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G = C5×C23.31D4order 320 = 26·5

Direct product of C5 and C23.31D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×C23.31D4, C4⋊C42C20, (C2×Q8)⋊1C20, C10.35C4≀C2, (Q8×C10)⋊13C4, (C2×C20).444D4, C22⋊C8.2C10, (C2×C10).11Q16, C23.30(C5×D4), C22⋊Q8.1C10, C22.2(C5×Q16), (C2×C10).23SD16, C22.2(C5×SD16), C10.51(C23⋊C4), (C22×C10).150D4, C10.22(Q8⋊C4), C2.C42.5C10, (C22×C20).385C22, (C5×C4⋊C4)⋊11C4, C2.5(C5×C4≀C2), (C2×C4).8(C2×C20), (C2×C4).96(C5×D4), C2.5(C5×C23⋊C4), (C5×C22⋊C8).4C2, C2.3(C5×Q8⋊C4), (C2×C20).348(C2×C4), (C5×C22⋊Q8).11C2, (C22×C4).15(C2×C10), C22.36(C5×C22⋊C4), (C2×C10).187(C22⋊C4), (C5×C2.C42).24C2, SmallGroup(320,133)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C23.31D4
C1C2C22C23C22×C4C22×C20C5×C2.C42 — C5×C23.31D4
C1C22C2×C4 — C5×C23.31D4
C1C2×C10C22×C20 — C5×C23.31D4

Generators and relations for C5×C23.31D4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bcde3 >

Subgroups: 162 in 80 conjugacy classes, 34 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C22 [×3], C22 [×2], C5, C8, C2×C4 [×2], C2×C4 [×7], Q8, C23, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C20 [×6], C2×C10 [×3], C2×C10 [×2], C2.C42, C22⋊C8, C22⋊Q8, C40, C2×C20 [×2], C2×C20 [×7], C5×Q8, C22×C10, C23.31D4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C22×C20, C22×C20, Q8×C10, C5×C2.C42, C5×C22⋊C8, C5×C22⋊Q8, C5×C23.31D4
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C2×C4, D4 [×2], C10 [×3], C22⋊C4, SD16, Q16, C20 [×2], C2×C10, C23⋊C4, Q8⋊C4, C4≀C2, C2×C20, C5×D4 [×2], C23.31D4, C5×C22⋊C4, C5×SD16, C5×Q16, C5×C23⋊C4, C5×Q8⋊C4, C5×C4≀C2, C5×C23.31D4

Smallest permutation representation of C5×C23.31D4
On 80 points
Generators in S80
(1 41 77 39 69)(2 42 78 40 70)(3 43 79 33 71)(4 44 80 34 72)(5 45 73 35 65)(6 46 74 36 66)(7 47 75 37 67)(8 48 76 38 68)(9 28 58 20 50)(10 29 59 21 51)(11 30 60 22 52)(12 31 61 23 53)(13 32 62 24 54)(14 25 63 17 55)(15 26 64 18 56)(16 27 57 19 49)
(2 54)(4 56)(6 50)(8 52)(9 46)(11 48)(13 42)(15 44)(18 72)(20 66)(22 68)(24 70)(26 80)(28 74)(30 76)(32 78)(34 64)(36 58)(38 60)(40 62)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 65)(24 66)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 73)(32 74)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)
(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)
(1 5)(2 8 54 52)(3 55)(4 50 56 6)(7 51)(9 15 46 44)(10 47)(11 42 48 13)(12 16)(14 43)(17 71)(18 66 72 20)(19 23)(21 67)(22 70 68 24)(25 79)(26 74 80 28)(27 31)(29 75)(30 78 76 32)(33 63)(34 58 64 36)(35 39)(37 59)(38 62 60 40)(41 45)(49 53)(57 61)(65 69)(73 77)

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

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

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

95 conjugacy classes

class 1 2A2B2C2D2E4A4B4C···4G4H4I5A5B5C5D8A8B8C8D10A···10L10M···10T20A···20H20I···20AB20AC···20AJ40A···40P
order122222444···4445555888810···1010···1020···2020···2020···2040···40
size111122224···488111144441···12···22···24···48···84···4

95 irreducible representations

dim111111111111222222222244
type++++++-+
imageC1C2C2C2C4C4C5C10C10C10C20C20D4D4SD16Q16C4≀C2C5×D4C5×D4C5×SD16C5×Q16C5×C4≀C2C23⋊C4C5×C23⋊C4
kernelC5×C23.31D4C5×C2.C42C5×C22⋊C8C5×C22⋊Q8C5×C4⋊C4Q8×C10C23.31D4C2.C42C22⋊C8C22⋊Q8C4⋊C4C2×Q8C2×C20C22×C10C2×C10C2×C10C10C2×C4C23C22C22C2C10C2
# reps1111224444881122444881614

Matrix representation of C5×C23.31D4 in GL4(𝔽41) generated by

18000
01800
0010
0001
,
1000
0100
0010
003540
,
40000
04000
0010
0001
,
40000
04000
00400
00040
,
261500
262600
00279
00214
,
40000
0100
00400
001732
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,35,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[26,26,0,0,15,26,0,0,0,0,27,2,0,0,9,14],[40,0,0,0,0,1,0,0,0,0,40,17,0,0,0,32] >;

C5×C23.31D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{31}D_4
% in TeX

G:=Group("C5xC2^3.31D4");
// GroupNames label

G:=SmallGroup(320,133);
// by ID

G=gap.SmallGroup(320,133);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,2803,2530,248,4911]);
// Polycyclic

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

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