D10.3M4(2) - GroupNames

G = D₁₀.₃M₄(2) order 320 = 2⁶·5

1^st non-split extension by D₁₀ of M₄(2) acting via M₄(2)/C₈=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₀.₃M₄(2), (C₂×F₅)⋊C₈, (C₂×C₈)⋊₃F₅, (C₂×C₄₀)⋊₁C₄, D₅.(C₄⋊C₈), C₂.₅(C₈×F₅), C₁₀.₄(C₄×C₈), D₁₀.₆(C₂×C₈), C₄.₂₈(C₄⋊F₅), D₅.(C₂²⋊C₈), C₂₀.₂₈(C₄⋊C₄), (C₄×D₅).₂₉Q₈, (C₄×D₅).₁₁₆D₄, C₂.₃(C₈⋊F₅), C₁₀.₄(C₈⋊C₄), D₁₀.₂₁(C₄⋊C₄), (C₂²×F₅).₂C₄, C₂².₁₆(C₄×F₅), (C₂×C₁₀).₁₁C₄², C₄.₃₇(C₂²⋊F₅), C₂₀.₃₅(C₂²⋊C₄), Dic₅.₂₃(C₄⋊C₄), D₁₀.₃₀(C₂²⋊C₄), C₂.₁(D₁₀.₃Q₈), C₅⋊₁(C₂².₇C₄²), C₁₀.₇(C₂.C₄²), Dic₅.₃₁(C₂²⋊C₄), (C₂×C₅⋊C₈)⋊₃C₄, (C₂×C₄×F₅).₇C₂, (D₅×C₂×C₈).₁₁C₂, (C₂×C₅⋊₂C₈)⋊₂₁C₄, (C₂×D₅⋊C₈).₇C₂, (C₂×C₄).₁₅₇(C₂×F₅), (C₂×C₂₀).₁₆₃(C₂×C₄), (C₂×C₄×D₅).₄₀₇C₂², (C₂²×D₅).₈₅(C₂×C₄), (C₂×Dic₅).₁₂₂(C₂×C₄), SmallGroup(320,230)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — D₁₀.₃M₄(2)

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₄×D₅ — C₂×C₄×D₅ — C₂×C₄×F₅ — D₁₀.₃M₄(2)

Lower central

C₅ — C₁₀ — D₁₀.₃M₄(2)

Upper central

C₁ — C₂×C₄ — C₂×C₈

Generators and relations for D₁₀.₃M₄(2)
G = < a,b,c,d | a¹⁰=b²=c⁸=1, d²=a^-1b, bab=a^-1, cac^-1=dad^-1=a³, cbc^-1=dbd^-1=a²b, dcd^-1=a⁵c⁵ >

Subgroups: 418 in 118 conjugacy classes, 50 normal (34 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂³, D₅, C₁₀, C₄², C₂×C₈, C₂×C₈, C₂²×C₄, Dic₅, C₂₀, F₅, D₁₀, D₁₀, C₂×C₁₀, C₂×C₄², C₂²×C₈, C₅⋊₂C₈, C₄₀, C₅⋊C₈, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂×F₅, C₂×F₅, C₂²×D₅, C₂².₇C₄², C₈×D₅, C₂×C₅⋊₂C₈, C₂×C₄₀, D₅⋊C₈, C₄×F₅, C₂×C₅⋊C₈, C₂×C₄×D₅, C₂²×F₅, D₅×C₂×C₈, C₂×D₅⋊C₈, C₂×C₄×F₅, D₁₀.₃M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, D₄, Q₈, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), F₅, C₂.C₄², C₄×C₈, C₈⋊C₄, C₂²⋊C₈, C₄⋊C₈, C₂×F₅, C₂².₇C₄², C₄×F₅, C₄⋊F₅, C₂²⋊F₅, C₈×F₅, C₈⋊F₅, D₁₀.₃Q₈, D₁₀.₃M₄(2)

Smallest permutation representation of D₁₀.₃M₄(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 24)(2 23)(3 22)(4 21)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 80)(19 79)(20 78)(31 41)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(51 61)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)

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

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

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

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

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

56 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	···	4P	5	8A	8B	8C	8D	8E	···	8P	10A	10B	10C	20A	20B	20C	20D	40A	···	40H
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	···	4	5	8	8	8	8	8	···	8	10	10	10	20	20	20	20	40	···	40
size	1	1	1	1	5	5	5	5	1	1	1	1	5	5	5	5	10	···	10	4	2	2	2	2	10	···	10	4	4	4	4	4	4	4	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	4	4	4	4	4	4	4
type	+	+	+	+						+	-		+	+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	C₄	C₈	D₄	Q₈	M₄(2)	F₅	C₂×F₅	C₄⋊F₅	C₂²⋊F₅	C₄×F₅	C₈×F₅	C₈⋊F₅
kernel	D₁₀.₃M₄(2)	D₅×C₂×C₈	C₂×D₅⋊C₈	C₂×C₄×F₅	C₂×C₅⋊₂C₈	C₂×C₄₀	C₂×C₅⋊C₈	C₂²×F₅	C₂×F₅	C₄×D₅	C₄×D₅	D₁₀	C₂×C₈	C₂×C₄	C₄	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	2	2	4	4	16	3	1	4	1	1	2	2	2	4	4

Matrix representation of D₁₀.₃M₄(2) ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	0	40
0	0	1	1	1	1
0	0	40	0	0	0
0	0	0	40	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	40	40	40	40
0	0	0	0	0	1

32	35	0	0	0	0
0	9	0	0	0	0
0	0	0	0	14	0
0	0	14	0	0	0
0	0	0	0	0	14
0	0	0	14	0	0

9	0	0	0	0	0
14	32	0	0	0	0
0	0	0	0	9	0
0	0	9	0	0	0
0	0	0	0	0	9
0	0	0	9	0	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,40,0,0,0,0,1,0,40,0,0,0,1,0,0,0,0,40,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,40,0,0,0,1,0,40,0,0,0,0,0,40,0,0,0,0,0,40,1],[32,0,0,0,0,0,35,9,0,0,0,0,0,0,0,14,0,0,0,0,0,0,0,14,0,0,14,0,0,0,0,0,0,0,14,0],[9,14,0,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,9,0,0,0,0,0,0,0,9,0] >;

D₁₀.₃M₄(2) in GAP, Magma, Sage, TeX

D_{10}._3M_4(2)

% in TeX

G:=Group("D10.3M4(2)");

// GroupNames label

G:=SmallGroup(320,230);

// by ID

G=gap.SmallGroup(320,230);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,136,6278,3156]);

// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^5*c^5>;

// generators/relations

G = D10.3M4(2) order 320 = 26·5

1st non-split extension by D10 of M4(2) acting via M4(2)/C8=C2

G = D₁₀.₃M₄(2) order 320 = 2⁶·5

1^st non-split extension by D₁₀ of M₄(2) acting via M₄(2)/C₈=C₂