direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C6×C3⋊S3, C62⋊14C6, C62⋊13S3, C33⋊6C23, C6⋊2(S3×C6), (C3×C6)⋊7D6, (C3×C62)⋊6C2, C32⋊8(C22×S3), (C32×C6)⋊5C22, C32⋊5(C22×C6), C3⋊2(S3×C2×C6), (C2×C6)⋊7(C3×S3), (C3×C6)⋊5(C2×C6), SmallGroup(216,175)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C2×C6×C3⋊S3 |
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, C2, C3, C3, C3, C22, C22, S3, C6, C6, C23, C32, C32, C32, D6, C2×C6, C2×C6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C22×S3, C22×C6, C33, S3×C6, C2×C3⋊S3, C62, C62, C62, C3×C3⋊S3, C32×C6, S3×C2×C6, C22×C3⋊S3, C6×C3⋊S3, C3×C62, C2×C6×C3⋊S3
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C3×S3, C3⋊S3, C22×S3, C22×C6, S3×C6, C2×C3⋊S3, C3×C3⋊S3, S3×C2×C6, C22×C3⋊S3, C6×C3⋊S3, 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 C62⋊11Dic3 C62⋊24D6 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