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## G = S3×C62order 216 = 23·33

### Direct product of C62 and S3

Aliases: S3×C62, C6⋊C62, C6213C6, C335C23, C3⋊(C2×C62), (C3×C62)⋊5C2, (C32×C6)⋊4C22, C324(C22×C6), (C3×C6)⋊4(C2×C6), (C2×C6)⋊5(C3×C6), SmallGroup(216,174)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C62
 Chief series C1 — C3 — C32 — C33 — S3×C32 — S3×C3×C6 — S3×C62
 Lower central C3 — S3×C62
 Upper central C1 — C62

Generators and relations for S3×C62
G = < a,b,c,d | a6=b6=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 404 in 232 conjugacy classes, 126 normal (10 characteristic)
C1, C2 [×3], C2 [×4], C3, C3 [×4], C3 [×4], C22, C22 [×6], S3 [×4], C6 [×15], C6 [×28], C23, C32, C32 [×4], C32 [×4], D6 [×6], C2×C6, C2×C6 [×4], C2×C6 [×28], C3×S3 [×16], C3×C6 [×15], C3×C6 [×16], C22×S3, C22×C6 [×4], C33, S3×C6 [×24], C62, C62 [×4], C62 [×10], S3×C32 [×4], C32×C6 [×3], S3×C2×C6 [×4], C2×C62, S3×C3×C6 [×6], C3×C62, S3×C62
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], S3, C6 [×28], C23, C32, D6 [×3], C2×C6 [×28], C3×S3 [×4], C3×C6 [×7], C22×S3, C22×C6 [×4], S3×C6 [×12], C62 [×7], S3×C32, S3×C2×C6 [×4], C2×C62, S3×C3×C6 [×3], S3×C62

Smallest permutation representation of S3×C62
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 59 37 45 13 31)(2 60 38 46 14 32)(3 55 39 47 15 33)(4 56 40 48 16 34)(5 57 41 43 17 35)(6 58 42 44 18 36)(7 25 54 65 70 23)(8 26 49 66 71 24)(9 27 50 61 72 19)(10 28 51 62 67 20)(11 29 52 63 68 21)(12 30 53 64 69 22)
(1 39 17)(2 40 18)(3 41 13)(4 42 14)(5 37 15)(6 38 16)(7 68 50)(8 69 51)(9 70 52)(10 71 53)(11 72 54)(12 67 49)(19 65 29)(20 66 30)(21 61 25)(22 62 26)(23 63 27)(24 64 28)(31 55 43)(32 56 44)(33 57 45)(34 58 46)(35 59 47)(36 60 48)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 67)(7 39)(8 40)(9 41)(10 42)(11 37)(12 38)(13 52)(14 53)(15 54)(16 49)(17 50)(18 51)(19 57)(20 58)(21 59)(22 60)(23 55)(24 56)(25 47)(26 48)(27 43)(28 44)(29 45)(30 46)(31 63)(32 64)(33 65)(34 66)(35 61)(36 62)

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

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

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

S3×C62 is a maximal subgroup of   C62.77D6

108 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3H 3I ··· 3Q 6A ··· 6X 6Y ··· 6AY 6AZ ··· 6CE order 1 2 2 2 2 2 2 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 1 1 3 3 3 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3

108 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 kernel S3×C62 S3×C3×C6 C3×C62 S3×C2×C6 S3×C6 C62 C62 C3×C6 C2×C6 C6 # reps 1 6 1 8 48 8 1 3 8 24

Matrix representation of S3×C62 in GL4(𝔽7) generated by

 6 0 0 0 0 2 0 0 0 0 4 0 0 0 0 4
,
 3 0 0 0 0 6 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 2 0 0 0 0 4
,
 6 0 0 0 0 6 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(7))| [6,0,0,0,0,2,0,0,0,0,4,0,0,0,0,4],[3,0,0,0,0,6,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,4],[6,0,0,0,0,6,0,0,0,0,0,1,0,0,1,0] >;

S3×C62 in GAP, Magma, Sage, TeX

S_3\times C_6^2
% in TeX

G:=Group("S3xC6^2");
// GroupNames label

G:=SmallGroup(216,174);
// by ID

G=gap.SmallGroup(216,174);
# by ID

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

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

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