non-abelian, supersoluble, monomial
Aliases: C62.31D6, (C6×C12)⋊2S3, (C3×C6).27D12, C32⋊6(D6⋊C4), He3⋊6(C22⋊C4), (C2×He3).33D4, C2.2(He3⋊7D4), C2.2(He3⋊5D4), C6.20(C12⋊S3), C6.30(C32⋊7D4), C3.2(C6.11D12), (C22×He3).24C22, (C2×C4×He3)⋊2C2, C6.31(C4×C3⋊S3), (C3×C6).24(C4×S3), (C2×C12).4(C3⋊S3), (C2×He3⋊C2)⋊2C4, (C2×He3⋊3C4)⋊2C2, C2.5(C4×He3⋊C2), (C2×C4)⋊1(He3⋊C2), (C3×C6).34(C3⋊D4), (C2×He3).25(C2×C4), C22.6(C2×He3⋊C2), (C22×He3⋊C2).2C2, (C2×C6).54(C2×C3⋊S3), SmallGroup(432,189)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.31D6
G = < a,b,c,d | a6=b6=1, c6=a3, d2=a3b3, ab=ba, cac-1=ab4, dad-1=a-1b2, bc=cb, bd=db, dcd-1=b3c5 >
Subgroups: 797 in 187 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3×C6, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, He3, C3×Dic3, C3×C12, S3×C6, C62, D6⋊C4, C3×C22⋊C4, He3⋊C2, C2×He3, C6×Dic3, C6×C12, S3×C2×C6, He3⋊3C4, C4×He3, C2×He3⋊C2, C2×He3⋊C2, C22×He3, C3×D6⋊C4, C2×He3⋊3C4, C2×C4×He3, C22×He3⋊C2, C62.31D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C3⋊S3, C4×S3, D12, C3⋊D4, C2×C3⋊S3, D6⋊C4, He3⋊C2, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C2×He3⋊C2, C6.11D12, C4×He3⋊C2, He3⋊5D4, He3⋊7D4, C62.31D6
►On 72 points
(1 7)(2 64 51 8 70 57)(3 58 71 9 52 65)(4 10)(5 67 54 11 61 60)(6 49 62 12 55 68)(13 19)(14 35 47 20 29 41)(15 42 30 21 48 36)(16 22)(17 26 38 23 32 44)(18 45 33 24 39 27)(25 31)(28 34)(37 43)(40 46)(50 56)(53 59)(63 69)(66 72)
(1 19 50 40 69 34)(2 20 51 41 70 35)(3 21 52 42 71 36)(4 22 53 43 72 25)(5 23 54 44 61 26)(6 24 55 45 62 27)(7 13 56 46 63 28)(8 14 57 47 64 29)(9 15 58 48 65 30)(10 16 59 37 66 31)(11 17 60 38 67 32)(12 18 49 39 68 33)
(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 6 46 39)(2 38 47 5)(3 4 48 37)(7 12 40 45)(8 44 41 11)(9 10 42 43)(13 18 69 62)(14 61 70 17)(15 16 71 72)(19 24 63 68)(20 67 64 23)(21 22 65 66)(25 58 59 36)(26 35 60 57)(27 56 49 34)(28 33 50 55)(29 54 51 32)(30 31 52 53)
G:=sub<Sym(72)| (1,7)(2,64,51,8,70,57)(3,58,71,9,52,65)(4,10)(5,67,54,11,61,60)(6,49,62,12,55,68)(13,19)(14,35,47,20,29,41)(15,42,30,21,48,36)(16,22)(17,26,38,23,32,44)(18,45,33,24,39,27)(25,31)(28,34)(37,43)(40,46)(50,56)(53,59)(63,69)(66,72), (1,19,50,40,69,34)(2,20,51,41,70,35)(3,21,52,42,71,36)(4,22,53,43,72,25)(5,23,54,44,61,26)(6,24,55,45,62,27)(7,13,56,46,63,28)(8,14,57,47,64,29)(9,15,58,48,65,30)(10,16,59,37,66,31)(11,17,60,38,67,32)(12,18,49,39,68,33), (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), (1,6,46,39)(2,38,47,5)(3,4,48,37)(7,12,40,45)(8,44,41,11)(9,10,42,43)(13,18,69,62)(14,61,70,17)(15,16,71,72)(19,24,63,68)(20,67,64,23)(21,22,65,66)(25,58,59,36)(26,35,60,57)(27,56,49,34)(28,33,50,55)(29,54,51,32)(30,31,52,53)>;
G:=Group( (1,7)(2,64,51,8,70,57)(3,58,71,9,52,65)(4,10)(5,67,54,11,61,60)(6,49,62,12,55,68)(13,19)(14,35,47,20,29,41)(15,42,30,21,48,36)(16,22)(17,26,38,23,32,44)(18,45,33,24,39,27)(25,31)(28,34)(37,43)(40,46)(50,56)(53,59)(63,69)(66,72), (1,19,50,40,69,34)(2,20,51,41,70,35)(3,21,52,42,71,36)(4,22,53,43,72,25)(5,23,54,44,61,26)(6,24,55,45,62,27)(7,13,56,46,63,28)(8,14,57,47,64,29)(9,15,58,48,65,30)(10,16,59,37,66,31)(11,17,60,38,67,32)(12,18,49,39,68,33), (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), (1,6,46,39)(2,38,47,5)(3,4,48,37)(7,12,40,45)(8,44,41,11)(9,10,42,43)(13,18,69,62)(14,61,70,17)(15,16,71,72)(19,24,63,68)(20,67,64,23)(21,22,65,66)(25,58,59,36)(26,35,60,57)(27,56,49,34)(28,33,50,55)(29,54,51,32)(30,31,52,53) );
G=PermutationGroup([[(1,7),(2,64,51,8,70,57),(3,58,71,9,52,65),(4,10),(5,67,54,11,61,60),(6,49,62,12,55,68),(13,19),(14,35,47,20,29,41),(15,42,30,21,48,36),(16,22),(17,26,38,23,32,44),(18,45,33,24,39,27),(25,31),(28,34),(37,43),(40,46),(50,56),(53,59),(63,69),(66,72)], [(1,19,50,40,69,34),(2,20,51,41,70,35),(3,21,52,42,71,36),(4,22,53,43,72,25),(5,23,54,44,61,26),(6,24,55,45,62,27),(7,13,56,46,63,28),(8,14,57,47,64,29),(9,15,58,48,65,30),(10,16,59,37,66,31),(11,17,60,38,67,32),(12,18,49,39,68,33)], [(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)], [(1,6,46,39),(2,38,47,5),(3,4,48,37),(7,12,40,45),(8,44,41,11),(9,10,42,43),(13,18,69,62),(14,61,70,17),(15,16,71,72),(19,24,63,68),(20,67,64,23),(21,22,65,66),(25,58,59,36),(26,35,60,57),(27,56,49,34),(28,33,50,55),(29,54,51,32),(30,31,52,53)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6R | 6S | 6T | 6U | 6V | 12A | 12B | 12C | 12D | 12E | ··· | 12T | 12U | 12V | 12W | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 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 | 1 | 1 | 18 | 18 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 18 | 18 | 1 | ··· | 1 | 6 | ··· | 6 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 18 | 18 | 18 | 18 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | C4×S3 | D12 | C3⋊D4 | He3⋊C2 | C2×He3⋊C2 | C4×He3⋊C2 | He3⋊5D4 | He3⋊7D4 |
| kernel | C62.31D6 | C2×He3⋊3C4 | C2×C4×He3 | C22×He3⋊C2 | C2×He3⋊C2 | C6×C12 | C2×He3 | C62 | C3×C6 | C3×C6 | C3×C6 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 8 | 8 | 8 | 4 | 4 | 8 | 2 | 2 |
Matrix representation of C62.31D6 ►in GL7(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 2 | 4 | 0 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 11 | 9 | 0 | 0 | 0 | 0 | 0 |
| 11 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
G:=sub<GL(7,GF(13))| [0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3],[2,9,0,0,0,0,0,4,11,0,0,0,0,0,0,0,3,10,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0],[11,11,0,0,0,0,0,9,2,0,0,0,0,0,0,0,10,3,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0] >;
C62.31D6 in GAP, Magma, Sage, TeX
C_6^2._{31}D_6 % in TeX
G:=Group("C6^2.31D6"); // GroupNames label
G:=SmallGroup(432,189);
// by ID
G=gap.SmallGroup(432,189);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,1124,4037,537]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=a^3,d^2=a^3*b^3,a*b=b*a,c*a*c^-1=a*b^4,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=b^3*c^5>;
// generators/relations