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## G = C3×C23.9D6order 288 = 25·32

### Direct product of C3 and C23.9D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C23.9D6
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C2×C12 — C3×C23.9D6
 Lower central C3 — C2×C6 — C3×C23.9D6
 Upper central C1 — C2×C6 — C3×C22⋊C4

Generators and relations for C3×C23.9D6
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e6=f2=c, 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=e5 >

Subgroups: 418 in 173 conjugacy classes, 62 normal (58 characteristic)
C1, C2 [×3], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×7], S3 [×2], C6 [×6], C6 [×9], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C32, Dic3 [×3], C12 [×9], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×15], C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, C3×S3 [×2], C3×C6 [×3], C3×C6, C4×S3 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×4], C2×C12 [×7], C3×D4 [×2], C22×S3, C22×C6 [×2], C22×C6 [×2], C22.D4, C3×Dic3 [×3], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, C62 [×3], Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4 [×2], C3×C22⋊C4 [×3], C3×C4⋊C4 [×2], S3×C2×C4, C2×C3⋊D4, C22×C12, C6×D4, S3×C12 [×2], C6×Dic3 [×3], C3×C3⋊D4 [×2], C6×C12 [×2], S3×C2×C6, C2×C62, C23.9D6, C3×C22.D4, C3×Dic3⋊C4, C3×C4⋊Dic3, C3×D6⋊C4, C3×C6.D4, C32×C22⋊C4, S3×C2×C12, C6×C3⋊D4, C3×C23.9D6
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, C22.D4, S3×C6 [×3], C4○D12, S3×D4, D42S3, C6×D4, C3×C4○D4 [×2], S3×C2×C6, C23.9D6, C3×C22.D4, C3×C4○D12, C3×S3×D4, C3×D42S3, C3×C23.9D6

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 6A ··· 6F 6G ··· 6O 6P ··· 6W 6X 6Y 6Z 6AA 12A 12B 12C 12D 12E ··· 12R 12S 12T 12U 12V 12W 12X 12Y 12Z order 1 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 12 12 12 size 1 1 1 1 4 6 6 1 1 2 2 2 2 2 4 6 6 12 12 1 ··· 1 2 ··· 2 4 ··· 4 6 6 6 6 2 2 2 2 4 ··· 4 6 6 6 6 12 12 12 12

72 irreducible representations

 dim 1 1 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 C2 C3 C6 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×C23.9D6 C3×Dic3⋊C4 C3×C4⋊Dic3 C3×D6⋊C4 C3×C6.D4 C32×C22⋊C4 S3×C2×C12 C6×C3⋊D4 C23.9D6 Dic3⋊C4 C4⋊Dic3 D6⋊C4 C6.D4 C3×C22⋊C4 S3×C2×C4 C2×C3⋊D4 C3×C22⋊C4 S3×C6 C2×C12 C22×C6 C3×C6 C22⋊C4 D6 C2×C4 C23 C6 C6 C2 C6 C6 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 2 2 1 4 2 4 4 2 4 8 8 1 1 2 2

Matrix representation of C3×C23.9D6 in GL4(𝔽13) generated by

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

C3×C23.9D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3._9D_6
% in TeX

G:=Group("C3xC2^3.9D6");
// GroupNames label

G:=SmallGroup(288,654);
// by ID

G=gap.SmallGroup(288,654);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,176,590,555,9414]);
// Polycyclic

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

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