direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C22⋊C8, C22⋊C40, C20.65D4, C23.2C20, C10.13M4(2), (C2×C10)⋊3C8, (C2×C8)⋊1C10, (C2×C40)⋊3C2, (C2×C4).3C20, C2.1(C2×C40), C4.16(C5×D4), C10.20(C2×C8), (C2×C20).17C4, (C22×C10).7C4, C22.9(C2×C20), (C22×C20).3C2, (C22×C4).2C10, C2.2(C5×M4(2)), C10.31(C22⋊C4), (C2×C20).135C22, C2.2(C5×C22⋊C4), (C2×C4).31(C2×C10), (C2×C10).58(C2×C4), SmallGroup(160,48)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22⋊C8
G = < a,b,c,d | a5=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Smallest permutation representation of C5×C22⋊C8
►On 80 points
Generators in S80
(1 14 55 63 23)(2 15 56 64 24)(3 16 49 57 17)(4 9 50 58 18)(5 10 51 59 19)(6 11 52 60 20)(7 12 53 61 21)(8 13 54 62 22)(25 45 73 33 65)(26 46 74 34 66)(27 47 75 35 67)(28 48 76 36 68)(29 41 77 37 69)(30 42 78 38 70)(31 43 79 39 71)(32 44 80 40 72)
(2 32)(4 26)(6 28)(8 30)(9 46)(11 48)(13 42)(15 44)(18 66)(20 68)(22 70)(24 72)(34 58)(36 60)(38 62)(40 64)(50 74)(52 76)(54 78)(56 80)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)
(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)| (1,14,55,63,23)(2,15,56,64,24)(3,16,49,57,17)(4,9,50,58,18)(5,10,51,59,19)(6,11,52,60,20)(7,12,53,61,21)(8,13,54,62,22)(25,45,73,33,65)(26,46,74,34,66)(27,47,75,35,67)(28,48,76,36,68)(29,41,77,37,69)(30,42,78,38,70)(31,43,79,39,71)(32,44,80,40,72), (2,32)(4,26)(6,28)(8,30)(9,46)(11,48)(13,42)(15,44)(18,66)(20,68)(22,70)(24,72)(34,58)(36,60)(38,62)(40,64)(50,74)(52,76)(54,78)(56,80), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80), (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( (1,14,55,63,23)(2,15,56,64,24)(3,16,49,57,17)(4,9,50,58,18)(5,10,51,59,19)(6,11,52,60,20)(7,12,53,61,21)(8,13,54,62,22)(25,45,73,33,65)(26,46,74,34,66)(27,47,75,35,67)(28,48,76,36,68)(29,41,77,37,69)(30,42,78,38,70)(31,43,79,39,71)(32,44,80,40,72), (2,32)(4,26)(6,28)(8,30)(9,46)(11,48)(13,42)(15,44)(18,66)(20,68)(22,70)(24,72)(34,58)(36,60)(38,62)(40,64)(50,74)(52,76)(54,78)(56,80), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80), (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([[(1,14,55,63,23),(2,15,56,64,24),(3,16,49,57,17),(4,9,50,58,18),(5,10,51,59,19),(6,11,52,60,20),(7,12,53,61,21),(8,13,54,62,22),(25,45,73,33,65),(26,46,74,34,66),(27,47,75,35,67),(28,48,76,36,68),(29,41,77,37,69),(30,42,78,38,70),(31,43,79,39,71),(32,44,80,40,72)], [(2,32),(4,26),(6,28),(8,30),(9,46),(11,48),(13,42),(15,44),(18,66),(20,68),(22,70),(24,72),(34,58),(36,60),(38,62),(40,64),(50,74),(52,76),(54,78),(56,80)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(49,73),(50,74),(51,75),(52,76),(53,77),(54,78),(55,79),(56,80)], [(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)]])
C5×C22⋊C8 is a maximal subgroup of
C23.30D20 C5⋊3(C23⋊C8) (C2×Dic5)⋊C8 C22.2D40 Dic5.14M4(2) Dic5.9M4(2) C40⋊8C4⋊C2 C23.34D20 C23.35D20 C23.10D20 C5⋊5(C8×D4) D10⋊7M4(2) C22⋊C8⋊D5 D10⋊4M4(2) Dic5⋊2M4(2) C5⋊2C8⋊26D4 D20.31D4 D20⋊13D4 D20.32D4 D20⋊14D4 C23.38D20 C22.D40 C23.13D20 Dic10⋊14D4 C22⋊Dic20 D4×C40
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C20 | C20 | C40 | D4 | M4(2) | C5×D4 | C5×M4(2) |
| kernel | C5×C22⋊C8 | C2×C40 | C22×C20 | C2×C20 | C22×C10 | C22⋊C8 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C20 | C10 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 8 | 4 | 8 | 8 | 32 | 2 | 2 | 8 | 8 |
Matrix representation of C5×C22⋊C8 ►in GL3(𝔽41) generated by
| 1 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 23 | 40 |
| 1 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 14 | 0 | 0 |
| 0 | 9 | 1 |
| 0 | 2 | 32 |
G:=sub<GL(3,GF(41))| [1,0,0,0,10,0,0,0,10],[40,0,0,0,1,23,0,0,40],[1,0,0,0,40,0,0,0,40],[14,0,0,0,9,2,0,1,32] >;
C5×C22⋊C8 in GAP, Magma, Sage, TeX
C_5\times C_2^2\rtimes C_8
% in TeX
G:=Group("C5xC2^2:C8"); // GroupNames label
G:=SmallGroup(160,48);
// by ID
G=gap.SmallGroup(160,48);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,88]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
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