direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C4×C11⋊C5, C44⋊4C10, C22⋊2C20, (C2×C44)⋊C5, C11⋊3(C2×C20), (C2×C22).2C10, C22.6(C2×C10), C22.(C2×C11⋊C5), C2.1(C22×C11⋊C5), (C22×C11⋊C5).2C2, (C2×C11⋊C5).6C22, SmallGroup(440,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C11 — C22 — C2×C11⋊C5 — C22×C11⋊C5 — C2×C4×C11⋊C5 |
| C11 — C2×C4×C11⋊C5 |
Generators and relations for C2×C4×C11⋊C5
G = < a,b,c,d | a2=b4=c11=d5=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Smallest permutation representation of C2×C4×C11⋊C5
►On 88 points
Generators in S88
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)
(1 23 12 34)(2 24 13 35)(3 25 14 36)(4 26 15 37)(5 27 16 38)(6 28 17 39)(7 29 18 40)(8 30 19 41)(9 31 20 42)(10 32 21 43)(11 33 22 44)(45 67 56 78)(46 68 57 79)(47 69 58 80)(48 70 59 81)(49 71 60 82)(50 72 61 83)(51 73 62 84)(52 74 63 85)(53 75 64 86)(54 76 65 87)(55 77 66 88)
(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)| (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77), (1,23,12,34)(2,24,13,35)(3,25,14,36)(4,26,15,37)(5,27,16,38)(6,28,17,39)(7,29,18,40)(8,30,19,41)(9,31,20,42)(10,32,21,43)(11,33,22,44)(45,67,56,78)(46,68,57,79)(47,69,58,80)(48,70,59,81)(49,71,60,82)(50,72,61,83)(51,73,62,84)(52,74,63,85)(53,75,64,86)(54,76,65,87)(55,77,66,88), (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( (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77), (1,23,12,34)(2,24,13,35)(3,25,14,36)(4,26,15,37)(5,27,16,38)(6,28,17,39)(7,29,18,40)(8,30,19,41)(9,31,20,42)(10,32,21,43)(11,33,22,44)(45,67,56,78)(46,68,57,79)(47,69,58,80)(48,70,59,81)(49,71,60,82)(50,72,61,83)(51,73,62,84)(52,74,63,85)(53,75,64,86)(54,76,65,87)(55,77,66,88), (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([[(1,56),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,73),(41,74),(42,75),(43,76),(44,77)], [(1,23,12,34),(2,24,13,35),(3,25,14,36),(4,26,15,37),(5,27,16,38),(6,28,17,39),(7,29,18,40),(8,30,19,41),(9,31,20,42),(10,32,21,43),(11,33,22,44),(45,67,56,78),(46,68,57,79),(47,69,58,80),(48,70,59,81),(49,71,60,82),(50,72,61,83),(51,73,62,84),(52,74,63,85),(53,75,64,86),(54,76,65,87),(55,77,66,88)], [(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 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 11A | 11B | 20A | ··· | 20P | 22A | ··· | 22F | 44A | ··· | 44H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 11 | 11 | 20 | ··· | 20 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 11 | ··· | 11 | 5 | 5 | 11 | ··· | 11 | 5 | ··· | 5 | 5 | ··· | 5 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | C11⋊C5 | C2×C11⋊C5 | C2×C11⋊C5 | C4×C11⋊C5 |
| kernel | C2×C4×C11⋊C5 | C4×C11⋊C5 | C22×C11⋊C5 | C2×C11⋊C5 | C2×C44 | C44 | C2×C22 | C22 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 2 | 4 | 2 | 8 |
Matrix representation of C2×C4×C11⋊C5 ►in GL6(𝔽661)
| 660 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 106 | 0 | 0 | 0 | 0 | 0 |
| 0 | 106 | 0 | 0 | 0 | 0 |
| 0 | 0 | 106 | 0 | 0 | 0 |
| 0 | 0 | 0 | 106 | 0 | 0 |
| 0 | 0 | 0 | 0 | 106 | 0 |
| 0 | 0 | 0 | 0 | 0 | 106 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 45 |
| 0 | 0 | 1 | 0 | 0 | 660 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 44 |
| 197 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 2 | 0 | 1 |
| 0 | 0 | 0 | 44 | 1 | 45 |
| 0 | 0 | 0 | 616 | 0 | 660 |
| 0 | 0 | 0 | 46 | 0 | 1 |
| 0 | 0 | 1 | 43 | 0 | 44 |
G:=sub<GL(6,GF(661))| [660,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[106,0,0,0,0,0,0,106,0,0,0,0,0,0,106,0,0,0,0,0,0,106,0,0,0,0,0,0,106,0,0,0,0,0,0,106],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,45,660,1,44],[197,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,2,44,616,46,43,0,0,1,0,0,0,0,1,45,660,1,44] >;
C2×C4×C11⋊C5 in GAP, Magma, Sage, TeX
C_2\times C_4\times C_{11}\rtimes C_5 % in TeX
G:=Group("C2xC4xC11:C5"); // GroupNames label
G:=SmallGroup(440,12);
// by ID
G=gap.SmallGroup(440,12);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-11,106,1014]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^11=d^5=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations
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