,
 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 12 0 0 0 0 0 7 0 0 0 7 11
,
 0 8 0 0 5 0 0 0 0 0 11 10 0 0 6 2
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,12,0,0,0,0,1,3,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,0,0,0,0,0,7,7,0,0,0,11],[0,5,0,0,8,0,0,0,0,0,11,6,0,0,10,2] >;

C3×C23.8D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3._8D_6
% in TeX

G:=Group("C3xC2^3.8D6");
// GroupNames label

G:=SmallGroup(288,650);
// by ID

G=gap.SmallGroup(288,650);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,176,1598,555,9414]);
// 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=e^5>;
// generators/relations

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