direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C4.10D4, C20.59D4, M4(2).1C10, (C2×C4).C20, C4.10(C5×D4), (C2×C20).13C4, (C2×Q8).1C10, (Q8×C10).6C2, C22.4(C2×C20), (C2×C20).60C22, (C5×M4(2)).3C2, C10.34(C22⋊C4), (C2×C4).2(C2×C10), C2.5(C5×C22⋊C4), (C2×C10).41(C2×C4), SmallGroup(160,51)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4.10D4
G = < a,b,c,d | a5=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >
Smallest permutation representation of C5×C4.10D4
►On 80 points
Generators in S80
(1 63 71 23 31)(2 64 72 24 32)(3 57 65 17 25)(4 58 66 18 26)(5 59 67 19 27)(6 60 68 20 28)(7 61 69 21 29)(8 62 70 22 30)(9 74 50 34 46)(10 75 51 35 47)(11 76 52 36 48)(12 77 53 37 41)(13 78 54 38 42)(14 79 55 39 43)(15 80 56 40 44)(16 73 49 33 45)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)(41 43 45 47)(42 48 46 44)(49 51 53 55)(50 56 54 52)(57 59 61 63)(58 64 62 60)(65 67 69 71)(66 72 70 68)(73 75 77 79)(74 80 78 76)
(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)
(1 36 7 38 5 40 3 34)(2 37 4 35 6 33 8 39)(9 71 11 69 13 67 15 65)(10 68 16 70 14 72 12 66)(17 74 23 76 21 78 19 80)(18 75 20 73 22 79 24 77)(25 50 31 52 29 54 27 56)(26 51 28 49 30 55 32 53)(41 58 47 60 45 62 43 64)(42 59 44 57 46 63 48 61)
G:=sub<Sym(80)| (1,63,71,23,31)(2,64,72,24,32)(3,57,65,17,25)(4,58,66,18,26)(5,59,67,19,27)(6,60,68,20,28)(7,61,69,21,29)(8,62,70,22,30)(9,74,50,34,46)(10,75,51,35,47)(11,76,52,36,48)(12,77,53,37,41)(13,78,54,38,42)(14,79,55,39,43)(15,80,56,40,44)(16,73,49,33,45), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20)(25,27,29,31)(26,32,30,28)(33,35,37,39)(34,40,38,36)(41,43,45,47)(42,48,46,44)(49,51,53,55)(50,56,54,52)(57,59,61,63)(58,64,62,60)(65,67,69,71)(66,72,70,68)(73,75,77,79)(74,80,78,76), (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), (1,36,7,38,5,40,3,34)(2,37,4,35,6,33,8,39)(9,71,11,69,13,67,15,65)(10,68,16,70,14,72,12,66)(17,74,23,76,21,78,19,80)(18,75,20,73,22,79,24,77)(25,50,31,52,29,54,27,56)(26,51,28,49,30,55,32,53)(41,58,47,60,45,62,43,64)(42,59,44,57,46,63,48,61)>;
G:=Group( (1,63,71,23,31)(2,64,72,24,32)(3,57,65,17,25)(4,58,66,18,26)(5,59,67,19,27)(6,60,68,20,28)(7,61,69,21,29)(8,62,70,22,30)(9,74,50,34,46)(10,75,51,35,47)(11,76,52,36,48)(12,77,53,37,41)(13,78,54,38,42)(14,79,55,39,43)(15,80,56,40,44)(16,73,49,33,45), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20)(25,27,29,31)(26,32,30,28)(33,35,37,39)(34,40,38,36)(41,43,45,47)(42,48,46,44)(49,51,53,55)(50,56,54,52)(57,59,61,63)(58,64,62,60)(65,67,69,71)(66,72,70,68)(73,75,77,79)(74,80,78,76), (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), (1,36,7,38,5,40,3,34)(2,37,4,35,6,33,8,39)(9,71,11,69,13,67,15,65)(10,68,16,70,14,72,12,66)(17,74,23,76,21,78,19,80)(18,75,20,73,22,79,24,77)(25,50,31,52,29,54,27,56)(26,51,28,49,30,55,32,53)(41,58,47,60,45,62,43,64)(42,59,44,57,46,63,48,61) );
G=PermutationGroup([[(1,63,71,23,31),(2,64,72,24,32),(3,57,65,17,25),(4,58,66,18,26),(5,59,67,19,27),(6,60,68,20,28),(7,61,69,21,29),(8,62,70,22,30),(9,74,50,34,46),(10,75,51,35,47),(11,76,52,36,48),(12,77,53,37,41),(13,78,54,38,42),(14,79,55,39,43),(15,80,56,40,44),(16,73,49,33,45)], [(1,3,5,7),(2,8,6,4),(9,15,13,11),(10,12,14,16),(17,19,21,23),(18,24,22,20),(25,27,29,31),(26,32,30,28),(33,35,37,39),(34,40,38,36),(41,43,45,47),(42,48,46,44),(49,51,53,55),(50,56,54,52),(57,59,61,63),(58,64,62,60),(65,67,69,71),(66,72,70,68),(73,75,77,79),(74,80,78,76)], [(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)], [(1,36,7,38,5,40,3,34),(2,37,4,35,6,33,8,39),(9,71,11,69,13,67,15,65),(10,68,16,70,14,72,12,66),(17,74,23,76,21,78,19,80),(18,75,20,73,22,79,24,77),(25,50,31,52,29,54,27,56),(26,51,28,49,30,55,32,53),(41,58,47,60,45,62,43,64),(42,59,44,57,46,63,48,61)]])
C5×C4.10D4 is a maximal subgroup of
(C2×C4).D20 (C2×Q8).D10 M4(2).21D10 D20.4D4 D20.5D4 D20.6D4 D20.7D4
55 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | D4 | C5×D4 | C4.10D4 | C5×C4.10D4 |
| kernel | C5×C4.10D4 | C5×M4(2) | Q8×C10 | C2×C20 | C4.10D4 | M4(2) | C2×Q8 | C2×C4 | C20 | C4 | C5 | C1 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 2 | 8 | 1 | 4 |
Matrix representation of C5×C4.10D4 ►in GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 9 | 0 | 0 | 0 |
| 21 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 20 | 9 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 21 | 32 | 0 | 0 |
| 0 | 0 | 23 | 33 |
| 0 | 0 | 15 | 18 |
| 39 | 31 | 0 | 0 |
| 21 | 2 | 0 | 0 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[9,21,0,0,0,32,0,0,0,0,32,20,0,0,0,9],[0,0,9,21,0,0,0,32,1,0,0,0,0,1,0,0],[0,0,39,21,0,0,31,2,23,15,0,0,33,18,0,0] >;
C5×C4.10D4 in GAP, Magma, Sage, TeX
C_5\times C_4._{10}D_4 % in TeX
G:=Group("C5xC4.10D4"); // GroupNames label
G:=SmallGroup(160,51);
// by ID
G=gap.SmallGroup(160,51);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,487,2403,1810,88]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=1,c^4=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations
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