D10:10M4(2) - GroupNames

G = D₁₀:₁₀M₄(2) order 320 = 2⁶·5

4^th semidirect product of D₁₀ and M₄(2) acting via M₄(2)/C₂xC₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₀:₁₀M₄(2), D₁₀:C₈:₆C₂, (C₄xD₅).₁₁₂D₄, (C₂xC₁₀):₃M₄(2), C₂²:₂(C₄.F₅), C₂³.₄₃(C₂xF₅), (C₂²xC₄).₂₁F₅, (C₂²xC₂₀).₂₃C₄, C₅:₂(C₂⁴.₄C₄), (C₂³xD₅).₁₆C₄, C₄.₃₁(C₂²:F₅), C₂³.₂F₅:₇C₂, C₂₀.₃₀(C₂²:C₄), Dic₅.₁₀₆(C₂xD₄), C₁₀.₂₂(C₂xM₄(2)), D₁₀.₃₉(C₂²:C₄), C₂².₈₅(C₂²xF₅), C₂.₁₆(D₅:M₄(2)), (C₂xDic₅).₃₄₇C₂³, (C₂²xDic₅).₂₇₅C₂², (C₂xC₅:C₈):₂C₂², (C₂xC₄xD₅).₃₇C₄, (C₂xC₄.F₅):₁₃C₂, C₂.₁₁(C₂xC₄.F₅), C₂.₉(C₂xC₂²:F₅), C₁₀.₆(C₂xC₂²:C₄), (C₂xC₄).₁₄₃(C₂xF₅), (D₅xC₂²xC₄).₂₈C₂, (C₂xC₂₀).₁₀₉(C₂xC₄), (C₂xC₄xD₅).₃₉₈C₂², (C₂²xC₁₀).₆₃(C₂xC₄), (C₂xC₁₀).₆₃(C₂²xC₄), (C₂xDic₅).₁₈₅(C₂xC₄), (C₂²xD₅).₁₂₆(C₂xC₄), SmallGroup(320,1094)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₀ — D₁₀:₁₀M₄(2)

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₂xDic₅ — C₂xC₅:C₈ — C₂³.₂F₅ — D₁₀:₁₀M₄(2)

Lower central

C₅ — C₂xC₁₀ — D₁₀:₁₀M₄(2)

Upper central

C₁ — C₂² — C₂²xC₄

Generators and relations for D₁₀:₁₀M₄(2)
G = < a,b,c,d | a¹⁰=b²=c⁸=d²=1, bab=dad=a^-1, cac^-1=a⁷, cbc^-1=ab, dbd=a⁸b, dcd=c⁵ >

Subgroups: 762 in 190 conjugacy classes, 58 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₅, C₈, C₂xC₄, C₂xC₄, C₂³, C₂³, D₅, C₁₀, C₁₀, C₂xC₈, M₄(2), C₂²xC₄, C₂²xC₄, C₂⁴, Dic₅, Dic₅, C₂₀, C₂₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₂xC₁₀, C₂²:C₈, C₂xM₄(2), C₂³xC₄, C₅:C₈, C₄xD₅, C₄xD₅, C₂xDic₅, C₂xDic₅, C₂xC₂₀, C₂xC₂₀, C₂²xD₅, C₂²xD₅, C₂²xC₁₀, C₂⁴.₄C₄, C₄.F₅, C₂xC₅:C₈, C₂xC₄xD₅, C₂xC₄xD₅, C₂²xDic₅, C₂²xC₂₀, C₂³xD₅, D₁₀:C₈, C₂³.₂F₅, C₂xC₄.F₅, D₅xC₂²xC₄, D₁₀:₁₀M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, C₂²:C₄, M₄(2), C₂²xC₄, C₂xD₄, F₅, C₂xC₂²:C₄, C₂xM₄(2), C₂xF₅, C₂⁴.₄C₄, C₄.F₅, C₂²:F₅, C₂²xF₅, C₂xC₄.F₅, D₅:M₄(2), C₂xC₂²:F₅, D₁₀:₁₀M₄(2)

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

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

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	4	4	4	4	4	4
type	+	+	+	+	+				+			+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	M₄(2)	M₄(2)	F₅	C₂xF₅	C₂xF₅	C₂²:F₅	C₄.F₅	D₅:M₄(2)
kernel	D₁₀:₁₀M₄(2)	D₁₀:C₈	C₂³.₂F₅	C₂xC₄.F₅	D₅xC₂²xC₄	C₂xC₄xD₅	C₂²xC₂₀	C₂³xD₅	C₄xD₅	D₁₀	C₂xC₁₀	C₂²xC₄	C₂xC₄	C₂³	C₄	C₂²	C₂
# reps	1	2	2	2	1	4	2	2	4	4	4	1	2	1	4	4	4

Matrix representation of D₁₀:₁₀M₄(2) ►in GL₆(F₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	35	35	28	28
0	0	6	40	0	32
0	0	0	0	40	8
0	0	0	0	40	7

40	0	0	0	0	0
37	1	0	0	0	0
0	0	40	6	0	13
0	0	0	1	9	0
0	0	0	0	40	0
0	0	0	0	40	1

32	25	0	0	0	0
5	9	0	0	0	0
0	0	0	40	24	26
0	0	0	40	30	19
0	0	13	23	1	35
0	0	9	32	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	6	40	26	21
0	0	35	35	13	17
0	0	0	0	7	33
0	0	0	0	6	34

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,28,0,40,40,0,0,28,32,8,7],[40,37,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,6,1,0,0,0,0,0,9,40,40,0,0,13,0,0,1],[32,5,0,0,0,0,25,9,0,0,0,0,0,0,0,0,13,9,0,0,40,40,23,32,0,0,24,30,1,0,0,0,26,19,35,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,35,0,0,0,0,40,35,0,0,0,0,26,13,7,6,0,0,21,17,33,34] >;

D₁₀:₁₀M₄(2) in GAP, Magma, Sage, TeX

D_{10}\rtimes_{10}M_4(2)

% in TeX

G:=Group("D10:10M4(2)");

// GroupNames label

G:=SmallGroup(320,1094);

// by ID

G=gap.SmallGroup(320,1094);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,758,184,136,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=a*b,d*b*d=a^8*b,d*c*d=c^5>;

// generators/relations

G = D10:10M4(2) order 320 = 26·5

4th semidirect product of D10 and M4(2) acting via M4(2)/C2xC4=C2

G = D₁₀:₁₀M₄(2) order 320 = 2⁶·5

4^th semidirect product of D₁₀ and M₄(2) acting via M₄(2)/C₂xC₄=C₂