direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary
Aliases: C8×3- 1+2, C72⋊C3, C9⋊2C24, C36.4C6, C32.C24, C18.2C12, C24.2C32, (C3×C24).C3, (C3×C6).4C12, C6.3(C3×C12), (C3×C12).7C6, C3.2(C3×C24), C12.11(C3×C6), C2.(C4×3- 1+2), C4.2(C2×3- 1+2), (C2×3- 1+2).2C4, (C4×3- 1+2).4C2, SmallGroup(216,20)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8×3- 1+2
G = < a,b,c | a8=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C8×3- 1+2
►On 72 points
Generators in S72
(1 21 63 36 42 66 53 10)(2 22 55 28 43 67 54 11)(3 23 56 29 44 68 46 12)(4 24 57 30 45 69 47 13)(5 25 58 31 37 70 48 14)(6 26 59 32 38 71 49 15)(7 27 60 33 39 72 50 16)(8 19 61 34 40 64 51 17)(9 20 62 35 41 65 52 18)
(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 8 5)(3 6 9)(11 17 14)(12 15 18)(19 25 22)(20 23 26)(28 34 31)(29 32 35)(37 43 40)(38 41 44)(46 49 52)(48 54 51)(55 61 58)(56 59 62)(64 70 67)(65 68 71)
G:=sub<Sym(72)| (1,21,63,36,42,66,53,10)(2,22,55,28,43,67,54,11)(3,23,56,29,44,68,46,12)(4,24,57,30,45,69,47,13)(5,25,58,31,37,70,48,14)(6,26,59,32,38,71,49,15)(7,27,60,33,39,72,50,16)(8,19,61,34,40,64,51,17)(9,20,62,35,41,65,52,18), (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), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,25,22)(20,23,26)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(46,49,52)(48,54,51)(55,61,58)(56,59,62)(64,70,67)(65,68,71)>;
G:=Group( (1,21,63,36,42,66,53,10)(2,22,55,28,43,67,54,11)(3,23,56,29,44,68,46,12)(4,24,57,30,45,69,47,13)(5,25,58,31,37,70,48,14)(6,26,59,32,38,71,49,15)(7,27,60,33,39,72,50,16)(8,19,61,34,40,64,51,17)(9,20,62,35,41,65,52,18), (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), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,25,22)(20,23,26)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(46,49,52)(48,54,51)(55,61,58)(56,59,62)(64,70,67)(65,68,71) );
G=PermutationGroup([[(1,21,63,36,42,66,53,10),(2,22,55,28,43,67,54,11),(3,23,56,29,44,68,46,12),(4,24,57,30,45,69,47,13),(5,25,58,31,37,70,48,14),(6,26,59,32,38,71,49,15),(7,27,60,33,39,72,50,16),(8,19,61,34,40,64,51,17),(9,20,62,35,41,65,52,18)], [(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)], [(2,8,5),(3,6,9),(11,17,14),(12,15,18),(19,25,22),(20,23,26),(28,34,31),(29,32,35),(37,43,40),(38,41,44),(46,49,52),(48,54,51),(55,61,58),(56,59,62),(64,70,67),(65,68,71)]])
C8×3- 1+2 is a maximal subgroup of
C9⋊C48 C72.C6 C72⋊C6 C72⋊2C6 D72⋊C3
88 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | ··· | 18F | 24A | ··· | 24H | 24I | ··· | 24P | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | C24 | 3- 1+2 | C2×3- 1+2 | C4×3- 1+2 | C8×3- 1+2 |
| kernel | C8×3- 1+2 | C4×3- 1+2 | C72 | C3×C24 | C2×3- 1+2 | C36 | C3×C12 | 3- 1+2 | C18 | C3×C6 | C9 | C32 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 6 | 2 | 2 | 6 | 2 | 4 | 12 | 4 | 24 | 8 | 2 | 2 | 4 | 8 |
Matrix representation of C8×3- 1+2 ►in GL4(𝔽73) generated by
| 10 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 |
| 64 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 64 |
G:=sub<GL(4,GF(73))| [10,0,0,0,0,27,0,0,0,0,27,0,0,0,0,27],[8,0,0,0,0,0,0,1,0,1,0,0,0,0,8,0],[64,0,0,0,0,1,0,0,0,0,8,0,0,0,0,64] >;
C8×3- 1+2 in GAP, Magma, Sage, TeX
C_8\times 3_-^{1+2} % in TeX
G:=Group("C8xES-(3,1)"); // GroupNames label
G:=SmallGroup(216,20);
// by ID
G=gap.SmallGroup(216,20);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-3,-2,108,223,386,165]);
// Polycyclic
G:=Group<a,b,c|a^8=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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