direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C11⋊F5, C22⋊F5, C110⋊1C4, D5⋊Dic11, D10.D11, C10⋊Dic11, D5.2D22, C55⋊2(C2×C4), C5⋊(C2×Dic11), C11⋊2(C2×F5), (D5×C11)⋊2C4, (D5×C22).2C2, (D5×C11).2C22, SmallGroup(440,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C55 — C2×C11⋊F5 |
Generators and relations for C2×C11⋊F5
G = < a,b,c,d | a2=b11=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Smallest permutation representation of C2×C11⋊F5
►On 110 points
Generators in S110
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)
(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)(89 90 91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108 109 110)
(1 45 34 23 12)(2 46 35 24 13)(3 47 36 25 14)(4 48 37 26 15)(5 49 38 27 16)(6 50 39 28 17)(7 51 40 29 18)(8 52 41 30 19)(9 53 42 31 20)(10 54 43 32 21)(11 55 44 33 22)(56 100 89 78 67)(57 101 90 79 68)(58 102 91 80 69)(59 103 92 81 70)(60 104 93 82 71)(61 105 94 83 72)(62 106 95 84 73)(63 107 96 85 74)(64 108 97 86 75)(65 109 98 87 76)(66 110 99 88 77)
(1 56)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 60)(9 59)(10 58)(11 57)(12 78 45 89)(13 88 46 99)(14 87 47 98)(15 86 48 97)(16 85 49 96)(17 84 50 95)(18 83 51 94)(19 82 52 93)(20 81 53 92)(21 80 54 91)(22 79 55 90)(23 100 34 67)(24 110 35 77)(25 109 36 76)(26 108 37 75)(27 107 38 74)(28 106 39 73)(29 105 40 72)(30 104 41 71)(31 103 42 70)(32 102 43 69)(33 101 44 68)
G:=sub<Sym(110)| (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110), (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)(89,90,91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108,109,110), (1,45,34,23,12)(2,46,35,24,13)(3,47,36,25,14)(4,48,37,26,15)(5,49,38,27,16)(6,50,39,28,17)(7,51,40,29,18)(8,52,41,30,19)(9,53,42,31,20)(10,54,43,32,21)(11,55,44,33,22)(56,100,89,78,67)(57,101,90,79,68)(58,102,91,80,69)(59,103,92,81,70)(60,104,93,82,71)(61,105,94,83,72)(62,106,95,84,73)(63,107,96,85,74)(64,108,97,86,75)(65,109,98,87,76)(66,110,99,88,77), (1,56)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,60)(9,59)(10,58)(11,57)(12,78,45,89)(13,88,46,99)(14,87,47,98)(15,86,48,97)(16,85,49,96)(17,84,50,95)(18,83,51,94)(19,82,52,93)(20,81,53,92)(21,80,54,91)(22,79,55,90)(23,100,34,67)(24,110,35,77)(25,109,36,76)(26,108,37,75)(27,107,38,74)(28,106,39,73)(29,105,40,72)(30,104,41,71)(31,103,42,70)(32,102,43,69)(33,101,44,68)>;
G:=Group( (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110), (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)(89,90,91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108,109,110), (1,45,34,23,12)(2,46,35,24,13)(3,47,36,25,14)(4,48,37,26,15)(5,49,38,27,16)(6,50,39,28,17)(7,51,40,29,18)(8,52,41,30,19)(9,53,42,31,20)(10,54,43,32,21)(11,55,44,33,22)(56,100,89,78,67)(57,101,90,79,68)(58,102,91,80,69)(59,103,92,81,70)(60,104,93,82,71)(61,105,94,83,72)(62,106,95,84,73)(63,107,96,85,74)(64,108,97,86,75)(65,109,98,87,76)(66,110,99,88,77), (1,56)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,60)(9,59)(10,58)(11,57)(12,78,45,89)(13,88,46,99)(14,87,47,98)(15,86,48,97)(16,85,49,96)(17,84,50,95)(18,83,51,94)(19,82,52,93)(20,81,53,92)(21,80,54,91)(22,79,55,90)(23,100,34,67)(24,110,35,77)(25,109,36,76)(26,108,37,75)(27,107,38,74)(28,106,39,73)(29,105,40,72)(30,104,41,71)(31,103,42,70)(32,102,43,69)(33,101,44,68) );
G=PermutationGroup([[(1,56),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,71),(17,72),(18,73),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110)], [(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),(89,90,91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108,109,110)], [(1,45,34,23,12),(2,46,35,24,13),(3,47,36,25,14),(4,48,37,26,15),(5,49,38,27,16),(6,50,39,28,17),(7,51,40,29,18),(8,52,41,30,19),(9,53,42,31,20),(10,54,43,32,21),(11,55,44,33,22),(56,100,89,78,67),(57,101,90,79,68),(58,102,91,80,69),(59,103,92,81,70),(60,104,93,82,71),(61,105,94,83,72),(62,106,95,84,73),(63,107,96,85,74),(64,108,97,86,75),(65,109,98,87,76),(66,110,99,88,77)], [(1,56),(2,66),(3,65),(4,64),(5,63),(6,62),(7,61),(8,60),(9,59),(10,58),(11,57),(12,78,45,89),(13,88,46,99),(14,87,47,98),(15,86,48,97),(16,85,49,96),(17,84,50,95),(18,83,51,94),(19,82,52,93),(20,81,53,92),(21,80,54,91),(22,79,55,90),(23,100,34,67),(24,110,35,77),(25,109,36,76),(26,108,37,75),(27,107,38,74),(28,106,39,73),(29,105,40,72),(30,104,41,71),(31,103,42,70),(32,102,43,69),(33,101,44,68)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 10 | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22O | 55A | ··· | 55J | 110A | ··· | 110J |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 5 | 5 | 55 | 55 | 55 | 55 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | D11 | Dic11 | D22 | Dic11 | F5 | C2×F5 | C11⋊F5 | C2×C11⋊F5 |
| kernel | C2×C11⋊F5 | C11⋊F5 | D5×C22 | D5×C11 | C110 | D10 | D5 | D5 | C10 | C22 | C11 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 1 | 1 | 10 | 10 |
Matrix representation of C2×C11⋊F5 ►in GL6(𝔽661)
| 660 | 0 | 0 | 0 | 0 | 0 |
| 0 | 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 |
| 18 | 660 | 0 | 0 | 0 | 0 |
| 556 | 520 | 0 | 0 | 0 | 0 |
| 0 | 0 | 217 | 660 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 217 | 660 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 660 | 0 | 506 | 370 |
| 0 | 0 | 0 | 660 | 291 | 154 |
| 422 | 49 | 0 | 0 | 0 | 0 |
| 480 | 239 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 217 | 660 | 0 | 0 |
| 0 | 0 | 155 | 291 | 155 | 291 |
| 0 | 0 | 215 | 506 | 215 | 506 |
G:=sub<GL(6,GF(661))| [660,0,0,0,0,0,0,660,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,556,0,0,0,0,660,520,0,0,0,0,0,0,217,1,0,0,0,0,660,0,0,0,0,0,0,0,217,1,0,0,0,0,660,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,660,0,0,1,0,506,291,0,0,0,1,370,154],[422,480,0,0,0,0,49,239,0,0,0,0,0,0,1,217,155,215,0,0,0,660,291,506,0,0,0,0,155,215,0,0,0,0,291,506] >;
C2×C11⋊F5 in GAP, Magma, Sage, TeX
C_2\times C_{11}\rtimes F_5 % in TeX
G:=Group("C2xC11:F5"); // GroupNames label
G:=SmallGroup(440,46);
// by ID
G=gap.SmallGroup(440,46);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-11,20,483,173,10004]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^11=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
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