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## G = (Q8×C10).C4order 320 = 26·5

### 2nd non-split extension by Q8×C10 of C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (Q8×C10).C4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C2×Dic10 — Dic5.D4 — (Q8×C10).C4
 Lower central C5 — C10 — C2×C10 — C2×C20 — (Q8×C10).C4
 Upper central C1 — C2 — C22 — C2×C4 — C2×Q8

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

Smallest permutation representation of (Q8×C10).C4
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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