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 [×2], C4 [×4], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×3], D4, Q8, C23, C10, C10 [×2], C42, C22⋊C4 [×2], M4(2) [×2], C2×D4, C2×Q8, Dic5 [×3], C20, C2×C10, C2×C10 [×3], C4.10D4 [×2], C4.4D4, C5⋊C8 [×2], Dic10, C2×Dic5 [×2], C2×Dic5, C2×C20, C5×D4, C22×C10, C42.C4, C4×Dic5, C23.D5 [×2], C22.F5 [×2], C2×Dic10, D4×C10, Dic5.D4 [×2], C20.17D4, (D4×C10).C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], 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 17 7 12)(2 18 8 13)(3 19 9 14)(4 20 10 15)(5 16 6 11)(21 32 27 37)(22 33 28 38)(23 34 29 39)(24 35 30 40)(25 31 26 36)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 78 66 73)(62 79 67 74)(63 80 68 75)(64 71 69 76)(65 72 70 77)
(1 33)(2 34)(3 35)(4 31)(5 32)(6 37)(7 38)(8 39)(9 40)(10 36)(11 27)(12 28)(13 29)(14 30)(15 26)(16 21)(17 22)(18 23)(19 24)(20 25)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 61)(51 79)(52 80)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)
(1 67 28 46 7 62 22 41)(2 69 27 44 8 64 21 49)(3 61 26 42 9 66 25 47)(4 63 30 50 10 68 24 45)(5 65 29 48 6 70 23 43)(11 72 39 58 16 77 34 53)(12 74 38 56 17 79 33 51)(13 76 37 54 18 71 32 59)(14 78 36 52 19 73 31 57)(15 80 40 60 20 75 35 55)
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), (1,17,7,12)(2,18,8,13)(3,19,9,14)(4,20,10,15)(5,16,6,11)(21,32,27,37)(22,33,28,38)(23,34,29,39)(24,35,30,40)(25,31,26,36)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,78,66,73)(62,79,67,74)(63,80,68,75)(64,71,69,76)(65,72,70,77), (1,33)(2,34)(3,35)(4,31)(5,32)(6,37)(7,38)(8,39)(9,40)(10,36)(11,27)(12,28)(13,29)(14,30)(15,26)(16,21)(17,22)(18,23)(19,24)(20,25)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,61)(51,79)(52,80)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78), (1,67,28,46,7,62,22,41)(2,69,27,44,8,64,21,49)(3,61,26,42,9,66,25,47)(4,63,30,50,10,68,24,45)(5,65,29,48,6,70,23,43)(11,72,39,58,16,77,34,53)(12,74,38,56,17,79,33,51)(13,76,37,54,18,71,32,59)(14,78,36,52,19,73,31,57)(15,80,40,60,20,75,35,55)>;
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), (1,17,7,12)(2,18,8,13)(3,19,9,14)(4,20,10,15)(5,16,6,11)(21,32,27,37)(22,33,28,38)(23,34,29,39)(24,35,30,40)(25,31,26,36)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,78,66,73)(62,79,67,74)(63,80,68,75)(64,71,69,76)(65,72,70,77), (1,33)(2,34)(3,35)(4,31)(5,32)(6,37)(7,38)(8,39)(9,40)(10,36)(11,27)(12,28)(13,29)(14,30)(15,26)(16,21)(17,22)(18,23)(19,24)(20,25)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,61)(51,79)(52,80)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78), (1,67,28,46,7,62,22,41)(2,69,27,44,8,64,21,49)(3,61,26,42,9,66,25,47)(4,63,30,50,10,68,24,45)(5,65,29,48,6,70,23,43)(11,72,39,58,16,77,34,53)(12,74,38,56,17,79,33,51)(13,76,37,54,18,71,32,59)(14,78,36,52,19,73,31,57)(15,80,40,60,20,75,35,55) );
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)], [(1,17,7,12),(2,18,8,13),(3,19,9,14),(4,20,10,15),(5,16,6,11),(21,32,27,37),(22,33,28,38),(23,34,29,39),(24,35,30,40),(25,31,26,36),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,78,66,73),(62,79,67,74),(63,80,68,75),(64,71,69,76),(65,72,70,77)], [(1,33),(2,34),(3,35),(4,31),(5,32),(6,37),(7,38),(8,39),(9,40),(10,36),(11,27),(12,28),(13,29),(14,30),(15,26),(16,21),(17,22),(18,23),(19,24),(20,25),(41,62),(42,63),(43,64),(44,65),(45,66),(46,67),(47,68),(48,69),(49,70),(50,61),(51,79),(52,80),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78)], [(1,67,28,46,7,62,22,41),(2,69,27,44,8,64,21,49),(3,61,26,42,9,66,25,47),(4,63,30,50,10,68,24,45),(5,65,29,48,6,70,23,43),(11,72,39,58,16,77,34,53),(12,74,38,56,17,79,33,51),(13,76,37,54,18,71,32,59),(14,78,36,52,19,73,31,57),(15,80,40,60,20,75,35,55)])
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
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