C2^4.F5 - GroupNames

G = C₂⁴.F₅ order 320 = 2⁶·5

1^st non-split extension by C₂⁴ of F₅ acting via F₅/C₅=C₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₁F₅, C₂³⋊(C₅⋊C₈), C₅⋊₂(C₂³⋊C₈), (C₂²×C₁₀)⋊₃C₈, (C₂³×C₁₀).₅C₄, C₂.₃(C₂³⋊F₅), C₂³.₃₆(C₂×F₅), C₂³.₂F₅⋊₂C₂, C₁₀.₁₅(C₂²⋊C₈), C₁₀.₂₀(C₂³⋊C₄), C₂.₃(C₂³.F₅), (C₂×Dic₅).₁₁₃D₄, C₁₀.₇(C₄.D₄), (C₂×C₁₀).₂₁M₄(2), (C₂²×Dic₅).₉C₄, C₂².₄₃(C₂²⋊F₅), C₂.₈(C₂³.₂F₅), C₂².₅(C₂².F₅), (C₂²×Dic₅).₁₇₅C₂², C₂².₄(C₂×C₅⋊C₈), (C₂×C₁₀).₃₀(C₂×C₈), (C₂²×C₁₀).₄₉(C₂×C₄), (C₂×C₂³.D₅).₂₄C₂, (C₂×C₁₀).₄₅(C₂²⋊C₄), SmallGroup(320,271)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₂⁴.F₅

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×Dic₅ — C₂²×Dic₅ — C₂³.₂F₅ — C₂⁴.F₅

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₂⁴.F₅

Upper central

C₁ — C₂² — C₂³ — C₂⁴

Generators and relations for C₂⁴.F₅
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁵=1, f⁴=c, ab=ba, ac=ca, ad=da, ae=ea, faf^-1=abd, bc=cb, fbf^-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e³ >

Subgroups: 386 in 98 conjugacy classes, 30 normal (22 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₂²⋊C₄, C₂×C₈, C₂²×C₄, C₂⁴, Dic₅, C₂×C₁₀, C₂×C₁₀, C₂²⋊C₈, C₂×C₂²⋊C₄, C₅⋊C₈, C₂×Dic₅, C₂×Dic₅, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂³⋊C₈, C₂³.D₅, C₂×C₅⋊C₈, C₂²×Dic₅, C₂³×C₁₀, C₂³.₂F₅, C₂×C₂³.D₅, C₂⁴.F₅
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, D₄, C₂²⋊C₄, C₂×C₈, M₄(2), F₅, C₂²⋊C₈, C₂³⋊C₄, C₄.D₄, C₅⋊C₈, C₂×F₅, C₂³⋊C₈, C₂×C₅⋊C₈, C₂².F₅, C₂²⋊F₅, C₂³⋊F₅, C₂³.₂F₅, C₂³.F₅, C₂⁴.F₅

Smallest permutation representation of C₂⁴.F₅
►On 80 points

Generators in S₈₀

(2 13)(3 10)(4 8)(6 9)(7 14)(11 15)(17 55)(18 52)(19 23)(21 51)(22 56)(25 29)(27 34)(28 39)(31 38)(32 35)(36 40)(41 63)(42 60)(43 47)(45 59)(46 64)(49 53)(57 61)(65 69)(67 78)(68 75)(71 74)(72 79)(76 80)

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

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

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

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

(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)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim	1	1	1	1	1	1	2	2	4	4	4	4	4	4	4	4	4
type	+	+	+				+		+	+	+	-	+	-	+
image	C₁	C₂	C₂	C₄	C₄	C₈	D₄	M₄(2)	F₅	C₂³⋊C₄	C₄.D₄	C₅⋊C₈	C₂×F₅	C₂².F₅	C₂²⋊F₅	C₂³⋊F₅	C₂³.F₅
kernel	C₂⁴.F₅	C₂³.₂F₅	C₂×C₂³.D₅	C₂²×Dic₅	C₂³×C₁₀	C₂²×C₁₀	C₂×Dic₅	C₂×C₁₀	C₂⁴	C₁₀	C₁₀	C₂³	C₂³	C₂²	C₂²	C₂	C₂
# reps	1	2	1	2	2	8	2	2	1	1	1	2	1	2	2	4	4

Matrix representation of C₂⁴.F₅ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	40	0	0	0	0
0	0	1	0	0	0
0	0	20	40	0	0
0	0	0	0	40	0
0	0	0	0	21	1

40	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	1	0
0	0	0	0	0	1

40	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	0
0	0	0	0	0	40

1	0	0	0	0	0
0	1	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	10	0	0	0
0	0	17	37	0	0
0	0	0	0	18	0
0	0	0	0	20	16

0	1	0	0	0	0
9	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	9	36	0	0
0	0	0	32	0	0

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,20,0,0,0,0,0,40,0,0,0,0,0,0,40,21,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,17,0,0,0,0,0,37,0,0,0,0,0,0,18,20,0,0,0,0,0,16],[0,9,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,36,32,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂⁴.F₅ in GAP, Magma, Sage, TeX

C_2^4.F_5

% in TeX

G:=Group("C2^4.F5");

// GroupNames label

G:=SmallGroup(320,271);

// by ID

G=gap.SmallGroup(320,271);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,387,100,1123,6278,3156]);

// Polycyclic

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

// generators/relations

G = C24.F5 order 320 = 26·5

1st non-split extension by C24 of F5 acting via F5/C5=C4

G = C₂⁴.F₅ order 320 = 2⁶·5

1^st non-split extension by C₂⁴ of F₅ acting via F₅/C₅=C₄