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## G = C3×C23.12D6order 288 = 25·32

### Direct product of C3 and C23.12D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C23.12D6
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — Dic3×C12 — C3×C23.12D6
 Lower central C3 — C2×C6 — C3×C23.12D6
 Upper central C1 — C2×C6 — C6×D4

Generators and relations for C3×C23.12D6
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, ebe-1=bc=cb, fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 394 in 179 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C3×C6, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C4.4D4, C3×Dic3, C3×C12, C62, C62, C4×Dic3, C6.D4, C4×C12, C3×C22⋊C4, C2×Dic6, C6×D4, C6×D4, C6×Q8, C3×Dic6, C6×Dic3, C6×C12, D4×C32, C2×C62, C23.12D6, C3×C4.4D4, Dic3×C12, C3×C6.D4, C6×Dic6, D4×C3×C6, C3×C23.12D6
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, C4.4D4, S3×C6, D42S3, C2×C3⋊D4, C6×D4, C3×C4○D4, C3×C3⋊D4, S3×C2×C6, C23.12D6, C3×C4.4D4, C3×D42S3, C6×C3⋊D4, C3×C23.12D6

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

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

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

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

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 ··· 6AE 12A 12B 12C 12D 12E ··· 12J 12K ··· 12R 12S 12T 12U 12V 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 12 12 12 12 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 4 4 1 1 2 2 2 2 2 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 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D4 D6 D6 C4○D4 C3×S3 C3⋊D4 C3×D4 S3×C6 S3×C6 C3×C4○D4 C3×C3⋊D4 D4⋊2S3 C3×D4⋊2S3 kernel C3×C23.12D6 Dic3×C12 C3×C6.D4 C6×Dic6 D4×C3×C6 C23.12D6 C4×Dic3 C6.D4 C2×Dic6 C6×D4 C6×D4 C3×C12 C2×C12 C22×C6 C3×C6 C2×D4 C12 C12 C2×C4 C23 C6 C4 C6 C2 # reps 1 1 4 1 1 2 2 8 2 2 1 2 1 2 4 2 4 4 2 4 8 8 2 4

Matrix representation of C3×C23.12D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 3 12 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 1 12
,
 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
,
 12 0 0 0 0 0 0 12 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 10 4 0 0 0 0 0 0 12 2 0 0 0 0 12 1
,
 8 12 0 0 0 0 0 5 0 0 0 0 0 0 4 8 0 0 0 0 3 9 0 0 0 0 0 0 5 3 0 0 0 0 5 8

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

C3×C23.12D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{12}D_6
% in TeX

G:=Group("C3xC2^3.12D6");
// GroupNames label

G:=SmallGroup(288,707);
// by ID

G=gap.SmallGroup(288,707);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,176,1598,303,9414]);
// 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,e*b*e^-1=b*c=c*b,f*b*f^-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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