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

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), (1,40,32,25)(2,36,33,21)(3,37,34,22)(4,38,35,23)(5,39,31,24)(6,14,16,29)(7,15,17,30)(8,11,18,26)(9,12,19,27)(10,13,20,28)(41,80,46,75)(42,71,47,76)(43,72,48,77)(44,73,49,78)(45,74,50,79)(51,63,56,68)(52,64,57,69)(53,65,58,70)(54,66,59,61)(55,67,60,62), (1,17,32,7)(2,18,33,8)(3,19,34,9)(4,20,35,10)(5,16,31,6)(11,21,26,36)(12,22,27,37)(13,23,28,38)(14,24,29,39)(15,25,30,40)(41,67,46,62)(42,68,47,63)(43,69,48,64)(44,70,49,65)(45,61,50,66)(51,71,56,76)(52,72,57,77)(53,73,58,78)(54,74,59,79)(55,75,60,80), (1,46,28,68,32,41,13,63)(2,48,27,66,33,43,12,61)(3,50,26,64,34,45,11,69)(4,42,30,62,35,47,15,67)(5,44,29,70,31,49,14,65)(6,53,39,73,16,58,24,78)(7,55,38,71,17,60,23,76)(8,57,37,79,18,52,22,74)(9,59,36,77,19,54,21,72)(10,51,40,75,20,56,25,80) );

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)], [(1,40,32,25),(2,36,33,21),(3,37,34,22),(4,38,35,23),(5,39,31,24),(6,14,16,29),(7,15,17,30),(8,11,18,26),(9,12,19,27),(10,13,20,28),(41,80,46,75),(42,71,47,76),(43,72,48,77),(44,73,49,78),(45,74,50,79),(51,63,56,68),(52,64,57,69),(53,65,58,70),(54,66,59,61),(55,67,60,62)], [(1,17,32,7),(2,18,33,8),(3,19,34,9),(4,20,35,10),(5,16,31,6),(11,21,26,36),(12,22,27,37),(13,23,28,38),(14,24,29,39),(15,25,30,40),(41,67,46,62),(42,68,47,63),(43,69,48,64),(44,70,49,65),(45,61,50,66),(51,71,56,76),(52,72,57,77),(53,73,58,78),(54,74,59,79),(55,75,60,80)], [(1,46,28,68,32,41,13,63),(2,48,27,66,33,43,12,61),(3,50,26,64,34,45,11,69),(4,42,30,62,35,47,15,67),(5,44,29,70,31,49,14,65),(6,53,39,73,16,58,24,78),(7,55,38,71,17,60,23,76),(8,57,37,79,18,52,22,74),(9,59,36,77,19,54,21,72),(10,51,40,75,20,56,25,80)]])

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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