Copied to
clipboard

G = C22.2D40order 320 = 26·5

1st non-split extension by C22 of D40 acting via D40/D20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22.2D40, C23.31D20, (C2×D20)⋊2C4, C4⋊Dic54C4, C22⋊C82D5, (C2×C10).1D8, C10.21C4≀C2, (C2×C20).440D4, C207D4.1C2, (C2×C10).2SD16, (C22×C10).40D4, (C22×C4).55D10, C54(C22.SD16), C2.7(D207C4), C2.3(D205C4), C10.27(C23⋊C4), C22.2(C40⋊C2), C10.26(D4⋊C4), (C22×C20).41C22, C10.10C4226C2, C2.7(C23.1D10), C22.59(D10⋊C4), (C5×C22⋊C8)⋊2C2, (C2×C4).13(C4×D5), (C2×C20).198(C2×C4), (C2×C4).211(C5⋊D4), (C2×C10).106(C22⋊C4), SmallGroup(320,28)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C22.2D40
C1C5C10C2×C10C2×C20C22×C20C207D4 — C22.2D40
C5C2×C10C2×C20 — C22.2D40
C1C22C22×C4C22⋊C8

Generators and relations for C22.2D40
 G = < a,b,c,d | a2=b2=c40=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >

Subgroups: 494 in 90 conjugacy classes, 29 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, D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4 [×2], Dic5 [×3], C20 [×2], D10 [×3], C2×C10 [×3], C2×C10 [×2], C2.C42, C22⋊C8, C4⋊D4, C40, D20, C2×Dic5 [×5], C5⋊D4 [×2], C2×C20 [×2], C2×C20, C22×D5, C22×C10, C22.SD16, C4⋊Dic5, D10⋊C4, C2×C40, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, C10.10C42, C5×C22⋊C8, C207D4, C22.2D40
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D8, SD16, D10, C23⋊C4, D4⋊C4, C4≀C2, C4×D5, D20, C5⋊D4, C22.SD16, C40⋊C2, D40, D10⋊C4, C23.1D10, D205C4, D207C4, C22.2D40

Smallest permutation representation of C22.2D40
On 80 points
Generators in S80
(1 65)(3 67)(5 69)(7 71)(9 73)(11 75)(13 77)(15 79)(17 41)(19 43)(21 45)(23 47)(25 49)(27 51)(29 53)(31 55)(33 57)(35 59)(37 61)(39 63)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)
(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 15 65 79)(2 14)(3 77 67 13)(4 76)(5 11 69 75)(6 10)(7 73 71 9)(8 72)(12 68)(16 64)(17 39 41 63)(18 38)(19 61 43 37)(20 60)(21 35 45 59)(22 34)(23 57 47 33)(24 56)(25 31 49 55)(26 30)(27 53 51 29)(28 52)(32 48)(36 44)(40 80)(42 62)(46 58)(50 54)(66 78)(70 74)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222224444444455888810···101010101020···202020202040···40
size1111224022420202020402244442···244442···244444···4

59 irreducible representations

dim111111222222222222444
type++++++++++++
imageC1C2C2C2C4C4D4D4D5D8SD16D10C4≀C2C4×D5C5⋊D4D20C40⋊C2D40C23⋊C4C23.1D10D207C4
kernelC22.2D40C10.10C42C5×C22⋊C8C207D4C4⋊Dic5C2×D20C2×C20C22×C10C22⋊C8C2×C10C2×C10C22×C4C10C2×C4C2×C4C23C22C22C10C2C2
# reps111122112222444488144

Matrix representation of C22.2D40 in GL4(𝔽41) generated by

1000
0100
004011
0001
,
1000
0100
00400
00040
,
63900
22000
001931
003722
,
142700
112700
00927
00040
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,11,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[6,2,0,0,39,20,0,0,0,0,19,37,0,0,31,22],[14,11,0,0,27,27,0,0,0,0,9,0,0,0,27,40] >;

C22.2D40 in GAP, Magma, Sage, TeX

C_2^2._2D_{40}
% in TeX

G:=Group("C2^2.2D40");
// GroupNames label

G:=SmallGroup(320,28);
// by ID

G=gap.SmallGroup(320,28);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,92,422,387,100,1123,570,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^40=1,d^2=a,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations

׿
×
𝔽