M4(2).F5 - GroupNames

G = M₄(2).F₅ order 320 = 2⁶·5

3^rd non-split extension by M₄(2) of F₅ acting via F₅/D₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₀.₄C₄², M₄(2).₃F₅, C₄.₉(C₄×F₅), D₅⋊C₈.₄C₄, C₄.F₅.₁C₄, D₁₀.₇(C₄⋊C₄), (C₄×D₅).₁₁₈D₄, D₅.(C₈.C₄), C₄.Dic₅.₁C₄, (C₂²×D₅).₈Q₈, C₅⋊₂(C₄.C₄²), D₅⋊M₄(2).₂C₂, (D₅×M₄(2)).₅C₂, (C₅×M₄(2)).₃C₄, C₂².₁₁(C₄⋊F₅), C₄.₄₁(C₂²⋊F₅), C₂₀.₃₉(C₂²⋊C₄), (C₂×Dic₅).₁₀₀D₄, Dic₅.₃₃(C₂²⋊C₄), C₂.₁₆(D₁₀.₃Q₈), C₁₀.₁₅(C₂.C₄²), (C₂×D₅⋊C₈).₂C₂, (C₂×C₄).₇₀(C₂×F₅), (C₂×C₁₀).₄(C₄⋊C₄), (C₂×C₂₀).₃₇(C₂×C₄), (C₄×D₅).₁₅(C₂×C₄), (C₂×C₄×D₅).₁₈₈C₂², SmallGroup(320,239)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — M₄(2).F₅

Chief series

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

Lower central

C₅ — C₁₀ — C₂₀ — M₄(2).F₅

Upper central

C₁ — C₄ — C₂×C₄ — M₄(2)

Generators and relations for M₄(2).F₅
G = < a,b,c,d | a⁸=b²=c⁵=1, d⁴=a⁴, bab=a⁵, ac=ca, dad^-1=a^-1b, bc=cb, dbd^-1=a⁴b, dcd^-1=c³ >

Subgroups: 322 in 90 conjugacy classes, 36 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂³, D₅, D₅, C₁₀, C₁₀, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, Dic₅, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₂²×C₈, C₂×M₄(2), C₅⋊₂C₈, C₄₀, C₅⋊C₈, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂²×D₅, C₄.C₄², C₈×D₅, C₈⋊D₅, C₄.Dic₅, C₅×M₄(2), D₅⋊C₈, D₅⋊C₈, C₄.F₅, C₂×C₅⋊C₈, C₂².F₅, C₂×C₄×D₅, D₅×M₄(2), C₂×D₅⋊C₈, D₅⋊M₄(2), M₄(2).F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₄², C₂²⋊C₄, C₄⋊C₄, F₅, C₂.C₄², C₈.C₄, C₂×F₅, C₄.C₄², C₄×F₅, C₄⋊F₅, C₂²⋊F₅, D₁₀.₃Q₈, M₄(2).F₅

Smallest permutation representation of M₄(2).F₅
►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)

(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)(66 70)(68 72)(74 78)(76 80)

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

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

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

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

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

38 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	5	8A	8B	8C	···	8J	8K	···	8P	10A	10B	20A	20B	20C	40A	40B	40C	40D
order	1	2	2	2	2	2	4	4	4	4	4	4	5	8	8	8	···	8	8	···	8	10	10	20	20	20	40	40	40	40
size	1	1	2	5	5	10	1	1	2	5	5	10	4	4	4	10	···	10	20	···	20	4	8	4	4	8	8	8	8	8

38 irreducible representations

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

Matrix representation of M₄(2).F₅ ►in GL₆(𝔽₄₁)

28	4	0	0	0	0
17	13	0	0	0	0
0	0	34	14	0	27
0	0	0	7	14	27
0	0	27	14	7	0
0	0	27	0	14	34

40	0	0	0	0	0
14	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

39	29	0	0	0	0
3	2	0	0	0	0
0	0	37	0	4	10
0	0	0	10	35	6
0	0	31	6	35	10
0	0	31	10	4	0

G:=sub<GL(6,GF(41))| [28,17,0,0,0,0,4,13,0,0,0,0,0,0,34,0,27,27,0,0,14,7,14,0,0,0,0,14,7,14,0,0,27,27,0,34],[40,14,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[39,3,0,0,0,0,29,2,0,0,0,0,0,0,37,0,31,31,0,0,0,10,6,10,0,0,4,35,35,4,0,0,10,6,10,0] >;

M₄(2).F₅ in GAP, Magma, Sage, TeX

M_4(2).F_5

% in TeX

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

// GroupNames label

G:=SmallGroup(320,239);

// by ID

G=gap.SmallGroup(320,239);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,136,1684,851,102,6278,3156]);

// Polycyclic

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

// generators/relations

G = M4(2).F5 order 320 = 26·5

3rd non-split extension by M4(2) of F5 acting via F5/D5=C2

G = M₄(2).F₅ order 320 = 2⁶·5

3^rd non-split extension by M₄(2) of F₅ acting via F₅/D₅=C₂