direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C5⋊2C8, C10⋊2C8, C20.6C4, C4.14D10, C4.3Dic5, C20.14C22, C22.2Dic5, C5⋊4(C2×C8), C4○(C5⋊2C8), (C2×C4).5D5, (C2×C10).4C4, (C2×C20).6C2, C10.13(C2×C4), C2.1(C2×Dic5), SmallGroup(80,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C2×C5⋊2C8 |
Generators and relations for C2×C5⋊2C8
G = < a,b,c | a2=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×C5⋊2C8
►Regular action on 80 points
Generators in S80
(1 60)(2 61)(3 62)(4 63)(5 64)(6 57)(7 58)(8 59)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(41 70)(42 71)(43 72)(44 65)(45 66)(46 67)(47 68)(48 69)(49 78)(50 79)(51 80)(52 73)(53 74)(54 75)(55 76)(56 77)
(1 11 25 67 75)(2 76 68 26 12)(3 13 27 69 77)(4 78 70 28 14)(5 15 29 71 79)(6 80 72 30 16)(7 9 31 65 73)(8 74 66 32 10)(17 46 54 60 38)(18 39 61 55 47)(19 48 56 62 40)(20 33 63 49 41)(21 42 50 64 34)(22 35 57 51 43)(23 44 52 58 36)(24 37 59 53 45)
(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)
G:=sub<Sym(80)| (1,60)(2,61)(3,62)(4,63)(5,64)(6,57)(7,58)(8,59)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(41,70)(42,71)(43,72)(44,65)(45,66)(46,67)(47,68)(48,69)(49,78)(50,79)(51,80)(52,73)(53,74)(54,75)(55,76)(56,77), (1,11,25,67,75)(2,76,68,26,12)(3,13,27,69,77)(4,78,70,28,14)(5,15,29,71,79)(6,80,72,30,16)(7,9,31,65,73)(8,74,66,32,10)(17,46,54,60,38)(18,39,61,55,47)(19,48,56,62,40)(20,33,63,49,41)(21,42,50,64,34)(22,35,57,51,43)(23,44,52,58,36)(24,37,59,53,45), (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)>;
G:=Group( (1,60)(2,61)(3,62)(4,63)(5,64)(6,57)(7,58)(8,59)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(41,70)(42,71)(43,72)(44,65)(45,66)(46,67)(47,68)(48,69)(49,78)(50,79)(51,80)(52,73)(53,74)(54,75)(55,76)(56,77), (1,11,25,67,75)(2,76,68,26,12)(3,13,27,69,77)(4,78,70,28,14)(5,15,29,71,79)(6,80,72,30,16)(7,9,31,65,73)(8,74,66,32,10)(17,46,54,60,38)(18,39,61,55,47)(19,48,56,62,40)(20,33,63,49,41)(21,42,50,64,34)(22,35,57,51,43)(23,44,52,58,36)(24,37,59,53,45), (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) );
G=PermutationGroup([[(1,60),(2,61),(3,62),(4,63),(5,64),(6,57),(7,58),(8,59),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(41,70),(42,71),(43,72),(44,65),(45,66),(46,67),(47,68),(48,69),(49,78),(50,79),(51,80),(52,73),(53,74),(54,75),(55,76),(56,77)], [(1,11,25,67,75),(2,76,68,26,12),(3,13,27,69,77),(4,78,70,28,14),(5,15,29,71,79),(6,80,72,30,16),(7,9,31,65,73),(8,74,66,32,10),(17,46,54,60,38),(18,39,61,55,47),(19,48,56,62,40),(20,33,63,49,41),(21,42,50,64,34),(22,35,57,51,43),(23,44,52,58,36),(24,37,59,53,45)], [(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)]])
C2×C5⋊2C8 is a maximal subgroup of
C42.D5 C20⋊3C8 C10.D8 C20.Q8 D20⋊6C4 C10.Q16 C8×Dic5 C20.8Q8 C40⋊8C4 D10⋊1C8 C20.53D4 C20.55D4 D4⋊Dic5 Q8⋊Dic5 C20.C8 D5×C2×C8 D20.2C4 D4.Dic5 D4.8D10 C20.14F5
C2×C5⋊2C8 is a maximal quotient of
C20⋊3C8 C20.4C8 C20.55D4 C20.14F5
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D5 | Dic5 | D10 | Dic5 | C5⋊2C8 |
| kernel | C2×C5⋊2C8 | C5⋊2C8 | C2×C20 | C20 | C2×C10 | C10 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C2×C5⋊2C8 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 34 | 40 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 31 | 3 |
| 0 | 32 | 10 |
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,34,1,0,40,0],[1,0,0,0,31,32,0,3,10] >;
C2×C5⋊2C8 in GAP, Magma, Sage, TeX
C_2\times C_5\rtimes_2C_8
% in TeX
G:=Group("C2xC5:2C8"); // GroupNames label
G:=SmallGroup(80,9);
// by ID
G=gap.SmallGroup(80,9);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,20,42,1604]);
// Polycyclic
G:=Group<a,b,c|a^2=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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