direct product, metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary
Aliases: C6×C11⋊C5, C66⋊C5, C22⋊C15, C33⋊4C10, C11⋊2C30, SmallGroup(330,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C6×C11⋊C5 |
Generators and relations for C6×C11⋊C5
G = < a,b,c | a6=b11=c5=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C6×C11⋊C5
►On 66 points
Generators in S66
(1 45 23 34 12 56)(2 46 24 35 13 57)(3 47 25 36 14 58)(4 48 26 37 15 59)(5 49 27 38 16 60)(6 50 28 39 17 61)(7 51 29 40 18 62)(8 52 30 41 19 63)(9 53 31 42 20 64)(10 54 32 43 21 65)(11 55 33 44 22 66)
(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)
(2 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)(24 27 28 32 26)(25 31 33 30 29)(35 38 39 43 37)(36 42 44 41 40)(46 49 50 54 48)(47 53 55 52 51)(57 60 61 65 59)(58 64 66 63 62)
G:=sub<Sym(66)| (1,45,23,34,12,56)(2,46,24,35,13,57)(3,47,25,36,14,58)(4,48,26,37,15,59)(5,49,27,38,16,60)(6,50,28,39,17,61)(7,51,29,40,18,62)(8,52,30,41,19,63)(9,53,31,42,20,64)(10,54,32,43,21,65)(11,55,33,44,22,66), (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), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40)(46,49,50,54,48)(47,53,55,52,51)(57,60,61,65,59)(58,64,66,63,62)>;
G:=Group( (1,45,23,34,12,56)(2,46,24,35,13,57)(3,47,25,36,14,58)(4,48,26,37,15,59)(5,49,27,38,16,60)(6,50,28,39,17,61)(7,51,29,40,18,62)(8,52,30,41,19,63)(9,53,31,42,20,64)(10,54,32,43,21,65)(11,55,33,44,22,66), (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), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40)(46,49,50,54,48)(47,53,55,52,51)(57,60,61,65,59)(58,64,66,63,62) );
G=PermutationGroup([[(1,45,23,34,12,56),(2,46,24,35,13,57),(3,47,25,36,14,58),(4,48,26,37,15,59),(5,49,27,38,16,60),(6,50,28,39,17,61),(7,51,29,40,18,62),(8,52,30,41,19,63),(9,53,31,42,20,64),(10,54,32,43,21,65),(11,55,33,44,22,66)], [(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)], [(2,5,6,10,4),(3,9,11,8,7),(13,16,17,21,15),(14,20,22,19,18),(24,27,28,32,26),(25,31,33,30,29),(35,38,39,43,37),(36,42,44,41,40),(46,49,50,54,48),(47,53,55,52,51),(57,60,61,65,59),(58,64,66,63,62)]])
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 11A | 11B | 15A | ··· | 15H | 22A | 22B | 30A | ··· | 30H | 33A | 33B | 33C | 33D | 66A | 66B | 66C | 66D |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 11 | 11 | 15 | ··· | 15 | 22 | 22 | 30 | ··· | 30 | 33 | 33 | 33 | 33 | 66 | 66 | 66 | 66 |
| size | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 1 | 1 | 11 | 11 | 11 | 11 | 5 | 5 | 11 | ··· | 11 | 5 | 5 | 11 | ··· | 11 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | C11⋊C5 | C2×C11⋊C5 | C3×C11⋊C5 | C6×C11⋊C5 |
| kernel | C6×C11⋊C5 | C3×C11⋊C5 | C2×C11⋊C5 | C66 | C11⋊C5 | C33 | C22 | C11 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of C6×C11⋊C5 ►in GL5(𝔽331)
| 300 | 0 | 0 | 0 | 0 |
| 0 | 300 | 0 | 0 | 0 |
| 0 | 0 | 300 | 0 | 0 |
| 0 | 0 | 0 | 300 | 0 |
| 0 | 0 | 0 | 0 | 300 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 227 |
| 0 | 1 | 0 | 0 | 330 |
| 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 1 | 226 |
| 103 | 1 | 0 | 2 | 0 |
| 106 | 0 | 0 | 226 | 1 |
| 226 | 0 | 0 | 104 | 0 |
| 104 | 0 | 0 | 228 | 0 |
| 2 | 0 | 1 | 225 | 0 |
G:=sub<GL(5,GF(331))| [300,0,0,0,0,0,300,0,0,0,0,0,300,0,0,0,0,0,300,0,0,0,0,0,300],[0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,227,330,1,226],[103,106,226,104,2,1,0,0,0,0,0,0,0,0,1,2,226,104,228,225,0,1,0,0,0] >;
C6×C11⋊C5 in GAP, Magma, Sage, TeX
C_6\times C_{11}\rtimes C_5 % in TeX
G:=Group("C6xC11:C5"); // GroupNames label
G:=SmallGroup(330,4);
// by ID
G=gap.SmallGroup(330,4);
# by ID
G:=PCGroup([4,-2,-3,-5,-11,331]);
// Polycyclic
G:=Group<a,b,c|a^6=b^11=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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