direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×M4(2).C4, M4(2).1C20, C8.4(C2×C20), C40.86(C2×C4), C4.78(D4×C10), C20.96(C4⋊C4), (C2×C20).44Q8, C8.C4⋊3C10, (C2×C20).523D4, C20.483(C2×D4), C23.5(C5×Q8), C22.2(Q8×C10), (C22×C10).5Q8, C4.29(C22×C20), (C5×M4(2)).7C4, (C2×C20).903C23, (C2×C40).270C22, C20.246(C22×C4), (C2×M4(2)).2C10, (C10×M4(2)).34C2, M4(2).10(C2×C10), (C22×C20).415C22, (C5×M4(2)).44C22, C4.16(C5×C4⋊C4), C10.95(C2×C4⋊C4), C2.16(C10×C4⋊C4), (C2×C4).7(C5×Q8), (C2×C8).17(C2×C10), (C2×C4).26(C2×C20), (C2×C4).126(C5×D4), C22.12(C5×C4⋊C4), (C2×C10).15(C2×Q8), (C5×C8.C4)⋊12C2, (C2×C10).56(C4⋊C4), (C2×C20).372(C2×C4), (C2×C4).78(C22×C10), (C22×C4).34(C2×C10), SmallGroup(320,931)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×M4(2).C4
G = < a,b,c,d | a5=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, dcd-1=b4c >
Subgroups: 130 in 102 conjugacy classes, 78 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, C10, C10, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C8.C4, C2×M4(2), C2×M4(2), C40, C40, C2×C20, C2×C20, C22×C10, M4(2).C4, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C5×C8.C4, C10×M4(2), C10×M4(2), C5×M4(2).C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C23, C10, C4⋊C4, C22×C4, C2×D4, C2×Q8, C20, C2×C10, C2×C4⋊C4, C2×C20, C5×D4, C5×Q8, C22×C10, M4(2).C4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C10×C4⋊C4, C5×M4(2).C4
►On 80 points
(1 45 25 33 12)(2 46 26 34 13)(3 47 27 35 14)(4 48 28 36 15)(5 41 29 37 16)(6 42 30 38 9)(7 43 31 39 10)(8 44 32 40 11)(17 67 76 55 64)(18 68 77 56 57)(19 69 78 49 58)(20 70 79 50 59)(21 71 80 51 60)(22 72 73 52 61)(23 65 74 53 62)(24 66 75 54 63)
(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 5)(3 7)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(58 62)(60 64)(65 69)(67 71)(74 78)(76 80)
(1 61 7 63 5 57 3 59)(2 60 8 62 6 64 4 58)(9 55 15 49 13 51 11 53)(10 54 16 56 14 50 12 52)(17 48 19 46 21 44 23 42)(18 47 20 45 22 43 24 41)(25 72 31 66 29 68 27 70)(26 71 32 65 30 67 28 69)(33 73 39 75 37 77 35 79)(34 80 40 74 38 76 36 78)
G:=sub<Sym(80)| (1,45,25,33,12)(2,46,26,34,13)(3,47,27,35,14)(4,48,28,36,15)(5,41,29,37,16)(6,42,30,38,9)(7,43,31,39,10)(8,44,32,40,11)(17,67,76,55,64)(18,68,77,56,57)(19,69,78,49,58)(20,70,79,50,59)(21,71,80,51,60)(22,72,73,52,61)(23,65,74,53,62)(24,66,75,54,63), (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), (1,5)(3,7)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(58,62)(60,64)(65,69)(67,71)(74,78)(76,80), (1,61,7,63,5,57,3,59)(2,60,8,62,6,64,4,58)(9,55,15,49,13,51,11,53)(10,54,16,56,14,50,12,52)(17,48,19,46,21,44,23,42)(18,47,20,45,22,43,24,41)(25,72,31,66,29,68,27,70)(26,71,32,65,30,67,28,69)(33,73,39,75,37,77,35,79)(34,80,40,74,38,76,36,78)>;
G:=Group( (1,45,25,33,12)(2,46,26,34,13)(3,47,27,35,14)(4,48,28,36,15)(5,41,29,37,16)(6,42,30,38,9)(7,43,31,39,10)(8,44,32,40,11)(17,67,76,55,64)(18,68,77,56,57)(19,69,78,49,58)(20,70,79,50,59)(21,71,80,51,60)(22,72,73,52,61)(23,65,74,53,62)(24,66,75,54,63), (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), (1,5)(3,7)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(58,62)(60,64)(65,69)(67,71)(74,78)(76,80), (1,61,7,63,5,57,3,59)(2,60,8,62,6,64,4,58)(9,55,15,49,13,51,11,53)(10,54,16,56,14,50,12,52)(17,48,19,46,21,44,23,42)(18,47,20,45,22,43,24,41)(25,72,31,66,29,68,27,70)(26,71,32,65,30,67,28,69)(33,73,39,75,37,77,35,79)(34,80,40,74,38,76,36,78) );
G=PermutationGroup([[(1,45,25,33,12),(2,46,26,34,13),(3,47,27,35,14),(4,48,28,36,15),(5,41,29,37,16),(6,42,30,38,9),(7,43,31,39,10),(8,44,32,40,11),(17,67,76,55,64),(18,68,77,56,57),(19,69,78,49,58),(20,70,79,50,59),(21,71,80,51,60),(22,72,73,52,61),(23,65,74,53,62),(24,66,75,54,63)], [(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)], [(1,5),(3,7),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(49,53),(51,55),(58,62),(60,64),(65,69),(67,71),(74,78),(76,80)], [(1,61,7,63,5,57,3,59),(2,60,8,62,6,64,4,58),(9,55,15,49,13,51,11,53),(10,54,16,56,14,50,12,52),(17,48,19,46,21,44,23,42),(18,47,20,45,22,43,24,41),(25,72,31,66,29,68,27,70),(26,71,32,65,30,67,28,69),(33,73,39,75,37,77,35,79),(34,80,40,74,38,76,36,78)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 8A | ··· | 8L | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 20A | ··· | 20H | 20I | ··· | 20T | 40A | ··· | 40AV |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | - | ||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | D4 | Q8 | Q8 | C5×D4 | C5×Q8 | C5×Q8 | M4(2).C4 | C5×M4(2).C4 |
| kernel | C5×M4(2).C4 | C5×C8.C4 | C10×M4(2) | C5×M4(2) | M4(2).C4 | C8.C4 | C2×M4(2) | M4(2) | C2×C20 | C2×C20 | C22×C10 | C2×C4 | C2×C4 | C23 | C5 | C1 |
| # reps | 1 | 4 | 3 | 8 | 4 | 16 | 12 | 32 | 2 | 1 | 1 | 8 | 4 | 4 | 2 | 8 |
Matrix representation of C5×M4(2).C4 ►in GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 0 | 9 | 25 | 25 |
| 1 | 0 | 16 | 18 |
| 0 | 0 | 32 | 36 |
| 0 | 0 | 18 | 9 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 21 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 40 |
| 0 | 25 | 1 | 17 |
| 0 | 16 | 0 | 24 |
| 32 | 32 | 0 | 0 |
| 0 | 18 | 0 | 25 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[0,1,0,0,9,0,0,0,25,16,32,18,25,18,36,9],[40,0,0,0,0,1,0,0,0,0,1,0,0,21,1,40],[0,0,32,0,25,16,32,18,1,0,0,0,17,24,0,25] >;
C5×M4(2).C4 in GAP, Magma, Sage, TeX
C_5\times M_4(2).C_4
% in TeX
G:=Group("C5xM4(2).C4"); // GroupNames label
G:=SmallGroup(320,931);
// by ID
G=gap.SmallGroup(320,931);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,288,1731,7004,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b^-1,d*c*d^-1=b^4*c>;
// generators/relations