(Q8xC10).C4 - GroupNames

G = (Q₈×C₁₀).C₄ order 320 = 2⁶·5

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

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

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

Export

Character table of (Q₈×C₁₀).C₄ in TeX

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

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 = (Q8×C10).C4 order 320 = 26·5

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

G = (Q₈×C₁₀).C₄ order 320 = 2⁶·5

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

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