metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×D4).2F5, (D4×C10).2C4, (C4×Dic5).4C4, C2.7(C23⋊F5), C5⋊2(C42.C4), (C2×Dic5).12D4, C10.16(C23⋊C4), Dic5.D4⋊2C2, C20.17D4.3C2, C22.20(C22⋊F5), (C2×Dic10).46C22, (C2×C4).2(C2×F5), (C2×C20).12(C2×C4), (C2×C10).36(C22⋊C4), SmallGroup(320,261)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (D4×C10).C4
G = < a,b,c,d | a10=b4=c2=1, d4=b2, ab=ba, ac=ca, dad-1=a3b2, cbc=b-1, dbd-1=a5b-1, dcd-1=b-1c >
Subgroups: 298 in 64 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C4.10D4, C4.4D4, C5⋊C8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C42.C4, C4×Dic5, C23.D5, C22.F5, C2×Dic10, D4×C10, Dic5.D4, C20.17D4, (D4×C10).C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C23⋊C4, C2×F5, C42.C4, C22⋊F5, C23⋊F5, (D4×C10).C4
Character table of (D4×C10).C4
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 20A | 20B | |
| size | 1 | 1 | 2 | 8 | 4 | 20 | 20 | 20 | 20 | 4 | 40 | 40 | 40 | 40 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | -i | i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | i | i | -i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 0 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×F5 |
| ρ12 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ13 | 4 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ14 | 4 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | √5 | -√5 | √5 | -√5 | 1 | 1 | orthogonal lifted from C22⋊F5 |
| ρ15 | 4 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -√5 | √5 | -√5 | √5 | 1 | 1 | orthogonal lifted from C22⋊F5 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42.C4 |
| ρ17 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | -√5 | √5 | complex lifted from C23⋊F5 |
| ρ18 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | √5 | -√5 | complex lifted from C23⋊F5 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42.C4 |
| ρ20 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | √5 | -√5 | complex lifted from C23⋊F5 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | -√5 | √5 | complex lifted from C23⋊F5 |
| ρ22 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2√5 | -2√5 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ23 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2√5 | 2√5 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 80 points
Generators in S80
(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 18 9 15)(2 19 10 11)(3 20 6 12)(4 16 7 13)(5 17 8 14)(21 34 29 38)(22 35 30 39)(23 31 26 40)(24 32 27 36)(25 33 28 37)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(1 32)(2 33)(3 34)(4 35)(5 31)(6 38)(7 39)(8 40)(9 36)(10 37)(11 28)(12 29)(13 30)(14 26)(15 27)(16 22)(17 23)(18 24)(19 25)(20 21)(41 69)(42 70)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 79)(52 80)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)
(1 64 27 46 9 69 24 41)(2 66 26 44 10 61 23 49)(3 68 30 42 6 63 22 47)(4 70 29 50 7 65 21 45)(5 62 28 48 8 67 25 43)(11 76 40 54 19 71 31 59)(12 78 39 52 20 73 35 57)(13 80 38 60 16 75 34 55)(14 72 37 58 17 77 33 53)(15 74 36 56 18 79 32 51)
G:=sub<Sym(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), (1,18,9,15)(2,19,10,11)(3,20,6,12)(4,16,7,13)(5,17,8,14)(21,34,29,38)(22,35,30,39)(23,31,26,40)(24,32,27,36)(25,33,28,37)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (1,32)(2,33)(3,34)(4,35)(5,31)(6,38)(7,39)(8,40)(9,36)(10,37)(11,28)(12,29)(13,30)(14,26)(15,27)(16,22)(17,23)(18,24)(19,25)(20,21)(41,69)(42,70)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,79)(52,80)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78), (1,64,27,46,9,69,24,41)(2,66,26,44,10,61,23,49)(3,68,30,42,6,63,22,47)(4,70,29,50,7,65,21,45)(5,62,28,48,8,67,25,43)(11,76,40,54,19,71,31,59)(12,78,39,52,20,73,35,57)(13,80,38,60,16,75,34,55)(14,72,37,58,17,77,33,53)(15,74,36,56,18,79,32,51)>;
G:=Group( (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), (1,18,9,15)(2,19,10,11)(3,20,6,12)(4,16,7,13)(5,17,8,14)(21,34,29,38)(22,35,30,39)(23,31,26,40)(24,32,27,36)(25,33,28,37)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (1,32)(2,33)(3,34)(4,35)(5,31)(6,38)(7,39)(8,40)(9,36)(10,37)(11,28)(12,29)(13,30)(14,26)(15,27)(16,22)(17,23)(18,24)(19,25)(20,21)(41,69)(42,70)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,79)(52,80)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78), (1,64,27,46,9,69,24,41)(2,66,26,44,10,61,23,49)(3,68,30,42,6,63,22,47)(4,70,29,50,7,65,21,45)(5,62,28,48,8,67,25,43)(11,76,40,54,19,71,31,59)(12,78,39,52,20,73,35,57)(13,80,38,60,16,75,34,55)(14,72,37,58,17,77,33,53)(15,74,36,56,18,79,32,51) );
G=PermutationGroup([[(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)], [(1,18,9,15),(2,19,10,11),(3,20,6,12),(4,16,7,13),(5,17,8,14),(21,34,29,38),(22,35,30,39),(23,31,26,40),(24,32,27,36),(25,33,28,37),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,71,66,76),(62,72,67,77),(63,73,68,78),(64,74,69,79),(65,75,70,80)], [(1,32),(2,33),(3,34),(4,35),(5,31),(6,38),(7,39),(8,40),(9,36),(10,37),(11,28),(12,29),(13,30),(14,26),(15,27),(16,22),(17,23),(18,24),(19,25),(20,21),(41,69),(42,70),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,79),(52,80),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78)], [(1,64,27,46,9,69,24,41),(2,66,26,44,10,61,23,49),(3,68,30,42,6,63,22,47),(4,70,29,50,7,65,21,45),(5,62,28,48,8,67,25,43),(11,76,40,54,19,71,31,59),(12,78,39,52,20,73,35,57),(13,80,38,60,16,75,34,55),(14,72,37,58,17,77,33,53),(15,74,36,56,18,79,32,51)]])
Matrix representation of (D4×C10).C4 ►in GL8(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 6 | 34 |
| 0 | 0 | 0 | 0 | 0 | 2 | 6 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 30 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 37 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 24 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 11 | 32 | 19 |
| 0 | 0 | 0 | 0 | 16 | 0 | 22 | 19 |
| 0 | 0 | 0 | 0 | 35 | 0 | 25 | 16 |
| 0 | 0 | 0 | 0 | 0 | 4 | 11 | 0 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,35,35,2,0,0,0,0,0,6,40,0,2,0,0,0,0,0,0,6,6,0,0,0,0,0,0,34,0],[32,10,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,10,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[36,30,0,0,0,0,0,0,32,5,0,0,0,0,0,0,0,0,37,26,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[0,0,37,24,0,0,0,0,0,0,1,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,16,35,0,0,0,0,0,11,0,0,4,0,0,0,0,32,22,25,11,0,0,0,0,19,19,16,0] >;
(D4×C10).C4 in GAP, Magma, Sage, TeX
(D_4\times C_{10}).C_4 % in TeX
G:=Group("(D4xC10).C4"); // GroupNames label
G:=SmallGroup(320,261);
// by ID
G=gap.SmallGroup(320,261);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,184,1571,570,297,136,1684,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^2=1,d^4=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^3*b^2,c*b*c=b^-1,d*b*d^-1=a^5*b^-1,d*c*d^-1=b^-1*c>;
// generators/relations
Export