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G = C5×C22.SD16order 320 = 26·5

Direct product of C5 and C22.SD16

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

Aliases: C5×C22.SD16, C4⋊C41C20, (C2×D4)⋊1C20, C10.34C4≀C2, (D4×C10)⋊15C4, C22⋊C82C10, (C2×C10).24D8, C22.2(C5×D8), (C2×C20).443D4, C4⋊D4.1C10, C23.29(C5×D4), (C2×C10).35SD16, C22.7(C5×SD16), C10.50(C23⋊C4), C2.C426C10, (C22×C10).149D4, C10.48(D4⋊C4), (C22×C20).384C22, (C5×C4⋊C4)⋊10C4, C2.4(C5×C4≀C2), (C5×C22⋊C8)⋊4C2, (C2×C4).7(C2×C20), (C2×C4).95(C5×D4), C2.4(C5×C23⋊C4), C2.3(C5×D4⋊C4), (C2×C20).347(C2×C4), (C5×C4⋊D4).11C2, (C22×C4).14(C2×C10), C22.35(C5×C22⋊C4), (C5×C2.C42)⋊22C2, (C2×C10).186(C22⋊C4), SmallGroup(320,132)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C22.SD16
C1C2C22C23C22×C4C22×C20C5×C2.C42 — C5×C22.SD16
C1C22C2×C4 — C5×C22.SD16
C1C2×C10C22×C20 — C5×C22.SD16

Generators and relations for C5×C22.SD16
 G = < a,b,c,d,e | a5=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=bcd3 >

Subgroups: 210 in 90 conjugacy classes, 34 normal (all characteristic)
C1, C2 [×3], C2 [×3], C4 [×5], C22 [×3], C22 [×5], C5, C8, C2×C4 [×2], C2×C4 [×6], D4 [×3], C23, C23, C10 [×3], C10 [×3], C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C20 [×5], C2×C10 [×3], C2×C10 [×5], C2.C42, C22⋊C8, C4⋊D4, C40, C2×C20 [×2], C2×C20 [×6], C5×D4 [×3], C22×C10, C22×C10, C22.SD16, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C22×C20, C22×C20, D4×C10, D4×C10, C5×C2.C42, C5×C22⋊C8, C5×C4⋊D4, C5×C22.SD16
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C2×C4, D4 [×2], C10 [×3], C22⋊C4, D8, SD16, C20 [×2], C2×C10, C23⋊C4, D4⋊C4, C4≀C2, C2×C20, C5×D4 [×2], C22.SD16, C5×C22⋊C4, C5×D8, C5×SD16, C5×C23⋊C4, C5×D4⋊C4, C5×C4≀C2, C5×C22.SD16

Smallest permutation representation of C5×C22.SD16
On 80 points
Generators in S80
(1 42 73 34 65)(2 43 74 35 66)(3 44 75 36 67)(4 45 76 37 68)(5 46 77 38 69)(6 47 78 39 70)(7 48 79 40 71)(8 41 80 33 72)(9 31 22 58 50)(10 32 23 59 51)(11 25 24 60 52)(12 26 17 61 53)(13 27 18 62 54)(14 28 19 63 55)(15 29 20 64 56)(16 30 21 57 49)
(1 5)(2 15)(3 7)(4 9)(6 11)(8 13)(10 14)(12 16)(17 21)(18 80)(19 23)(20 74)(22 76)(24 78)(25 47)(26 30)(27 41)(28 32)(29 43)(31 45)(33 62)(34 38)(35 64)(36 40)(37 58)(39 60)(42 46)(44 48)(49 53)(50 68)(51 55)(52 70)(54 72)(56 66)(57 61)(59 63)(65 69)(67 71)(73 77)(75 79)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 73)(24 74)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 41)(32 42)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)(49 71)(50 72)(51 65)(52 66)(53 67)(54 68)(55 69)(56 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)
(1 2)(3 9)(4 16)(5 6)(7 13)(8 12)(10 11)(14 15)(17 80)(18 79)(19 20)(21 76)(22 75)(23 24)(25 32)(26 41)(27 48)(28 29)(30 45)(31 44)(33 61)(34 35)(36 58)(37 57)(38 39)(40 62)(42 43)(46 47)(49 68)(50 67)(51 52)(53 72)(54 71)(55 56)(59 60)(63 64)(65 66)(69 70)(73 74)(77 78)

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

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

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

95 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C···4G4H5A5B5C5D8A8B8C8D10A···10L10M···10T10U10V10W10X20A···20H20I···20AB20AC20AD20AE20AF40A···40P
order1222222444···445555888810···1010···101010101020···2020···202020202040···40
size1111228224···48111144441···12···288882···24···488884···4

95 irreducible representations

dim111111111111222222222244
type++++++++
imageC1C2C2C2C4C4C5C10C10C10C20C20D4D4D8SD16C4≀C2C5×D4C5×D4C5×D8C5×SD16C5×C4≀C2C23⋊C4C5×C23⋊C4
kernelC5×C22.SD16C5×C2.C42C5×C22⋊C8C5×C4⋊D4C5×C4⋊C4D4×C10C22.SD16C2.C42C22⋊C8C4⋊D4C4⋊C4C2×D4C2×C20C22×C10C2×C10C2×C10C10C2×C4C23C22C22C2C10C2
# reps1111224444881122444881614

Matrix representation of C5×C22.SD16 in GL4(𝔽41) generated by

1000
0100
00370
00037
,
40000
04000
00400
0001
,
1000
0100
00400
00040
,
291200
292900
0009
0010
,
292900
291200
0001
0010
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[29,29,0,0,12,29,0,0,0,0,0,1,0,0,9,0],[29,29,0,0,29,12,0,0,0,0,0,1,0,0,1,0] >;

C5×C22.SD16 in GAP, Magma, Sage, TeX

C_5\times C_2^2.{\rm SD}_{16}
% in TeX

G:=Group("C5xC2^2.SD16");
// GroupNames label

G:=SmallGroup(320,132);
// by ID

G=gap.SmallGroup(320,132);
# by ID

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

G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=b*c*d^3>;
// generators/relations

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