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G = C5×C24.4C4order 320 = 26·5

Direct product of C5 and C24.4C4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C24.4C4, C24.4C20, C4.70(D4×C10), C22⋊C811C10, (C2×C40)⋊37C22, C20.475(C2×D4), (C2×C20).513D4, (C23×C4).8C10, (C2×M4(2))⋊6C10, (C2×C10)⋊13M4(2), C23.29(C2×C20), (C23×C20).23C2, (C23×C10).12C4, (C22×C20).48C4, (C22×C4).12C20, C2.6(C10×M4(2)), C222(C5×M4(2)), (C10×M4(2))⋊24C2, (C2×C20).982C23, C10.82(C2×M4(2)), C20.154(C22⋊C4), C22.41(C22×C20), (C22×C20).496C22, (C2×C8)⋊7(C2×C10), (C5×C22⋊C8)⋊28C2, (C2×C4).71(C2×C20), (C2×C4).118(C5×D4), C4.21(C5×C22⋊C4), (C2×C20).506(C2×C4), C2.10(C10×C22⋊C4), C10.139(C2×C22⋊C4), (C22×C4).92(C2×C10), C22.17(C5×C22⋊C4), (C2×C10).333(C22×C4), (C2×C4).150(C22×C10), (C22×C10).183(C2×C4), (C2×C10).144(C22⋊C4), SmallGroup(320,908)

Series: Derived Chief Lower central Upper central

C1C22 — C5×C24.4C4
C1C2C4C2×C4C2×C20C2×C40C5×C22⋊C8 — C5×C24.4C4
C1C22 — C5×C24.4C4
C1C2×C20 — C5×C24.4C4

Generators and relations for C5×C24.4C4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=e2=1, f4=e, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 290 in 190 conjugacy classes, 90 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×2], C22, C22 [×6], C22 [×14], C5, C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×10], C23, C23 [×2], C23 [×6], C10, C10 [×2], C10 [×6], C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C20 [×4], C20 [×2], C2×C10, C2×C10 [×6], C2×C10 [×14], C22⋊C8 [×4], C2×M4(2) [×2], C23×C4, C40 [×4], C2×C20 [×2], C2×C20 [×6], C2×C20 [×10], C22×C10, C22×C10 [×2], C22×C10 [×6], C24.4C4, C2×C40 [×4], C5×M4(2) [×4], C22×C20 [×2], C22×C20 [×4], C22×C20 [×4], C23×C10, C5×C22⋊C8 [×4], C10×M4(2) [×2], C23×C20, C5×C24.4C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C2×C22⋊C4, C2×M4(2) [×2], C2×C20 [×6], C5×D4 [×4], C22×C10, C24.4C4, C5×C22⋊C4 [×4], C5×M4(2) [×4], C22×C20, D4×C10 [×2], C10×C22⋊C4, C10×M4(2) [×2], C5×C24.4C4

Smallest permutation representation of C5×C24.4C4
On 80 points
Generators in S80
(1 14 55 63 23)(2 15 56 64 24)(3 16 49 57 17)(4 9 50 58 18)(5 10 51 59 19)(6 11 52 60 20)(7 12 53 61 21)(8 13 54 62 22)(25 45 73 33 65)(26 46 74 34 66)(27 47 75 35 67)(28 48 76 36 68)(29 41 77 37 69)(30 42 78 38 70)(31 43 79 39 71)(32 44 80 40 72)
(2 32)(4 26)(6 28)(8 30)(9 46)(11 48)(13 42)(15 44)(18 66)(20 68)(22 70)(24 72)(34 58)(36 60)(38 62)(40 64)(50 74)(52 76)(54 78)(56 80)
(1 5)(3 7)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(65 69)(67 71)(73 77)(75 79)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 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 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)| (1,14,55,63,23)(2,15,56,64,24)(3,16,49,57,17)(4,9,50,58,18)(5,10,51,59,19)(6,11,52,60,20)(7,12,53,61,21)(8,13,54,62,22)(25,45,73,33,65)(26,46,74,34,66)(27,47,75,35,67)(28,48,76,36,68)(29,41,77,37,69)(30,42,78,38,70)(31,43,79,39,71)(32,44,80,40,72), (2,32)(4,26)(6,28)(8,30)(9,46)(11,48)(13,42)(15,44)(18,66)(20,68)(22,70)(24,72)(34,58)(36,60)(38,62)(40,64)(50,74)(52,76)(54,78)(56,80), (1,5)(3,7)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80), (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), (1,2,3,4,5,6,7,8)(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( (1,14,55,63,23)(2,15,56,64,24)(3,16,49,57,17)(4,9,50,58,18)(5,10,51,59,19)(6,11,52,60,20)(7,12,53,61,21)(8,13,54,62,22)(25,45,73,33,65)(26,46,74,34,66)(27,47,75,35,67)(28,48,76,36,68)(29,41,77,37,69)(30,42,78,38,70)(31,43,79,39,71)(32,44,80,40,72), (2,32)(4,26)(6,28)(8,30)(9,46)(11,48)(13,42)(15,44)(18,66)(20,68)(22,70)(24,72)(34,58)(36,60)(38,62)(40,64)(50,74)(52,76)(54,78)(56,80), (1,5)(3,7)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80), (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), (1,2,3,4,5,6,7,8)(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([(1,14,55,63,23),(2,15,56,64,24),(3,16,49,57,17),(4,9,50,58,18),(5,10,51,59,19),(6,11,52,60,20),(7,12,53,61,21),(8,13,54,62,22),(25,45,73,33,65),(26,46,74,34,66),(27,47,75,35,67),(28,48,76,36,68),(29,41,77,37,69),(30,42,78,38,70),(31,43,79,39,71),(32,44,80,40,72)], [(2,32),(4,26),(6,28),(8,30),(9,46),(11,48),(13,42),(15,44),(18,66),(20,68),(22,70),(24,72),(34,58),(36,60),(38,62),(40,64),(50,74),(52,76),(54,78),(56,80)], [(1,5),(3,7),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(49,53),(51,55),(57,61),(59,63),(65,69),(67,71),(73,77),(75,79)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(49,73),(50,74),(51,75),(52,76),(53,77),(54,78),(55,79),(56,80)], [(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)], [(1,2,3,4,5,6,7,8),(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)])

140 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J5A5B5C5D8A···8H10A···10L10M···10AJ20A···20P20Q···20AN40A···40AF
order12222···244444···455558···810···1010···1020···2020···2040···40
size11112···211112···211114···41···12···21···12···24···4

140 irreducible representations

dim1111111111112222
type+++++
imageC1C2C2C2C4C4C5C10C10C10C20C20D4M4(2)C5×D4C5×M4(2)
kernelC5×C24.4C4C5×C22⋊C8C10×M4(2)C23×C20C22×C20C23×C10C24.4C4C22⋊C8C2×M4(2)C23×C4C22×C4C24C2×C20C2×C10C2×C4C22
# reps14216241684248481632

Matrix representation of C5×C24.4C4 in GL4(𝔽41) generated by

10000
01000
00180
00018
,
1000
04000
0012
00040
,
40000
0100
0010
0001
,
40000
04000
00400
00040
,
40000
04000
0010
0001
,
0100
9000
0010
004040
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,2,40],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[0,9,0,0,1,0,0,0,0,0,1,40,0,0,0,40] >;

C5×C24.4C4 in GAP, Magma, Sage, TeX

C_5\times C_2^4._4C_4
% in TeX

G:=Group("C5xC2^4.4C4");
// GroupNames label

G:=SmallGroup(320,908);
// by ID

G=gap.SmallGroup(320,908);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,3446,124]);
// Polycyclic

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

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