D40.C4 - GroupNames

G = D₄₀.C₄ order 320 = 2⁶·5

1^st non-split extension by D₄₀ of C₄ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₈.₂F₅, D₄₀.₁C₄, Dic₅.₂D₈, D₁₀.₁₃SD₁₆, C₈.₂(C₂×F₅), C₄₀.₂(C₂×C₄), C₅⋊(M₅(2)⋊C₂), (C₅×D₈).₁C₄, (D₅×D₈).₂C₂, (C₄×D₅).₂₃D₄, C₅⋊₂C₈.₁₄D₄, C₄₀.C₄⋊₁C₂, C₈.F₅⋊₂C₂, C₄.₄(C₂²⋊F₅), C₂₀.₄(C₂²⋊C₄), C₂.₉(D₂₀⋊C₄), (C₈×D₅).₁₀C₂², C₁₀.₈(D₄⋊C₄), SmallGroup(320,244)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄₀ — D₄₀.C₄

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₄×D₅ — C₈×D₅ — C₄₀.C₄ — D₄₀.C₄

Lower central

C₅ — C₁₀ — C₂₀ — C₄₀ — D₄₀.C₄

Upper central

C₁ — C₂ — C₄ — C₈ — D₈

Generators and relations for D₄₀.C₄
G = < a,b,c | a⁴⁰=b²=1, c⁴=a²⁰, bab=a^-1, cac^-1=a³, cbc^-1=a³⁷b >

Subgroups: 410 in 62 conjugacy classes, 20 normal (all characteristic)
C₁, C₂, C₂^[×3], C₄, C₄, C₂²^[×5], C₅, C₈, C₈^[×2], C₂×C₄, D₄^[×3], C₂³, D₅^[×2], C₁₀, C₁₀, C₁₆, C₂×C₈, M₄(2), D₈, D₈^[×2], C₂×D₄, Dic₅, C₂₀, D₁₀, D₁₀^[×3], C₂×C₁₀, C₈.C₄, M₅(2), C₂×D₈, C₅⋊₂C₈, C₄₀, C₅⋊C₈, C₄×D₅, D₂₀, C₅⋊D₄, C₅×D₄, C₂²×D₅, M₅(2)⋊C₂, C₅⋊C₁₆, C₈×D₅, D₄₀, D₄⋊D₅, C₅×D₈, C₄.F₅, D₄×D₅, C₈.F₅, C₄₀.C₄, D₅×D₈, D₄₀.C₄
Quotients: C₁, C₂^[×3], C₄^[×2], C₂², C₂×C₄, D₄^[×2], C₂²⋊C₄, D₈, SD₁₆, F₅, D₄⋊C₄, C₂×F₅, M₅(2)⋊C₂, C₂²⋊F₅, D₂₀⋊C₄, D₄₀.C₄

Character table of D₄₀.C₄

class	1	2A	2B	2C	2D	4A	4B	5	8A	8B	8C	8D	8E	10A	10B	10C	16A	16B	16C	16D	20	40A	40B
size	1	1	8	10	40	2	10	4	4	10	10	40	40	4	16	16	20	20	20	20	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	-1	-i	i	1	1	1	-i	i	i	-i	1	1	1	linear of order 4
ρ₆	1	1	-1	-1	1	1	-1	1	1	-1	-1	-i	i	1	-1	-1	i	-i	-i	i	1	1	1	linear of order 4
ρ₇	1	1	1	-1	-1	1	-1	1	1	-1	-1	i	-i	1	1	1	i	-i	-i	i	1	1	1	linear of order 4
ρ₈	1	1	-1	-1	1	1	-1	1	1	-1	-1	i	-i	1	-1	-1	-i	i	i	-i	1	1	1	linear of order 4
ρ₉	2	2	0	2	0	2	2	2	-2	-2	-2	0	0	2	0	0	0	0	0	0	2	-2	-2	orthogonal lifted from D₄
ρ₁₀	2	2	0	-2	0	2	-2	2	-2	2	2	0	0	2	0	0	0	0	0	0	2	-2	-2	orthogonal lifted from D₄
ρ₁₁	2	2	0	-2	0	-2	2	2	0	0	0	0	0	2	0	0	√2	-√2	√2	-√2	-2	0	0	orthogonal lifted from D₈
ρ₁₂	2	2	0	-2	0	-2	2	2	0	0	0	0	0	2	0	0	-√2	√2	-√2	√2	-2	0	0	orthogonal lifted from D₈
ρ₁₃	2	2	0	2	0	-2	-2	2	0	0	0	0	0	2	0	0	√-2	√-2	-√-2	-√-2	-2	0	0	complex lifted from SD₁₆
ρ₁₄	2	2	0	2	0	-2	-2	2	0	0	0	0	0	2	0	0	-√-2	-√-2	√-2	√-2	-2	0	0	complex lifted from SD₁₆
ρ₁₅	4	4	4	0	0	4	0	-1	4	0	0	0	0	-1	-1	-1	0	0	0	0	-1	-1	-1	orthogonal lifted from F₅
ρ₁₆	4	4	-4	0	0	4	0	-1	4	0	0	0	0	-1	1	1	0	0	0	0	-1	-1	-1	orthogonal lifted from C₂×F₅
ρ₁₇	4	4	0	0	0	4	0	-1	-4	0	0	0	0	-1	-√5	√5	0	0	0	0	-1	1	1	orthogonal lifted from C₂²⋊F₅
ρ₁₈	4	4	0	0	0	4	0	-1	-4	0	0	0	0	-1	√5	-√5	0	0	0	0	-1	1	1	orthogonal lifted from C₂²⋊F₅
ρ₁₉	4	-4	0	0	0	0	0	4	0	2√2	-2√2	0	0	-4	0	0	0	0	0	0	0	0	0	orthogonal lifted from M₅(2)⋊C₂
ρ₂₀	4	-4	0	0	0	0	0	4	0	-2√2	2√2	0	0	-4	0	0	0	0	0	0	0	0	0	orthogonal lifted from M₅(2)⋊C₂
ρ₂₁	8	8	0	0	0	-8	0	-2	0	0	0	0	0	-2	0	0	0	0	0	0	2	0	0	orthogonal lifted from D₂₀⋊C₄, Schur index 2
ρ₂₂	8	-8	0	0	0	0	0	-2	0	0	0	0	0	2	0	0	0	0	0	0	0	-√10	√10	orthogonal faithful, Schur index 2
ρ₂₃	8	-8	0	0	0	0	0	-2	0	0	0	0	0	2	0	0	0	0	0	0	0	√10	-√10	orthogonal faithful, Schur index 2

Smallest permutation representation of D₄₀.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 10)(2 9)(3 8)(4 7)(5 6)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(41 43)(44 80)(45 79)(46 78)(47 77)(48 76)(49 75)(50 74)(51 73)(52 72)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(61 63)

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

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

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

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

Matrix representation of D₄₀.C₄ ►in GL₈(𝔽₂₄₁)

240	52	0	0	0	0	0	0
189	52	0	0	0	0	0	0
0	141	240	190	0	0	0	0
100	141	51	190	0	0	0	0
0	0	0	0	0	71	0	0
0	0	0	0	112	22	0	0
0	0	0	0	0	0	219	71
0	0	0	0	0	0	112	0

1	0	0	0	0	0	0	0
52	240	0	0	0	0	0	0
0	0	1	0	0	0	0	0
141	0	190	240	0	0	0	0
0	0	0	0	0	170	0	0
0	0	0	0	112	0	0	0
0	0	0	0	0	0	1	125
0	0	0	0	0	0	0	240

222	236	166	31	0	0	0	0
222	0	166	0	0	0	0	0
143	7	19	5	0	0	0	0
143	0	19	0	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	240	116	0	0
0	0	0	0	54	1	0	0

G:=sub<GL(8,GF(241))| [240,189,0,100,0,0,0,0,52,52,141,141,0,0,0,0,0,0,240,51,0,0,0,0,0,0,190,190,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,71,22,0,0,0,0,0,0,0,0,219,112,0,0,0,0,0,0,71,0],[1,52,0,141,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,190,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,170,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,125,240],[222,222,143,143,0,0,0,0,236,0,7,0,0,0,0,0,166,166,19,19,0,0,0,0,31,0,5,0,0,0,0,0,0,0,0,0,0,0,240,54,0,0,0,0,0,0,116,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

D₄₀.C₄ in GAP, Magma, Sage, TeX

D_{40}.C_4

% in TeX

G:=Group("D40.C4");

// GroupNames label

G:=SmallGroup(320,244);

// by ID

G=gap.SmallGroup(320,244);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,387,184,675,794,80,1684,851,102,6278,3156]);

// Polycyclic

G:=Group<a,b,c|a^40=b^2=1,c^4=a^20,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^37*b>;

// generators/relations

Export

Character table of D₄₀.C₄ in TeX

240	52	0	0	0	0	0	0
189	52	0	0	0	0	0	0
0	141	240	190	0	0	0	0
100	141	51	190	0	0	0	0
0	0	0	0	0	71	0	0
0	0	0	0	112	22	0	0
0	0	0	0	0	0	219	71
0	0	0	0	0	0	112	0

1	0	0	0	0	0	0	0
52	240	0	0	0	0	0	0
0	0	1	0	0	0	0	0
141	0	190	240	0	0	0	0
0	0	0	0	0	170	0	0
0	0	0	0	112	0	0	0
0	0	0	0	0	0	1	125
0	0	0	0	0	0	0	240

222	236	166	31	0	0	0	0
222	0	166	0	0	0	0	0
143	7	19	5	0	0	0	0
143	0	19	0	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	240	116	0	0
0	0	0	0	54	1	0	0

240	52	0	0	0	0	0	0
189	52	0	0	0	0	0	0
0	141	240	190	0	0	0	0
100	141	51	190	0	0	0	0
0	0	0	0	0	71	0	0
0	0	0	0	112	22	0	0
0	0	0	0	0	0	219	71
0	0	0	0	0	0	112	0

1	0	0	0	0	0	0	0
52	240	0	0	0	0	0	0
0	0	1	0	0	0	0	0
141	0	190	240	0	0	0	0
0	0	0	0	0	170	0	0
0	0	0	0	112	0	0	0
0	0	0	0	0	0	1	125
0	0	0	0	0	0	0	240

222	236	166	31	0	0	0	0
222	0	166	0	0	0	0	0
143	7	19	5	0	0	0	0
143	0	19	0	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	240	116	0	0
0	0	0	0	54	1	0	0

G = D40.C4 order 320 = 26·5

1st non-split extension by D40 of C4 acting faithfully

G = D₄₀.C₄ order 320 = 2⁶·5

1^st non-split extension by D₄₀ of C₄ acting faithfully

240	52	0	0	0	0	0	0
189	52	0	0	0	0	0	0
0	141	240	190	0	0	0	0
100	141	51	190	0	0	0	0
0	0	0	0	0	71	0	0
0	0	0	0	112	22	0	0
0	0	0	0	0	0	219	71
0	0	0	0	0	0	112	0

1	0	0	0	0	0	0	0
52	240	0	0	0	0	0	0
0	0	1	0	0	0	0	0
141	0	190	240	0	0	0	0
0	0	0	0	0	170	0	0
0	0	0	0	112	0	0	0
0	0	0	0	0	0	1	125
0	0	0	0	0	0	0	240

222	236	166	31	0	0	0	0
222	0	166	0	0	0	0	0
143	7	19	5	0	0	0	0
143	0	19	0	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	240	116	0	0
0	0	0	0	54	1	0	0