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## G = C2×C6×C3⋊S3order 216 = 23·33

### Direct product of C2×C6 and C3⋊S3

Aliases: C2×C6×C3⋊S3, C6214C6, C6213S3, C336C23, C62(S3×C6), (C3×C6)⋊7D6, (C3×C62)⋊6C2, C328(C22×S3), (C32×C6)⋊5C22, C325(C22×C6), C32(S3×C2×C6), (C2×C6)⋊7(C3×S3), (C3×C6)⋊5(C2×C6), SmallGroup(216,175)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C6×C3⋊S3
 Chief series C1 — C3 — C32 — C33 — C3×C3⋊S3 — C6×C3⋊S3 — C2×C6×C3⋊S3
 Lower central C32 — C2×C6×C3⋊S3
 Upper central C1 — C2×C6

Generators and relations for C2×C6×C3⋊S3
G = < a,b,c,d,e | a2=b6=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 624 in 232 conjugacy classes, 82 normal (10 characteristic)
C1, C2 [×3], C2 [×4], C3, C3 [×4], C3 [×4], C22, C22 [×6], S3 [×16], C6 [×15], C6 [×16], C23, C32, C32 [×4], C32 [×4], D6 [×24], C2×C6, C2×C6 [×4], C2×C6 [×10], C3×S3 [×16], C3⋊S3 [×4], C3×C6 [×15], C3×C6 [×12], C22×S3 [×4], C22×C6, C33, S3×C6 [×24], C2×C3⋊S3 [×6], C62, C62 [×4], C62 [×4], C3×C3⋊S3 [×4], C32×C6 [×3], S3×C2×C6 [×4], C22×C3⋊S3, C6×C3⋊S3 [×6], C3×C62, C2×C6×C3⋊S3
Quotients: C1, C2 [×7], C3, C22 [×7], S3 [×4], C6 [×7], C23, D6 [×12], C2×C6 [×7], C3×S3 [×4], C3⋊S3, C22×S3 [×4], C22×C6, S3×C6 [×12], C2×C3⋊S3 [×3], C3×C3⋊S3, S3×C2×C6 [×4], C22×C3⋊S3, C6×C3⋊S3 [×3], C2×C6×C3⋊S3

Smallest permutation representation of C2×C6×C3⋊S3
On 72 points
Generators in S72
(1 40)(2 41)(3 42)(4 37)(5 38)(6 39)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 60)(14 55)(15 56)(16 57)(17 58)(18 59)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(49 67)(50 68)(51 69)(52 70)(53 71)(54 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 25 22)(2 26 23)(3 27 24)(4 28 19)(5 29 20)(6 30 21)(7 17 70)(8 18 71)(9 13 72)(10 14 67)(11 15 68)(12 16 69)(31 37 46)(32 38 47)(33 39 48)(34 40 43)(35 41 44)(36 42 45)(49 64 55)(50 65 56)(51 66 57)(52 61 58)(53 62 59)(54 63 60)
(1 27 20)(2 28 21)(3 29 22)(4 30 23)(5 25 24)(6 26 19)(7 15 72)(8 16 67)(9 17 68)(10 18 69)(11 13 70)(12 14 71)(31 39 44)(32 40 45)(33 41 46)(34 42 47)(35 37 48)(36 38 43)(49 62 57)(50 63 58)(51 64 59)(52 65 60)(53 66 55)(54 61 56)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)

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

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

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

C2×C6×C3⋊S3 is a maximal subgroup of   C62.78D6  C62.84D6  C6211Dic3  C6224D6  S32×C2×C6

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3N 6A ··· 6F 6G ··· 6AP 6AQ ··· 6AX order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 1 1 9 9 9 9 1 1 2 ··· 2 1 ··· 1 2 ··· 2 9 ··· 9

72 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 C2×C6×C3⋊S3 C6×C3⋊S3 C3×C62 C22×C3⋊S3 C2×C3⋊S3 C62 C62 C3×C6 C2×C6 C6 # reps 1 6 1 2 12 2 4 12 8 24

Matrix representation of C2×C6×C3⋊S3 in GL5(𝔽7)

 6 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 5 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 6 0 0 0 0 0 6
,
 1 0 0 0 0 0 2 0 0 0 0 0 4 0 0 0 0 0 2 0 0 0 0 0 4
,
 1 0 0 0 0 0 4 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 4
,
 6 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(7))| [6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,6,0,0,0,0,0,6],[1,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,4],[6,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0] >;

C2×C6×C3⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_3\rtimes S_3
% in TeX

G:=Group("C2xC6xC3:S3");
// GroupNames label

G:=SmallGroup(216,175);
// by ID

G=gap.SmallGroup(216,175);
# by ID

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

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

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