metacyclic, supersoluble, monomial
Aliases: C72⋊4C6, D18.C12, Dic9.C12, 3- 1+2⋊1M4(2), C9⋊C8⋊4C6, C9⋊C12.C4, C8⋊D9⋊C3, C9⋊C24⋊4C2, C8⋊3(C9⋊C6), (C4×D9).2C6, C24.22(C3×S3), C6.10(S3×C12), C12.89(S3×C6), C18.2(C2×C12), C36.14(C2×C6), (C3×C24).10S3, (C3×C12).59D6, C9⋊1(C3×M4(2)), C32.(C8⋊S3), (C8×3- 1+2)⋊4C2, (C4×3- 1+2).13C22, (C2×C9⋊C6).C4, C2.3(C4×C9⋊C6), (C4×C9⋊C6).2C2, C4.13(C2×C9⋊C6), C3.3(C3×C8⋊S3), (C3×C6).13(C4×S3), (C2×3- 1+2).2(C2×C4), SmallGroup(432,121)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C72⋊C6
G = < a,b | a72=b6=1, bab-1=a29 >
Subgroups: 222 in 64 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, M4(2), D9, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, 3- 1+2, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, C8⋊S3, C3×M4(2), C9⋊C6, C2×3- 1+2, C9⋊C8, C72, C72, C4×D9, C3×C3⋊C8, C3×C24, S3×C12, C9⋊C12, C4×3- 1+2, C2×C9⋊C6, C8⋊D9, C3×C8⋊S3, C9⋊C24, C8×3- 1+2, C4×C9⋊C6, C72⋊C6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, D6, C2×C6, M4(2), C3×S3, C4×S3, C2×C12, S3×C6, C8⋊S3, C3×M4(2), C9⋊C6, S3×C12, C2×C9⋊C6, C3×C8⋊S3, C4×C9⋊C6, C72⋊C6
►On 72 points
(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)
(2 6 26 54 50 30)(3 11 51 35 27 59)(4 16)(5 21 29 69 53 45)(7 31)(8 36 32 12 56 60)(9 41 57 65 33 17)(10 46)(13 61)(14 66 38 42 62 18)(15 71 63 23 39 47)(20 24 44 72 68 48)(22 34)(25 49)(28 64)(40 52)(43 67)(58 70)
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), (2,6,26,54,50,30)(3,11,51,35,27,59)(4,16)(5,21,29,69,53,45)(7,31)(8,36,32,12,56,60)(9,41,57,65,33,17)(10,46)(13,61)(14,66,38,42,62,18)(15,71,63,23,39,47)(20,24,44,72,68,48)(22,34)(25,49)(28,64)(40,52)(43,67)(58,70)>;
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), (2,6,26,54,50,30)(3,11,51,35,27,59)(4,16)(5,21,29,69,53,45)(7,31)(8,36,32,12,56,60)(9,41,57,65,33,17)(10,46)(13,61)(14,66,38,42,62,18)(15,71,63,23,39,47)(20,24,44,72,68,48)(22,34)(25,49)(28,64)(40,52)(43,67)(58,70) );
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)], [(2,6,26,54,50,30),(3,11,51,35,27,59),(4,16),(5,21,29,69,53,45),(7,31),(8,36,32,12,56,60),(9,41,57,65,33,17),(10,46),(13,61),(14,66,38,42,62,18),(15,71,63,23,39,47),(20,24,44,72,68,48),(22,34),(25,49),(28,64),(40,52),(43,67),(58,70)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | 18B | 18C | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 24I | 24J | 24K | 24L | 36A | ··· | 36F | 72A | ··· | 72L |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 18 | 2 | 3 | 3 | 1 | 1 | 18 | 2 | 3 | 3 | 18 | 18 | 2 | 2 | 18 | 18 | 6 | 6 | 6 | 2 | 2 | 3 | 3 | 3 | 3 | 18 | 18 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 6 | ··· | 6 | 6 | ··· | 6 |
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 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | S3 | D6 | M4(2) | C3×S3 | C4×S3 | S3×C6 | C3×M4(2) | C8⋊S3 | S3×C12 | C3×C8⋊S3 | C9⋊C6 | C2×C9⋊C6 | C4×C9⋊C6 | C72⋊C6 |
| kernel | C72⋊C6 | C9⋊C24 | C8×3- 1+2 | C4×C9⋊C6 | C8⋊D9 | C9⋊C12 | C2×C9⋊C6 | C9⋊C8 | C72 | C4×D9 | Dic9 | D18 | C3×C24 | C3×C12 | 3- 1+2 | C24 | C3×C6 | C12 | C9 | C32 | C6 | C3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 1 | 2 | 4 |
Matrix representation of C72⋊C6 ►in GL6(𝔽73)
| 0 | 0 | 0 | 0 | 3 | 67 |
| 0 | 0 | 0 | 0 | 6 | 70 |
| 67 | 3 | 0 | 0 | 0 | 0 |
| 70 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 67 | 3 | 0 | 0 |
| 0 | 0 | 70 | 70 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
G:=sub<GL(6,GF(73))| [0,0,67,70,0,0,0,0,3,70,0,0,0,0,0,0,67,70,0,0,0,0,3,70,3,6,0,0,0,0,67,70,0,0,0,0],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,1,0,0,0,0,0,72,72,0,0] >;
C72⋊C6 in GAP, Magma, Sage, TeX
C_{72}\rtimes C_6 % in TeX
G:=Group("C72:C6"); // GroupNames label
G:=SmallGroup(432,121);
// by ID
G=gap.SmallGroup(432,121);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,80,10085,2035,292,14118]);
// Polycyclic
G:=Group<a,b|a^72=b^6=1,b*a*b^-1=a^29>;
// generators/relations