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G = C24.F5order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.1F5, C23⋊(C5⋊C8), C52(C23⋊C8), (C22×C10)⋊3C8, (C23×C10).5C4, C2.3(C23⋊F5), C23.36(C2×F5), C23.2F52C2, 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

C1C2×C10 — C24.F5
C1C5C10C2×C10C2×Dic5C22×Dic5C23.2F5 — C24.F5
C5C10C2×C10 — C24.F5
C1C22C23C24

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 [×3], C2 [×4], C4 [×3], C22 [×3], C22 [×10], C5, C8 [×2], C2×C4 [×5], C23, C23 [×2], C23 [×4], C10 [×3], C10 [×4], C22⋊C4 [×2], C2×C8 [×2], C22×C4 [×2], C24, Dic5 [×3], C2×C10 [×3], C2×C10 [×10], C22⋊C8 [×2], C2×C22⋊C4, C5⋊C8 [×2], C2×Dic5 [×2], C2×Dic5 [×3], C22×C10, C22×C10 [×2], C22×C10 [×4], C23⋊C8, C23.D5 [×2], C2×C5⋊C8 [×2], C22×Dic5 [×2], C23×C10, C23.2F5 [×2], C2×C23.D5, C24.F5
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C23⋊C4, C4.D4, C5⋊C8 [×2], 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 23)(3 20)(4 8)(6 19)(7 24)(9 13)(11 53)(12 50)(15 49)(16 54)(17 21)(25 29)(27 57)(28 62)(31 61)(32 58)(34 45)(35 42)(36 40)(38 41)(39 46)(43 47)(51 55)(59 63)(65 69)(67 78)(68 75)(71 74)(72 79)(76 80)
(1 5)(2 23)(3 7)(4 17)(6 19)(8 21)(9 51)(10 14)(11 53)(12 16)(13 55)(15 49)(18 22)(20 24)(25 63)(26 30)(27 57)(28 32)(29 59)(31 61)(33 37)(34 45)(35 39)(36 47)(38 41)(40 43)(42 46)(44 48)(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 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(65 80)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)
(1 48 14 26 77)(2 27 41 78 15)(3 79 28 16 42)(4 9 80 43 29)(5 44 10 30 73)(6 31 45 74 11)(7 75 32 12 46)(8 13 76 47 25)(17 51 69 40 59)(18 33 52 60 70)(19 61 34 71 53)(20 72 62 54 35)(21 55 65 36 63)(22 37 56 64 66)(23 57 38 67 49)(24 68 58 50 39)
(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)| (2,23)(3,20)(4,8)(6,19)(7,24)(9,13)(11,53)(12,50)(15,49)(16,54)(17,21)(25,29)(27,57)(28,62)(31,61)(32,58)(34,45)(35,42)(36,40)(38,41)(39,46)(43,47)(51,55)(59,63)(65,69)(67,78)(68,75)(71,74)(72,79)(76,80), (1,5)(2,23)(3,7)(4,17)(6,19)(8,21)(9,51)(10,14)(11,53)(12,16)(13,55)(15,49)(18,22)(20,24)(25,63)(26,30)(27,57)(28,32)(29,59)(31,61)(33,37)(34,45)(35,39)(36,47)(38,41)(40,43)(42,46)(44,48)(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), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(65,80)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (1,48,14,26,77)(2,27,41,78,15)(3,79,28,16,42)(4,9,80,43,29)(5,44,10,30,73)(6,31,45,74,11)(7,75,32,12,46)(8,13,76,47,25)(17,51,69,40,59)(18,33,52,60,70)(19,61,34,71,53)(20,72,62,54,35)(21,55,65,36,63)(22,37,56,64,66)(23,57,38,67,49)(24,68,58,50,39), (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( (2,23)(3,20)(4,8)(6,19)(7,24)(9,13)(11,53)(12,50)(15,49)(16,54)(17,21)(25,29)(27,57)(28,62)(31,61)(32,58)(34,45)(35,42)(36,40)(38,41)(39,46)(43,47)(51,55)(59,63)(65,69)(67,78)(68,75)(71,74)(72,79)(76,80), (1,5)(2,23)(3,7)(4,17)(6,19)(8,21)(9,51)(10,14)(11,53)(12,16)(13,55)(15,49)(18,22)(20,24)(25,63)(26,30)(27,57)(28,32)(29,59)(31,61)(33,37)(34,45)(35,39)(36,47)(38,41)(40,43)(42,46)(44,48)(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), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(65,80)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (1,48,14,26,77)(2,27,41,78,15)(3,79,28,16,42)(4,9,80,43,29)(5,44,10,30,73)(6,31,45,74,11)(7,75,32,12,46)(8,13,76,47,25)(17,51,69,40,59)(18,33,52,60,70)(19,61,34,71,53)(20,72,62,54,35)(21,55,65,36,63)(22,37,56,64,66)(23,57,38,67,49)(24,68,58,50,39), (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([(2,23),(3,20),(4,8),(6,19),(7,24),(9,13),(11,53),(12,50),(15,49),(16,54),(17,21),(25,29),(27,57),(28,62),(31,61),(32,58),(34,45),(35,42),(36,40),(38,41),(39,46),(43,47),(51,55),(59,63),(65,69),(67,78),(68,75),(71,74),(72,79),(76,80)], [(1,5),(2,23),(3,7),(4,17),(6,19),(8,21),(9,51),(10,14),(11,53),(12,16),(13,55),(15,49),(18,22),(20,24),(25,63),(26,30),(27,57),(28,32),(29,59),(31,61),(33,37),(34,45),(35,39),(36,47),(38,41),(40,43),(42,46),(44,48),(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)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,55),(10,56),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,57),(32,58),(33,48),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47),(65,80),(66,73),(67,74),(68,75),(69,76),(70,77),(71,78),(72,79)], [(1,48,14,26,77),(2,27,41,78,15),(3,79,28,16,42),(4,9,80,43,29),(5,44,10,30,73),(6,31,45,74,11),(7,75,32,12,46),(8,13,76,47,25),(17,51,69,40,59),(18,33,52,60,70),(19,61,34,71,53),(20,72,62,54,35),(21,55,65,36,63),(22,37,56,64,66),(23,57,38,67,49),(24,68,58,50,39)], [(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 2A2B2C2D2E2F2G4A4B4C4D4E4F 5 8A···8H10A···10O
order1222222244444458···810···10
size11112244101010102020420···204···4

38 irreducible representations

dim11111122444444444
type+++++++-+-+
imageC1C2C2C4C4C8D4M4(2)F5C23⋊C4C4.D4C5⋊C8C2×F5C22.F5C22⋊F5C23⋊F5C23.F5
kernelC24.F5C23.2F5C2×C23.D5C22×Dic5C23×C10C22×C10C2×Dic5C2×C10C24C10C10C23C23C22C22C2C2
# reps12122822111212244

Matrix representation of C24.F5 in GL6(𝔽41)

100000
0400000
001000
00204000
0000400
0000211
,
4000000
0400000
0040000
0004000
000010
000001
,
4000000
0400000
0040000
0004000
0000400
0000040
,
100000
010000
0040000
0004000
0000400
0000040
,
100000
010000
0010000
00173700
0000180
00002016
,
010000
900000
000010
000001
0093600
0003200

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

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