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## G = C3×C12.59D6order 432 = 24·33

### Direct product of C3 and C12.59D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C12.59D6
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C6×C3⋊S3 — C12×C3⋊S3 — C3×C12.59D6
 Lower central C32 — C3×C6 — C3×C12.59D6
 Upper central C1 — C12 — C2×C12

Generators and relations for C3×C12.59D6
G = < a,b,c,d | a3=b12=c6=1, d2=b6, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=b6c-1 >

Subgroups: 884 in 304 conjugacy classes, 94 normal (30 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, C62, C62, C4○D12, C3×C4○D4, C3×C3⋊S3, C32×C6, C32×C6, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, C6×C12, C6×C12, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C3×C62, C3×C4○D12, C12.59D6, C3×C324Q8, C12×C3⋊S3, C3×C12⋊S3, C3×C327D4, C3×C6×C12, C3×C12.59D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C22×S3, C22×C6, S3×C6, C2×C3⋊S3, C4○D12, C3×C4○D4, C3×C3⋊S3, S3×C2×C6, C22×C3⋊S3, C6×C3⋊S3, C3×C4○D12, C12.59D6, C2×C6×C3⋊S3, C3×C12.59D6

Smallest permutation representation of C3×C12.59D6
On 72 points
Generators in S72
(1 24 29)(2 13 30)(3 14 31)(4 15 32)(5 16 33)(6 17 34)(7 18 35)(8 19 36)(9 20 25)(10 21 26)(11 22 27)(12 23 28)(37 52 71)(38 53 72)(39 54 61)(40 55 62)(41 56 63)(42 57 64)(43 58 65)(44 59 66)(45 60 67)(46 49 68)(47 50 69)(48 51 70)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 24 29)(2 13 30)(3 14 31)(4 15 32)(5 16 33)(6 17 34)(7 18 35)(8 19 36)(9 20 25)(10 21 26)(11 22 27)(12 23 28)(37 65 52 43 71 58)(38 66 53 44 72 59)(39 67 54 45 61 60)(40 68 55 46 62 49)(41 69 56 47 63 50)(42 70 57 48 64 51)
(1 47 7 41)(2 40 8 46)(3 45 9 39)(4 38 10 44)(5 43 11 37)(6 48 12 42)(13 55 19 49)(14 60 20 54)(15 53 21 59)(16 58 22 52)(17 51 23 57)(18 56 24 50)(25 61 31 67)(26 66 32 72)(27 71 33 65)(28 64 34 70)(29 69 35 63)(30 62 36 68)

G:=sub<Sym(72)| (1,24,29)(2,13,30)(3,14,31)(4,15,32)(5,16,33)(6,17,34)(7,18,35)(8,19,36)(9,20,25)(10,21,26)(11,22,27)(12,23,28)(37,52,71)(38,53,72)(39,54,61)(40,55,62)(41,56,63)(42,57,64)(43,58,65)(44,59,66)(45,60,67)(46,49,68)(47,50,69)(48,51,70), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,24,29)(2,13,30)(3,14,31)(4,15,32)(5,16,33)(6,17,34)(7,18,35)(8,19,36)(9,20,25)(10,21,26)(11,22,27)(12,23,28)(37,65,52,43,71,58)(38,66,53,44,72,59)(39,67,54,45,61,60)(40,68,55,46,62,49)(41,69,56,47,63,50)(42,70,57,48,64,51), (1,47,7,41)(2,40,8,46)(3,45,9,39)(4,38,10,44)(5,43,11,37)(6,48,12,42)(13,55,19,49)(14,60,20,54)(15,53,21,59)(16,58,22,52)(17,51,23,57)(18,56,24,50)(25,61,31,67)(26,66,32,72)(27,71,33,65)(28,64,34,70)(29,69,35,63)(30,62,36,68)>;

G:=Group( (1,24,29)(2,13,30)(3,14,31)(4,15,32)(5,16,33)(6,17,34)(7,18,35)(8,19,36)(9,20,25)(10,21,26)(11,22,27)(12,23,28)(37,52,71)(38,53,72)(39,54,61)(40,55,62)(41,56,63)(42,57,64)(43,58,65)(44,59,66)(45,60,67)(46,49,68)(47,50,69)(48,51,70), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,24,29)(2,13,30)(3,14,31)(4,15,32)(5,16,33)(6,17,34)(7,18,35)(8,19,36)(9,20,25)(10,21,26)(11,22,27)(12,23,28)(37,65,52,43,71,58)(38,66,53,44,72,59)(39,67,54,45,61,60)(40,68,55,46,62,49)(41,69,56,47,63,50)(42,70,57,48,64,51), (1,47,7,41)(2,40,8,46)(3,45,9,39)(4,38,10,44)(5,43,11,37)(6,48,12,42)(13,55,19,49)(14,60,20,54)(15,53,21,59)(16,58,22,52)(17,51,23,57)(18,56,24,50)(25,61,31,67)(26,66,32,72)(27,71,33,65)(28,64,34,70)(29,69,35,63)(30,62,36,68) );

G=PermutationGroup([[(1,24,29),(2,13,30),(3,14,31),(4,15,32),(5,16,33),(6,17,34),(7,18,35),(8,19,36),(9,20,25),(10,21,26),(11,22,27),(12,23,28),(37,52,71),(38,53,72),(39,54,61),(40,55,62),(41,56,63),(42,57,64),(43,58,65),(44,59,66),(45,60,67),(46,49,68),(47,50,69),(48,51,70)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,24,29),(2,13,30),(3,14,31),(4,15,32),(5,16,33),(6,17,34),(7,18,35),(8,19,36),(9,20,25),(10,21,26),(11,22,27),(12,23,28),(37,65,52,43,71,58),(38,66,53,44,72,59),(39,67,54,45,61,60),(40,68,55,46,62,49),(41,69,56,47,63,50),(42,70,57,48,64,51)], [(1,47,7,41),(2,40,8,46),(3,45,9,39),(4,38,10,44),(5,43,11,37),(6,48,12,42),(13,55,19,49),(14,60,20,54),(15,53,21,59),(16,58,22,52),(17,51,23,57),(18,56,24,50),(25,61,31,67),(26,66,32,72),(27,71,33,65),(28,64,34,70),(29,69,35,63),(30,62,36,68)]])

126 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C ··· 3N 4A 4B 4C 4D 4E 6A 6B 6C ··· 6AN 6AO 6AP 6AQ 6AR 12A 12B 12C 12D 12E ··· 12BB 12BC 12BD 12BE 12BF order 1 2 2 2 2 3 3 3 ··· 3 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 2 18 18 1 1 2 ··· 2 1 1 2 18 18 1 1 2 ··· 2 18 18 18 18 1 1 1 1 2 ··· 2 18 18 18 18

126 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 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 kernel C3×C12.59D6 C3×C32⋊4Q8 C12×C3⋊S3 C3×C12⋊S3 C3×C32⋊7D4 C3×C6×C12 C12.59D6 C32⋊4Q8 C4×C3⋊S3 C12⋊S3 C32⋊7D4 C6×C12 C6×C12 C3×C12 C62 C33 C2×C12 C12 C2×C6 C32 C32 C3 # reps 1 1 2 1 2 1 2 2 4 2 4 2 4 8 4 2 8 16 8 16 4 32

Matrix representation of C3×C12.59D6 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
,
 3 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 3 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 9 0 0 0 0 0 0 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 0 12
,
 0 3 0 0 0 0 9 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0

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],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[9,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[0,9,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;

C3×C12.59D6 in GAP, Magma, Sage, TeX

C_3\times C_{12}._{59}D_6
% in TeX

G:=Group("C3xC12.59D6");
// GroupNames label

G:=SmallGroup(432,713);
// by ID

G=gap.SmallGroup(432,713);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^6=1,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=b^6*c^-1>;
// generators/relations

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