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G = C3×D6.6D6order 432 = 24·33

Direct product of C3 and D6.6D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×D6.6D6
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×C3⋊D12 — C3×D6.6D6
 Lower central C32 — C3×C6 — C3×D6.6D6
 Upper central C1 — C6 — C12

Generators and relations for C3×D6.6D6
G = < a,b,c,d,e | a3=b6=c2=1, d6=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=b3d5 >

Subgroups: 744 in 210 conjugacy classes, 64 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C33, C3×Dic3, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, C4○D12, Q83S3, C3×C4○D4, S3×C32, C3×C3⋊S3, C32×C6, C6.D6, C3⋊D12, C3×Dic6, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×C3⋊D4, C12⋊S3, C6×C12, Q8×C32, C32×Dic3, C32×Dic3, C32×C12, S3×C3×C6, C6×C3⋊S3, D6.6D6, C3×C4○D12, C3×Q83S3, C3×C6.D6, C3×C3⋊D12, C32×Dic6, S3×C3×C12, C3×C12⋊S3, C3×D6.6D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S32, S3×C6, C4○D12, Q83S3, C3×C4○D4, C2×S32, S3×C2×C6, C3×S32, D6.6D6, C3×C4○D12, C3×Q83S3, S32×C6, C3×D6.6D6

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

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

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

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

81 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 4E 6A 6B 6C ··· 6H 6I 6J 6K 6L ··· 6S 6T 6U 6V 6W 12A ··· 12H 12I 12J 12K 12L 12M ··· 12U 12V ··· 12AE 12AF ··· 12AK order 1 2 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 6 6 6 ··· 6 6 6 6 6 12 ··· 12 12 12 12 12 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 6 18 18 1 1 2 ··· 2 4 4 4 2 3 3 6 6 1 1 2 ··· 2 4 4 4 6 ··· 6 18 18 18 18 2 ··· 2 3 3 3 3 4 ··· 4 6 ··· 6 12 ··· 12

81 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 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 S3 D6 D6 D6 C4○D4 C3×S3 C3×S3 S3×C6 S3×C6 S3×C6 C4○D12 C3×C4○D4 C3×C4○D12 S32 Q8⋊3S3 C2×S32 C3×S32 D6.6D6 C3×Q8⋊3S3 S32×C6 C3×D6.6D6 kernel C3×D6.6D6 C3×C6.D6 C3×C3⋊D12 C32×Dic6 S3×C3×C12 C3×C12⋊S3 D6.6D6 C6.D6 C3⋊D12 C3×Dic6 S3×C12 C12⋊S3 C3×Dic6 S3×C12 C3×Dic3 C3×C12 S3×C6 C33 Dic6 C4×S3 Dic3 C12 D6 C32 C32 C3 C12 C32 C6 C4 C3 C3 C2 C1 # reps 1 2 2 1 1 1 2 4 4 2 2 2 1 1 3 2 1 2 2 2 6 4 2 4 4 8 1 1 1 2 2 2 2 4

Matrix representation of C3×D6.6D6 in GL6(𝔽13)

 3 0 0 0 0 0 0 3 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12 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 12 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 11 0 0 0 0 0 5 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 10 8 0 0 0 0 2 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 0 0 0 12

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

C3×D6.6D6 in GAP, Magma, Sage, TeX

C_3\times D_6._6D_6
% in TeX

G:=Group("C3xD6.6D6");
// GroupNames label

G:=SmallGroup(432,647);
// by ID

G=gap.SmallGroup(432,647);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,365,176,303,142,2028,14118]);
// Polycyclic

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

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