M4(2):4F5 - GroupNames

G = M₄(2)⋊₄F₅ order 320 = 2⁶·5

4^th semidirect product of M₄(2) and F₅ acting via F₅/D₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M₄(2)⋊₄F₅, C₂₀.₄(C₄⋊C₄), (C₄×D₅).₆Q₈, C₄.₁₉(C₄⋊F₅), C₄.Dic₅⋊₃C₄, C₂²⋊F₅.₂C₄, C₂².₉(C₄×F₅), (C₄×D₅).₁₁₁D₄, (C₅×M₄(2))⋊₄C₄, (C₂×C₁₀).₄C₄², D₁₀.₂₃(C₄⋊C₄), Dic₅.₇(C₄⋊C₄), (D₅×M₄(2)).₆C₂, C₄.₂₉(C₂²⋊F₅), C₂₀.₂₉(C₂²⋊C₄), C₅⋊₃(M₄(2)⋊₄C₄), D₁₀.₇(C₂²⋊C₄), Dic₅.₃₄(C₂²⋊C₄), C₂.₁₇(D₁₀.₃Q₈), C₁₀.₁₆(C₂.C₄²), D₁₀.C₂³.₂C₂, (C₂×C₅⋊C₈)⋊₂C₄, (C₂×C₄).₁₃(C₂×F₅), (C₂×C₄.F₅).₂C₂, (C₂×C₂₀).₃₈(C₂×C₄), (C₂×C₄×D₅).₁₈₉C₂², (C₂×Dic₅).₄₄(C₂×C₄), (C₂²×D₅).₃₇(C₂×C₄), SmallGroup(320,240)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — M₄(2)⋊₄F₅

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₄×D₅ — C₂×C₄×D₅ — C₂×C₄.F₅ — M₄(2)⋊₄F₅

Lower central

C₅ — C₁₀ — C₂×C₁₀ — M₄(2)⋊₄F₅

Upper central

C₁ — C₂ — C₂×C₄ — M₄(2)

Generators and relations for M₄(2)⋊₄F₅
G = < a,b,c,d | a⁸=b²=c⁵=d⁴=1, bab=a⁵, ac=ca, dad^-1=a⁵b, bc=cb, dbd^-1=a⁴b, dcd^-1=c³ >

Subgroups: 370 in 90 conjugacy classes, 34 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂³, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, Dic₅, C₂₀, F₅, D₁₀, D₁₀, C₂×C₁₀, C₄²⋊C₂, C₂×M₄(2), C₅⋊₂C₈, C₄₀, C₅⋊C₈, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂×F₅, C₂²×D₅, M₄(2)⋊₄C₄, C₈×D₅, C₈⋊D₅, C₄.Dic₅, C₅×M₄(2), C₄.F₅, C₄×F₅, C₄⋊F₅, C₂×C₅⋊C₈, C₂²⋊F₅, C₂×C₄×D₅, D₅×M₄(2), C₂×C₄.F₅, D₁₀.C₂³, M₄(2)⋊₄F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₄², C₂²⋊C₄, C₄⋊C₄, F₅, C₂.C₄², C₂×F₅, M₄(2)⋊₄C₄, C₄×F₅, C₄⋊F₅, C₂²⋊F₅, D₁₀.₃Q₈, M₄(2)⋊₄F₅

Smallest permutation representation of M₄(2)⋊₄F₅
►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 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)(17 58)(18 63)(19 60)(20 57)(21 62)(22 59)(23 64)(24 61)(25 44)(26 41)(27 46)(28 43)(29 48)(30 45)(31 42)(32 47)(33 80)(34 77)(35 74)(36 79)(37 76)(38 73)(39 78)(40 75)(49 72)(50 69)(51 66)(52 71)(53 68)(54 65)(55 70)(56 67)

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	4	4	4	4	4	4	8
type	+	+	+	+					+	-	+	+			+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	Q₈	F₅	C₂×F₅	M₄(2)⋊₄C₄	C₄⋊F₅	C₂²⋊F₅	C₄×F₅	M₄(2)⋊₄F₅
kernel	M₄(2)⋊₄F₅	D₅×M₄(2)	C₂×C₄.F₅	D₁₀.C₂³	C₄.Dic₅	C₅×M₄(2)	C₂×C₅⋊C₈	C₂²⋊F₅	C₄×D₅	C₄×D₅	M₄(2)	C₂×C₄	C₅	C₄	C₄	C₂²	C₁
# reps	1	1	1	1	2	2	4	4	3	1	1	1	2	2	2	2	2

Matrix representation of M₄(2)⋊₄F₅ ►in GL₈(𝔽₄₁)

40	0	4	4	0	0	0	0
0	0	1	0	0	0	0	0
18	9	1	1	0	0	0	0
0	32	0	0	0	0	0	0
0	0	0	0	34	0	27	27
0	0	0	0	14	7	14	0
0	0	0	0	0	14	7	14
0	0	0	0	27	27	0	34

1	0	32	32	0	0	0	0
39	40	9	9	0	0	0	0
0	0	0	40	0	0	0	0
0	0	40	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	40	0
0	0	0	0	0	0	0	40

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	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	1	0

32	32	40	0	0	0	0	0
18	9	1	1	0	0	0	0
0	0	9	0	0	0	0	0
0	0	0	32	0	0	0	0
0	0	0	0	40	0	0	0
0	0	0	0	0	0	0	40
0	0	0	0	0	40	0	0
0	0	0	0	1	1	1	1

G:=sub<GL(8,GF(41))| [40,0,18,0,0,0,0,0,0,0,9,32,0,0,0,0,4,1,1,0,0,0,0,0,4,0,1,0,0,0,0,0,0,0,0,0,34,14,0,27,0,0,0,0,0,7,14,27,0,0,0,0,27,14,7,0,0,0,0,0,27,0,14,34],[1,39,0,0,0,0,0,0,0,40,0,0,0,0,0,0,32,9,0,40,0,0,0,0,32,9,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[32,18,0,0,0,0,0,0,32,9,0,0,0,0,0,0,40,1,9,0,0,0,0,0,0,1,0,32,0,0,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,1,0,0,0,0,0,40,0,1] >;

M₄(2)⋊₄F₅ in GAP, Magma, Sage, TeX

M_4(2)\rtimes_4F_5

% in TeX

G:=Group("M4(2):4F5");

// GroupNames label

G:=SmallGroup(320,240);

// by ID

G=gap.SmallGroup(320,240);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,1123,136,851,6278,3156]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^5,a*c=c*a,d*a*d^-1=a^5*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^3>;

// generators/relations

40	0	4	4	0	0	0	0
0	0	1	0	0	0	0	0
18	9	1	1	0	0	0	0
0	32	0	0	0	0	0	0
0	0	0	0	34	0	27	27
0	0	0	0	14	7	14	0
0	0	0	0	0	14	7	14
0	0	0	0	27	27	0	34

1	0	32	32	0	0	0	0
39	40	9	9	0	0	0	0
0	0	0	40	0	0	0	0
0	0	40	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	40	0
0	0	0	0	0	0	0	40

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	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	1	0

32	32	40	0	0	0	0	0
18	9	1	1	0	0	0	0
0	0	9	0	0	0	0	0
0	0	0	32	0	0	0	0
0	0	0	0	40	0	0	0
0	0	0	0	0	0	0	40
0	0	0	0	0	40	0	0
0	0	0	0	1	1	1	1

40	0	4	4	0	0	0	0
0	0	1	0	0	0	0	0
18	9	1	1	0	0	0	0
0	32	0	0	0	0	0	0
0	0	0	0	34	0	27	27
0	0	0	0	14	7	14	0
0	0	0	0	0	14	7	14
0	0	0	0	27	27	0	34

1	0	32	32	0	0	0	0
39	40	9	9	0	0	0	0
0	0	0	40	0	0	0	0
0	0	40	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	40	0
0	0	0	0	0	0	0	40

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	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	1	0

32	32	40	0	0	0	0	0
18	9	1	1	0	0	0	0
0	0	9	0	0	0	0	0
0	0	0	32	0	0	0	0
0	0	0	0	40	0	0	0
0	0	0	0	0	0	0	40
0	0	0	0	0	40	0	0
0	0	0	0	1	1	1	1

G = M4(2)⋊4F5 order 320 = 26·5

4th semidirect product of M4(2) and F5 acting via F5/D5=C2

G = M₄(2)⋊₄F₅ order 320 = 2⁶·5

4^th semidirect product of M₄(2) and F₅ acting via F₅/D₅=C₂

40	0	4	4	0	0	0	0
0	0	1	0	0	0	0	0
18	9	1	1	0	0	0	0
0	32	0	0	0	0	0	0
0	0	0	0	34	0	27	27
0	0	0	0	14	7	14	0
0	0	0	0	0	14	7	14
0	0	0	0	27	27	0	34

1	0	32	32	0	0	0	0
39	40	9	9	0	0	0	0
0	0	0	40	0	0	0	0
0	0	40	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	40	0
0	0	0	0	0	0	0	40

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	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	1	0

32	32	40	0	0	0	0	0
18	9	1	1	0	0	0	0
0	0	9	0	0	0	0	0
0	0	0	32	0	0	0	0
0	0	0	0	40	0	0	0
0	0	0	0	0	0	0	40
0	0	0	0	0	40	0	0
0	0	0	0	1	1	1	1