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G = C24.4F5order 320 = 26·5

2nd non-split extension by C24 of F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.4F5, (C2×C10)⋊8M4(2), (C23×C10).7C4, C23.51(C2×F5), C53(C24.4C4), Dic5.117(C2×D4), (C2×Dic5).264D4, C10.34(C2×M4(2)), C23.2F513C2, C222(C22.F5), (C22×Dic5).37C4, (C23×Dic5).13C2, C22.30(C22⋊F5), C22.100(C22×F5), Dic5.52(C22⋊C4), (C2×Dic5).360C23, (C22×Dic5).280C22, (C2×C5⋊C8)⋊3C22, (C2×C22.F5)⋊8C2, C2.39(C2×C22⋊F5), C10.39(C2×C22⋊C4), C2.12(C2×C22.F5), (C22×C10).74(C2×C4), (C2×C10).92(C22×C4), (C2×C10).63(C22⋊C4), (C2×Dic5).196(C2×C4), SmallGroup(320,1136)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.4F5
C1C5C10Dic5C2×Dic5C2×C5⋊C8C23.2F5 — C24.4F5
C5C2×C10 — C24.4F5
C1C22C24

Generators and relations for C24.4F5
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=1, f4=d, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 602 in 190 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×6], C22, C22 [×6], C22 [×14], C5, C8 [×4], C2×C4 [×18], C23, C23 [×2], C23 [×6], C10, C10 [×2], C10 [×6], C2×C8 [×4], M4(2) [×4], C22×C4 [×10], C24, Dic5 [×4], Dic5 [×2], C2×C10, C2×C10 [×6], C2×C10 [×14], C22⋊C8 [×4], C2×M4(2) [×2], C23×C4, C5⋊C8 [×4], C2×Dic5 [×2], C2×Dic5 [×6], C2×Dic5 [×10], C22×C10, C22×C10 [×2], C22×C10 [×6], C24.4C4, C2×C5⋊C8 [×4], C22.F5 [×4], C22×Dic5 [×2], C22×Dic5 [×4], C22×Dic5 [×4], C23×C10, C23.2F5 [×4], C2×C22.F5 [×2], C23×Dic5, C24.4F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C2×M4(2) [×2], C2×F5 [×3], C24.4C4, C22.F5 [×4], C22⋊F5 [×2], C22×F5, C2×C22.F5 [×2], C2×C22⋊F5, C24.4F5

Smallest permutation representation of C24.4F5
On 80 points
Generators in S80
(2 44)(4 46)(6 48)(8 42)(9 26)(11 28)(13 30)(15 32)(17 50)(19 52)(21 54)(23 56)(34 57)(36 59)(38 61)(40 63)(66 74)(68 76)(70 78)(72 80)
(1 5)(3 7)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(58 62)(60 64)(65 69)(67 71)(73 77)(75 79)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 49)(33 64)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)
(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 12 18 64 73)(2 57 13 74 19)(3 75 58 20 14)(4 21 76 15 59)(5 16 22 60 77)(6 61 9 78 23)(7 79 62 24 10)(8 17 80 11 63)(25 55 37 69 47)(26 70 56 48 38)(27 41 71 39 49)(28 40 42 50 72)(29 51 33 65 43)(30 66 52 44 34)(31 45 67 35 53)(32 36 46 54 68)
(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)| (2,44)(4,46)(6,48)(8,42)(9,26)(11,28)(13,30)(15,32)(17,50)(19,52)(21,54)(23,56)(34,57)(36,59)(38,61)(40,63)(66,74)(68,76)(70,78)(72,80), (1,5)(3,7)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(58,62)(60,64)(65,69)(67,71)(73,77)(75,79), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,49)(33,64)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80), (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), (1,12,18,64,73)(2,57,13,74,19)(3,75,58,20,14)(4,21,76,15,59)(5,16,22,60,77)(6,61,9,78,23)(7,79,62,24,10)(8,17,80,11,63)(25,55,37,69,47)(26,70,56,48,38)(27,41,71,39,49)(28,40,42,50,72)(29,51,33,65,43)(30,66,52,44,34)(31,45,67,35,53)(32,36,46,54,68), (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( (2,44)(4,46)(6,48)(8,42)(9,26)(11,28)(13,30)(15,32)(17,50)(19,52)(21,54)(23,56)(34,57)(36,59)(38,61)(40,63)(66,74)(68,76)(70,78)(72,80), (1,5)(3,7)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(58,62)(60,64)(65,69)(67,71)(73,77)(75,79), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,49)(33,64)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80), (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), (1,12,18,64,73)(2,57,13,74,19)(3,75,58,20,14)(4,21,76,15,59)(5,16,22,60,77)(6,61,9,78,23)(7,79,62,24,10)(8,17,80,11,63)(25,55,37,69,47)(26,70,56,48,38)(27,41,71,39,49)(28,40,42,50,72)(29,51,33,65,43)(30,66,52,44,34)(31,45,67,35,53)(32,36,46,54,68), (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([(2,44),(4,46),(6,48),(8,42),(9,26),(11,28),(13,30),(15,32),(17,50),(19,52),(21,54),(23,56),(34,57),(36,59),(38,61),(40,63),(66,74),(68,76),(70,78),(72,80)], [(1,5),(3,7),(10,14),(12,16),(18,22),(20,24),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(49,53),(51,55),(58,62),(60,64),(65,69),(67,71),(73,77),(75,79)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,41),(8,42),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,49),(33,64),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80)], [(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)], [(1,12,18,64,73),(2,57,13,74,19),(3,75,58,20,14),(4,21,76,15,59),(5,16,22,60,77),(6,61,9,78,23),(7,79,62,24,10),(8,17,80,11,63),(25,55,37,69,47),(26,70,56,48,38),(27,41,71,39,49),(28,40,42,50,72),(29,51,33,65,43),(30,66,52,44,34),(31,45,67,35,53),(32,36,46,54,68)], [(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)])

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J 5 8A···8H10A···10O
order12222···244444···458···810···10
size11112···2555510···10420···204···4

44 irreducible representations

dim111111224444
type+++++++-+
imageC1C2C2C2C4C4D4M4(2)F5C2×F5C22.F5C22⋊F5
kernelC24.4F5C23.2F5C2×C22.F5C23×Dic5C22×Dic5C23×C10C2×Dic5C2×C10C24C23C22C22
# reps142162481384

Matrix representation of C24.4F5 in GL6(𝔽41)

100000
0400000
001000
000100
0000400
0000040
,
4000000
0400000
0040000
0004000
000010
000001
,
4000000
0400000
0040000
0004000
0000400
0000040
,
100000
010000
0040000
0004000
0000400
0000040
,
100000
010000
00344000
001000
000077
00003440
,
010000
4000000
000010
000001
0038300
0024300

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,7,34,0,0,0,0,7,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,38,24,0,0,0,0,3,3,0,0,1,0,0,0,0,0,0,1,0,0] >;

C24.4F5 in GAP, Magma, Sage, TeX

C_2^4._4F_5
% in TeX

G:=Group("C2^4.4F5");
// GroupNames label

G:=SmallGroup(320,1136);
// by ID

G=gap.SmallGroup(320,1136);
# by ID

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

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

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