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## G = C6×C3⋊Dic3order 216 = 23·33

### Direct product of C6 and C3⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C6×C3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C3×C3⋊Dic3 — C6×C3⋊Dic3
 Lower central C32 — C6×C3⋊Dic3
 Upper central C1 — C2×C6

Generators and relations for C6×C3⋊Dic3
G = < a,b,c,d | a6=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 272 in 136 conjugacy classes, 66 normal (14 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4 [×2], C22, C6, C6 [×14], C6 [×12], C2×C4, C32, C32 [×4], C32 [×4], Dic3 [×8], C12 [×2], C2×C6, C2×C6 [×4], C2×C6 [×4], C3×C6, C3×C6 [×14], C3×C6 [×12], C2×Dic3 [×4], C2×C12, C33, C3×Dic3 [×8], C3⋊Dic3 [×2], C62, C62 [×4], C62 [×4], C32×C6, C32×C6 [×2], C6×Dic3 [×4], C2×C3⋊Dic3, C3×C3⋊Dic3 [×2], C3×C62, C6×C3⋊Dic3
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3 [×4], C6 [×3], C2×C4, Dic3 [×8], C12 [×2], D6 [×4], C2×C6, C3×S3 [×4], C3⋊S3, C2×Dic3 [×4], C2×C12, C3×Dic3 [×8], C3⋊Dic3 [×2], S3×C6 [×4], C2×C3⋊S3, C3×C3⋊S3, C6×Dic3 [×4], C2×C3⋊Dic3, C3×C3⋊Dic3 [×2], C6×C3⋊S3, C6×C3⋊Dic3

Smallest permutation representation of C6×C3⋊Dic3
On 72 points
Generators in S72
(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 57 44)(2 58 45)(3 59 46)(4 60 47)(5 55 48)(6 56 43)(7 36 63)(8 31 64)(9 32 65)(10 33 66)(11 34 61)(12 35 62)(13 50 23)(14 51 24)(15 52 19)(16 53 20)(17 54 21)(18 49 22)(25 37 70)(26 38 71)(27 39 72)(28 40 67)(29 41 68)(30 42 69)
(1 51 59 20 48 18)(2 52 60 21 43 13)(3 53 55 22 44 14)(4 54 56 23 45 15)(5 49 57 24 46 16)(6 50 58 19 47 17)(7 26 34 42 65 67)(8 27 35 37 66 68)(9 28 36 38 61 69)(10 29 31 39 62 70)(11 30 32 40 63 71)(12 25 33 41 64 72)
(1 32 20 71)(2 33 21 72)(3 34 22 67)(4 35 23 68)(5 36 24 69)(6 31 19 70)(7 14 42 55)(8 15 37 56)(9 16 38 57)(10 17 39 58)(11 18 40 59)(12 13 41 60)(25 43 64 52)(26 44 65 53)(27 45 66 54)(28 46 61 49)(29 47 62 50)(30 48 63 51)

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

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

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

C6×C3⋊Dic3 is a maximal subgroup of
C3×Dic32  C62.77D6  C62.79D6  C62.80D6  C62.81D6  C62.82D6  C336C42  C62.84D6  C62.85D6  C3312M4(2)  S3×C6×Dic3  C62.90D6  C62.96D6  C3⋊S3×C2×C12

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3N 4A 4B 4C 4D 6A ··· 6F 6G ··· 6AP 12A ··· 12H order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 1 1 2 ··· 2 9 9 9 9 1 ··· 1 2 ··· 2 9 ··· 9

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 Dic3 D6 C3×S3 C3×Dic3 S3×C6 kernel C6×C3⋊Dic3 C3×C3⋊Dic3 C3×C62 C2×C3⋊Dic3 C32×C6 C3⋊Dic3 C62 C3×C6 C62 C3×C6 C3×C6 C2×C6 C6 C6 # reps 1 2 1 2 4 4 2 8 4 8 4 8 16 8

Matrix representation of C6×C3⋊Dic3 in GL5(𝔽13)

 9 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 10 0 0 0 0 0 10
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 9
,
 12 0 0 0 0 0 4 11 0 0 0 0 10 0 0 0 0 0 12 0 0 0 0 0 12
,
 5 0 0 0 0 0 5 0 0 0 0 11 8 0 0 0 0 0 0 1 0 0 0 12 0

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

C6×C3⋊Dic3 in GAP, Magma, Sage, TeX

C_6\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("C6xC3:Dic3");
// GroupNames label

G:=SmallGroup(216,143);
// by ID

G=gap.SmallGroup(216,143);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,1444,5189]);
// Polycyclic

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

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