M4(2):5F5 - GroupNames

G = M₄(2)⋊₅F₅ order 320 = 2⁶·5

The semidirect product of M₄(2) and F₅ acting through Inn(M₄(2))

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M₄(2)⋊₅F₅, C₂₀.₇C₄², D₁₀.₃C₄², Dic₅.₃C₄², C₄⋊F₅.C₄, D₅⋊C₈⋊₄C₄, (C₈×F₅)⋊₈C₂, C₈⋊D₅⋊₃C₄, C₈⋊F₅⋊₇C₂, C₈.₁₉(C₂×F₅), C₄.₁₂(C₄×F₅), C₄₀.₁₉(C₂×C₄), C₄.Dic₅⋊₆C₄, C₂²⋊F₅.₃C₄, D₅.₂(C₈○D₄), (C₅×M₄(2))⋊₆C₄, (C₂×C₁₀).₇C₄², C₂².₁₂(C₄×F₅), C₄.₅₃(C₂²×F₅), C₅⋊₄(C₈○₂M₄(2)), C₂₀.₉₃(C₂²×C₄), C₁₀.₁₅(C₂×C₄²), D₅⋊C₈.₂₀C₂², (C₄×D₅).₈₉C₂³, (C₈×D₅).₃₇C₂², (C₄×F₅).₁₉C₂², D₁₀.₃₅(C₂²×C₄), (D₅×M₄(2)).₁₁C₂, Dic₅.₃₄(C₂²×C₄), D₁₀.C₂³.₄C₂, (C₂×C₅⋊C₈)⋊₇C₄, C₅⋊C₈.₂(C₂×C₄), C₂.₁₆(C₂×C₄×F₅), (C₂×D₅⋊C₈).₅C₂, (C₂×F₅).₄(C₂×C₄), (C₂×C₄).₇₇(C₂×F₅), (C₂×C₂₀).₄₇(C₂×C₄), C₅⋊₂C₈.₂₁(C₂×C₄), (C₄×D₅).₁₈(C₂×C₄), (C₂×C₄×D₅).₁₉₄C₂², (C₂×Dic₅).₆₇(C₂×C₄), (C₂²×D₅).₅₃(C₂×C₄), SmallGroup(320,1066)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — M₄(2)⋊₅F₅

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₄×D₅ — D₅⋊C₈ — C₂×D₅⋊C₈ — M₄(2)⋊₅F₅

Lower central

C₅ — C₁₀ — M₄(2)⋊₅F₅

Upper central

C₁ — C₄ — M₄(2)

Generators and relations for M₄(2)⋊₅F₅
G = < a,b,c,d | a⁸=b²=c⁵=d⁴=1, bab=dad^-1=a⁵, ac=ca, bc=cb, dbd^-1=a⁴b, dcd^-1=c³ >

Subgroups: 394 in 130 conjugacy classes, 66 normal (32 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂×C₄, C₂×C₄, C₂³, D₅, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, Dic₅, C₂₀, F₅, D₁₀, D₁₀, C₂×C₁₀, C₄×C₈, C₈⋊C₄, C₄²⋊C₂, C₂²×C₈, C₂×M₄(2), C₅⋊₂C₈, C₄₀, C₅⋊C₈, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂×F₅, C₂²×D₅, C₈○₂M₄(2), C₈×D₅, C₈⋊D₅, C₄.Dic₅, C₅×M₄(2), D₅⋊C₈, D₅⋊C₈, C₄×F₅, C₄⋊F₅, C₂×C₅⋊C₈, C₂²⋊F₅, C₂×C₄×D₅, C₈×F₅, C₈⋊F₅, D₅×M₄(2), C₂×D₅⋊C₈, D₁₀.C₂³, M₄(2)⋊₅F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₄², C₂²×C₄, F₅, C₂×C₄², C₈○D₄, C₂×F₅, C₈○₂M₄(2), C₄×F₅, C₂²×F₅, C₂×C₄×F₅, M₄(2)⋊₅F₅

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

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

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

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

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

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

50 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	···	4N	5	8A	8B	8C	8D	8E	···	8L	8M	···	8T	10A	10B	20A	20B	20C	40A	40B	40C	40D
order	1	2	2	2	2	2	4	4	4	4	4	4	···	4	5	8	8	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	···	10	4	2	2	2	2	5	···	5	10	···	10	4	8	4	4	8	8	8	8	8

50 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	2	4	4	4	4	4	8
type	+	+	+	+	+	+									+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	C₄	C₄	C₄	C₈○D₄	F₅	C₂×F₅	C₂×F₅	C₄×F₅	C₄×F₅	M₄(2)⋊₅F₅
kernel	M₄(2)⋊₅F₅	C₈×F₅	C₈⋊F₅	D₅×M₄(2)	C₂×D₅⋊C₈	D₁₀.C₂³	C₈⋊D₅	C₄.Dic₅	C₅×M₄(2)	D₅⋊C₈	C₄⋊F₅	C₂×C₅⋊C₈	C₂²⋊F₅	D₅	M₄(2)	C₈	C₂×C₄	C₄	C₂²	C₁
# reps	1	2	2	1	1	1	4	2	2	4	4	4	4	8	1	2	1	2	2	2

Matrix representation of M₄(2)⋊₅F₅ ►in GL₆(𝔽₄₁)

14	0	0	0	0	0
0	27	0	0	0	0
0	0	32	0	0	0
0	0	0	32	0	0
0	0	0	0	32	0
0	0	0	0	0	32

0	38	0	0	0	0
27	0	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	40	0
0	0	0	0	0	40

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

0	27	0	0	0	0
38	0	0	0	0	0
0	0	40	0	0	0
0	0	0	0	0	40
0	0	0	40	0	0
0	0	1	1	1	1

G:=sub<GL(6,GF(41))| [14,0,0,0,0,0,0,27,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[0,27,0,0,0,0,38,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[0,38,0,0,0,0,27,0,0,0,0,0,0,0,40,0,0,1,0,0,0,0,40,1,0,0,0,0,0,1,0,0,0,40,0,1] >;

M₄(2)⋊₅F₅ in GAP, Magma, Sage, TeX

M_4(2)\rtimes_5F_5

% in TeX

G:=Group("M4(2):5F5");

// GroupNames label

G:=SmallGroup(320,1066);

// by ID

G=gap.SmallGroup(320,1066);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,136,102,6278,1595]);

// Polycyclic

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

// generators/relations

G = M4(2)⋊5F5 order 320 = 26·5

The semidirect product of M4(2) and F5 acting through Inn(M4(2))

G = M₄(2)⋊₅F₅ order 320 = 2⁶·5

The semidirect product of M₄(2) and F₅ acting through Inn(M₄(2))