direct product, metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary
Aliases: C8×C11⋊C5, C88⋊C5, C11⋊2C40, C44.4C10, C22.2C20, C2.(C4×C11⋊C5), C4.2(C2×C11⋊C5), (C4×C11⋊C5).4C2, (C2×C11⋊C5).2C4, SmallGroup(440,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C8×C11⋊C5 |
Generators and relations for C8×C11⋊C5
G = < a,b,c | a8=b11=c5=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C8×C11⋊C5
►On 88 points
Generators in S88
(1 78 34 56 12 67 23 45)(2 79 35 57 13 68 24 46)(3 80 36 58 14 69 25 47)(4 81 37 59 15 70 26 48)(5 82 38 60 16 71 27 49)(6 83 39 61 17 72 28 50)(7 84 40 62 18 73 29 51)(8 85 41 63 19 74 30 52)(9 86 42 64 20 75 31 53)(10 87 43 65 21 76 32 54)(11 88 44 66 22 77 33 55)
(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 73 74 75 76 77)(78 79 80 81 82 83 84 85 86 87 88)
(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)(68 71 72 76 70)(69 75 77 74 73)(79 82 83 87 81)(80 86 88 85 84)
G:=sub<Sym(88)| (1,78,34,56,12,67,23,45)(2,79,35,57,13,68,24,46)(3,80,36,58,14,69,25,47)(4,81,37,59,15,70,26,48)(5,82,38,60,16,71,27,49)(6,83,39,61,17,72,28,50)(7,84,40,62,18,73,29,51)(8,85,41,63,19,74,30,52)(9,86,42,64,20,75,31,53)(10,87,43,65,21,76,32,54)(11,88,44,66,22,77,33,55), (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)(67,68,69,70,71,72,73,74,75,76,77)(78,79,80,81,82,83,84,85,86,87,88), (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)(68,71,72,76,70)(69,75,77,74,73)(79,82,83,87,81)(80,86,88,85,84)>;
G:=Group( (1,78,34,56,12,67,23,45)(2,79,35,57,13,68,24,46)(3,80,36,58,14,69,25,47)(4,81,37,59,15,70,26,48)(5,82,38,60,16,71,27,49)(6,83,39,61,17,72,28,50)(7,84,40,62,18,73,29,51)(8,85,41,63,19,74,30,52)(9,86,42,64,20,75,31,53)(10,87,43,65,21,76,32,54)(11,88,44,66,22,77,33,55), (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)(67,68,69,70,71,72,73,74,75,76,77)(78,79,80,81,82,83,84,85,86,87,88), (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)(68,71,72,76,70)(69,75,77,74,73)(79,82,83,87,81)(80,86,88,85,84) );
G=PermutationGroup([[(1,78,34,56,12,67,23,45),(2,79,35,57,13,68,24,46),(3,80,36,58,14,69,25,47),(4,81,37,59,15,70,26,48),(5,82,38,60,16,71,27,49),(6,83,39,61,17,72,28,50),(7,84,40,62,18,73,29,51),(8,85,41,63,19,74,30,52),(9,86,42,64,20,75,31,53),(10,87,43,65,21,76,32,54),(11,88,44,66,22,77,33,55)], [(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),(67,68,69,70,71,72,73,74,75,76,77),(78,79,80,81,82,83,84,85,86,87,88)], [(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),(68,71,72,76,70),(69,75,77,74,73),(79,82,83,87,81),(80,86,88,85,84)]])
56 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 11A | 11B | 20A | ··· | 20H | 22A | 22B | 40A | ··· | 40P | 44A | 44B | 44C | 44D | 88A | ··· | 88H |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 11 | 11 | 20 | ··· | 20 | 22 | 22 | 40 | ··· | 40 | 44 | 44 | 44 | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 5 | 5 | 11 | ··· | 11 | 5 | 5 | 11 | ··· | 11 | 5 | 5 | 5 | 5 | 5 | ··· | 5 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 |
| type | + | + | ||||||||||
| image | C1 | C2 | C4 | C5 | C8 | C10 | C20 | C40 | C11⋊C5 | C2×C11⋊C5 | C4×C11⋊C5 | C8×C11⋊C5 |
| kernel | C8×C11⋊C5 | C4×C11⋊C5 | C2×C11⋊C5 | C88 | C11⋊C5 | C44 | C22 | C11 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 16 | 2 | 2 | 4 | 8 |
Matrix representation of C8×C11⋊C5 ►in GL5(𝔽881)
| 219 | 0 | 0 | 0 | 0 |
| 0 | 219 | 0 | 0 | 0 |
| 0 | 0 | 219 | 0 | 0 |
| 0 | 0 | 0 | 219 | 0 |
| 0 | 0 | 0 | 0 | 219 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 171 |
| 0 | 1 | 0 | 0 | 880 |
| 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 1 | 170 |
| 1 | 0 | 2 | 0 | 1 |
| 0 | 0 | 170 | 1 | 171 |
| 0 | 0 | 710 | 0 | 880 |
| 0 | 0 | 172 | 0 | 1 |
| 0 | 1 | 169 | 0 | 170 |
G:=sub<GL(5,GF(881))| [219,0,0,0,0,0,219,0,0,0,0,0,219,0,0,0,0,0,219,0,0,0,0,0,219],[0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,171,880,1,170],[1,0,0,0,0,0,0,0,0,1,2,170,710,172,169,0,1,0,0,0,1,171,880,1,170] >;
C8×C11⋊C5 in GAP, Magma, Sage, TeX
C_8\times C_{11}\rtimes C_5 % in TeX
G:=Group("C8xC11:C5"); // GroupNames label
G:=SmallGroup(440,2);
// by ID
G=gap.SmallGroup(440,2);
# by ID
G:=PCGroup([5,-2,-5,-2,-2,-11,50,42,2009]);
// Polycyclic
G:=Group<a,b,c|a^8=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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