metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×Q8).2F5, (Q8×C10).2C4, (C4×Dic5).5C4, C5⋊2(C42.3C4), (C2×Dic5).13D4, C2.10(C23⋊F5), C10.19(C23⋊C4), Dic5⋊Q8.2C2, Dic5.D4.2C2, C22.22(C22⋊F5), (C2×Dic10).47C22, (C2×C4).4(C2×F5), (C2×C20).17(C2×C4), (C2×C10).42(C22⋊C4), SmallGroup(320,267)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (Q8×C10).C4
G = < a,b,c,d | a10=b4=1, c2=d4=b2, ab=ba, ac=ca, dad-1=a3b2, cbc-1=b-1, dbd-1=a5b, dcd-1=b-1c >
Subgroups: 266 in 60 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C4 [×6], C22, C5, C8 [×2], C2×C4, C2×C4 [×4], Q8 [×2], C10, C10, C42, C4⋊C4 [×2], M4(2) [×2], C2×Q8, C2×Q8, Dic5 [×4], C20 [×2], C2×C10, C4.10D4 [×2], C4⋊Q8, C5⋊C8 [×2], Dic10, C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20, C5×Q8, C42.3C4, C4×Dic5, C10.D4 [×2], C22.F5 [×2], C2×Dic10, Q8×C10, Dic5.D4 [×2], Dic5⋊Q8, (Q8×C10).C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5, C23⋊C4, C2×F5, C42.3C4, C22⋊F5, C23⋊F5, (Q8×C10).C4
Character table of (Q8×C10).C4
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 20E | 20F | |
| size | 1 | 1 | 2 | 4 | 8 | 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 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | -2 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 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 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | √5 | 1 | -√5 | -√5 | 1 | √5 | orthogonal lifted from C22⋊F5 |
| ρ14 | 4 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -√5 | 1 | √5 | √5 | 1 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ15 | 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 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C42.3C4, Schur index 2 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C42.3C4, Schur index 2 |
| ρ18 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ54+2ζ53+1 | √5 | 2ζ54+2ζ52+1 | 2ζ53+2ζ5+1 | -√5 | 2ζ52+2ζ5+1 | complex lifted from C23⋊F5 |
| ρ19 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ52+2ζ5+1 | √5 | 2ζ53+2ζ5+1 | 2ζ54+2ζ52+1 | -√5 | 2ζ54+2ζ53+1 | complex lifted from C23⋊F5 |
| ρ20 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ53+2ζ5+1 | -√5 | 2ζ54+2ζ53+1 | 2ζ52+2ζ5+1 | √5 | 2ζ54+2ζ52+1 | complex lifted from C23⋊F5 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ54+2ζ52+1 | -√5 | 2ζ52+2ζ5+1 | 2ζ54+2ζ53+1 | √5 | 2ζ53+2ζ5+1 | 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 20 8 13)(2 16 9 14)(3 17 10 15)(4 18 6 11)(5 19 7 12)(21 38 28 31)(22 39 29 32)(23 40 30 33)(24 36 26 34)(25 37 27 35)(41 51 46 56)(42 52 47 57)(43 53 48 58)(44 54 49 59)(45 55 50 60)(61 75 66 80)(62 76 67 71)(63 77 68 72)(64 78 69 73)(65 79 70 74)
(1 21 8 28)(2 22 9 29)(3 23 10 30)(4 24 6 26)(5 25 7 27)(11 36 18 34)(12 37 19 35)(13 38 20 31)(14 39 16 32)(15 40 17 33)(41 78 46 73)(42 79 47 74)(43 80 48 75)(44 71 49 76)(45 72 50 77)(51 64 56 69)(52 65 57 70)(53 66 58 61)(54 67 59 62)(55 68 60 63)
(1 58 40 65 8 53 33 70)(2 60 39 63 9 55 32 68)(3 52 38 61 10 57 31 66)(4 54 37 69 6 59 35 64)(5 56 36 67 7 51 34 62)(11 44 27 73 18 49 25 78)(12 46 26 71 19 41 24 76)(13 48 30 79 20 43 23 74)(14 50 29 77 16 45 22 72)(15 42 28 75 17 47 21 80)
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), (1,20,8,13)(2,16,9,14)(3,17,10,15)(4,18,6,11)(5,19,7,12)(21,38,28,31)(22,39,29,32)(23,40,30,33)(24,36,26,34)(25,37,27,35)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,75,66,80)(62,76,67,71)(63,77,68,72)(64,78,69,73)(65,79,70,74), (1,21,8,28)(2,22,9,29)(3,23,10,30)(4,24,6,26)(5,25,7,27)(11,36,18,34)(12,37,19,35)(13,38,20,31)(14,39,16,32)(15,40,17,33)(41,78,46,73)(42,79,47,74)(43,80,48,75)(44,71,49,76)(45,72,50,77)(51,64,56,69)(52,65,57,70)(53,66,58,61)(54,67,59,62)(55,68,60,63), (1,58,40,65,8,53,33,70)(2,60,39,63,9,55,32,68)(3,52,38,61,10,57,31,66)(4,54,37,69,6,59,35,64)(5,56,36,67,7,51,34,62)(11,44,27,73,18,49,25,78)(12,46,26,71,19,41,24,76)(13,48,30,79,20,43,23,74)(14,50,29,77,16,45,22,72)(15,42,28,75,17,47,21,80)>;
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), (1,20,8,13)(2,16,9,14)(3,17,10,15)(4,18,6,11)(5,19,7,12)(21,38,28,31)(22,39,29,32)(23,40,30,33)(24,36,26,34)(25,37,27,35)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,75,66,80)(62,76,67,71)(63,77,68,72)(64,78,69,73)(65,79,70,74), (1,21,8,28)(2,22,9,29)(3,23,10,30)(4,24,6,26)(5,25,7,27)(11,36,18,34)(12,37,19,35)(13,38,20,31)(14,39,16,32)(15,40,17,33)(41,78,46,73)(42,79,47,74)(43,80,48,75)(44,71,49,76)(45,72,50,77)(51,64,56,69)(52,65,57,70)(53,66,58,61)(54,67,59,62)(55,68,60,63), (1,58,40,65,8,53,33,70)(2,60,39,63,9,55,32,68)(3,52,38,61,10,57,31,66)(4,54,37,69,6,59,35,64)(5,56,36,67,7,51,34,62)(11,44,27,73,18,49,25,78)(12,46,26,71,19,41,24,76)(13,48,30,79,20,43,23,74)(14,50,29,77,16,45,22,72)(15,42,28,75,17,47,21,80) );
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)], [(1,20,8,13),(2,16,9,14),(3,17,10,15),(4,18,6,11),(5,19,7,12),(21,38,28,31),(22,39,29,32),(23,40,30,33),(24,36,26,34),(25,37,27,35),(41,51,46,56),(42,52,47,57),(43,53,48,58),(44,54,49,59),(45,55,50,60),(61,75,66,80),(62,76,67,71),(63,77,68,72),(64,78,69,73),(65,79,70,74)], [(1,21,8,28),(2,22,9,29),(3,23,10,30),(4,24,6,26),(5,25,7,27),(11,36,18,34),(12,37,19,35),(13,38,20,31),(14,39,16,32),(15,40,17,33),(41,78,46,73),(42,79,47,74),(43,80,48,75),(44,71,49,76),(45,72,50,77),(51,64,56,69),(52,65,57,70),(53,66,58,61),(54,67,59,62),(55,68,60,63)], [(1,58,40,65,8,53,33,70),(2,60,39,63,9,55,32,68),(3,52,38,61,10,57,31,66),(4,54,37,69,6,59,35,64),(5,56,36,67,7,51,34,62),(11,44,27,73,18,49,25,78),(12,46,26,71,19,41,24,76),(13,48,30,79,20,43,23,74),(14,50,29,77,16,45,22,72),(15,42,28,75,17,47,21,80)])
Matrix representation of (Q8×C10).C4 ►in GL8(𝔽41)
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 40 | 40 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 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 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 0 | 0 | 32 | 0 |
| 7 | 0 | 14 | 14 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 7 | 0 | 0 | 0 | 0 |
| 27 | 34 | 27 | 0 | 0 | 0 | 0 | 0 |
| 34 | 7 | 7 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
G:=sub<GL(8,GF(41))| [0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0],[7,14,27,34,0,0,0,0,0,14,34,7,0,0,0,0,14,0,27,7,0,0,0,0,14,7,0,34,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
(Q8×C10).C4 in GAP, Magma, Sage, TeX
(Q_8\times C_{10}).C_4 % in TeX
G:=Group("(Q8xC10).C4"); // GroupNames label
G:=SmallGroup(320,267);
// by ID
G=gap.SmallGroup(320,267);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,184,1571,570,297,136,1684,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=1,c^2=d^4=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^3*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^5*b,d*c*d^-1=b^-1*c>;
// generators/relations
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