metabelian, supersoluble, monomial
Aliases: C62.36D6, (C6×C12)⋊3S3, (C6×C12)⋊4C6, C12⋊S3⋊6C6, C12.98(S3×C6), He3⋊7(C4○D4), C12.59D6⋊C3, He3⋊6D4⋊7C2, (C3×C12).50D6, C32⋊7D4⋊3C6, C32⋊4Q8⋊6C6, He3⋊4D4⋊11C2, He3⋊3Q8⋊11C2, C62.12(C2×C6), C32⋊5(C4○D12), (C2×He3).22C23, (C4×He3).49C22, C32⋊C12.11C22, (C22×He3).29C22, (C4×C3⋊S3)⋊4C6, C6.26(S3×C2×C6), (C2×C4×He3)⋊4C2, (C4×C32⋊C6)⋊9C2, (C2×C6).56(S3×C6), C3.2(C3×C4○D12), C32⋊1(C3×C4○D4), (C2×C4)⋊3(C32⋊C6), (C3×C12).12(C2×C6), (C2×C12).26(C3×S3), C4.16(C2×C32⋊C6), C3⋊Dic3.2(C2×C6), (C3×C6).4(C22×C6), (C3×C6).22(C22×S3), C22.2(C2×C32⋊C6), C2.5(C22×C32⋊C6), (C2×C32⋊C6).10C22, (C2×C3⋊S3).1(C2×C6), SmallGroup(432,351)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.36D6
G = < a,b,c,d | a6=b6=1, c6=d2=b3, ab=ba, cac-1=ab2, dad-1=ab-1, bc=cb, dbd-1=b-1, dcd-1=c5 >
Subgroups: 677 in 156 conjugacy classes, 52 normal (36 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, C62, C4○D12, C3×C4○D4, C32⋊C6, C2×He3, C2×He3, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C6×C12, C32⋊C12, C4×He3, C2×C32⋊C6, C22×He3, C3×C4○D12, C12.59D6, He3⋊3Q8, C4×C32⋊C6, He3⋊4D4, He3⋊6D4, C2×C4×He3, C62.36D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S3×C6, C4○D12, C3×C4○D4, C32⋊C6, S3×C2×C6, C2×C32⋊C6, C3×C4○D12, C22×C32⋊C6, C62.36D6
►On 72 points
(2 70 60)(3 49 71)(5 61 51)(6 52 62)(8 64 54)(9 55 65)(11 67 57)(12 58 68)(13 30 45 19 36 39)(14 40 25 20 46 31)(15 21)(16 33 48 22 27 42)(17 43 28 23 37 34)(18 24)(26 32)(29 35)(38 44)(41 47)
(1 53 69 7 59 63)(2 54 70 8 60 64)(3 55 71 9 49 65)(4 56 72 10 50 66)(5 57 61 11 51 67)(6 58 62 12 52 68)(13 39 36 19 45 30)(14 40 25 20 46 31)(15 41 26 21 47 32)(16 42 27 22 48 33)(17 43 28 23 37 34)(18 44 29 24 38 35)
(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 34 7 28)(2 27 8 33)(3 32 9 26)(4 25 10 31)(5 30 11 36)(6 35 12 29)(13 61 19 67)(14 66 20 72)(15 71 21 65)(16 64 22 70)(17 69 23 63)(18 62 24 68)(37 59 43 53)(38 52 44 58)(39 57 45 51)(40 50 46 56)(41 55 47 49)(42 60 48 54)
G:=sub<Sym(72)| (2,70,60)(3,49,71)(5,61,51)(6,52,62)(8,64,54)(9,55,65)(11,67,57)(12,58,68)(13,30,45,19,36,39)(14,40,25,20,46,31)(15,21)(16,33,48,22,27,42)(17,43,28,23,37,34)(18,24)(26,32)(29,35)(38,44)(41,47), (1,53,69,7,59,63)(2,54,70,8,60,64)(3,55,71,9,49,65)(4,56,72,10,50,66)(5,57,61,11,51,67)(6,58,62,12,52,68)(13,39,36,19,45,30)(14,40,25,20,46,31)(15,41,26,21,47,32)(16,42,27,22,48,33)(17,43,28,23,37,34)(18,44,29,24,38,35), (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), (1,34,7,28)(2,27,8,33)(3,32,9,26)(4,25,10,31)(5,30,11,36)(6,35,12,29)(13,61,19,67)(14,66,20,72)(15,71,21,65)(16,64,22,70)(17,69,23,63)(18,62,24,68)(37,59,43,53)(38,52,44,58)(39,57,45,51)(40,50,46,56)(41,55,47,49)(42,60,48,54)>;
G:=Group( (2,70,60)(3,49,71)(5,61,51)(6,52,62)(8,64,54)(9,55,65)(11,67,57)(12,58,68)(13,30,45,19,36,39)(14,40,25,20,46,31)(15,21)(16,33,48,22,27,42)(17,43,28,23,37,34)(18,24)(26,32)(29,35)(38,44)(41,47), (1,53,69,7,59,63)(2,54,70,8,60,64)(3,55,71,9,49,65)(4,56,72,10,50,66)(5,57,61,11,51,67)(6,58,62,12,52,68)(13,39,36,19,45,30)(14,40,25,20,46,31)(15,41,26,21,47,32)(16,42,27,22,48,33)(17,43,28,23,37,34)(18,44,29,24,38,35), (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), (1,34,7,28)(2,27,8,33)(3,32,9,26)(4,25,10,31)(5,30,11,36)(6,35,12,29)(13,61,19,67)(14,66,20,72)(15,71,21,65)(16,64,22,70)(17,69,23,63)(18,62,24,68)(37,59,43,53)(38,52,44,58)(39,57,45,51)(40,50,46,56)(41,55,47,49)(42,60,48,54) );
G=PermutationGroup([[(2,70,60),(3,49,71),(5,61,51),(6,52,62),(8,64,54),(9,55,65),(11,67,57),(12,58,68),(13,30,45,19,36,39),(14,40,25,20,46,31),(15,21),(16,33,48,22,27,42),(17,43,28,23,37,34),(18,24),(26,32),(29,35),(38,44),(41,47)], [(1,53,69,7,59,63),(2,54,70,8,60,64),(3,55,71,9,49,65),(4,56,72,10,50,66),(5,57,61,11,51,67),(6,58,62,12,52,68),(13,39,36,19,45,30),(14,40,25,20,46,31),(15,41,26,21,47,32),(16,42,27,22,48,33),(17,43,28,23,37,34),(18,44,29,24,38,35)], [(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)], [(1,34,7,28),(2,27,8,33),(3,32,9,26),(4,25,10,31),(5,30,11,36),(6,35,12,29),(13,61,19,67),(14,66,20,72),(15,71,21,65),(16,64,22,70),(17,69,23,63),(18,62,24,68),(37,59,43,53),(38,52,44,58),(39,57,45,51),(40,50,46,56),(41,55,47,49),(42,60,48,54)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | ··· | 6P | 6Q | 6R | 6S | 6T | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12V | 12W | 12X | 12Y | 12Z |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 18 | 18 | 2 | 3 | 3 | 6 | 6 | 6 | 1 | 1 | 2 | 18 | 18 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 18 | 18 | 18 | 18 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D6 | D6 | C4○D4 | C3×S3 | S3×C6 | S3×C6 | C4○D12 | C3×C4○D4 | C3×C4○D12 | C32⋊C6 | C2×C32⋊C6 | C2×C32⋊C6 | C62.36D6 |
| kernel | C62.36D6 | He3⋊3Q8 | C4×C32⋊C6 | He3⋊4D4 | He3⋊6D4 | C2×C4×He3 | C12.59D6 | C32⋊4Q8 | C4×C3⋊S3 | C12⋊S3 | C32⋊7D4 | C6×C12 | C6×C12 | C3×C12 | C62 | He3 | C2×C12 | C12 | C2×C6 | C32 | C32 | C3 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 1 | 2 | 1 | 4 |
Matrix representation of C62.36D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,12,0,0,0,0,0,0,4,0,0,0,0,0,0,10],[10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,8,0,0,0,0,0,0,8,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,8,0,0,0,0,0,0,8,0],[0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,8,0,0,0,0,8,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0] >;
C62.36D6 in GAP, Magma, Sage, TeX
C_6^2._{36}D_6 % in TeX
G:=Group("C6^2.36D6"); // GroupNames label
G:=SmallGroup(432,351);
// by ID
G=gap.SmallGroup(432,351);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,4037,1034,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=b^3,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a*b^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations