Copied to
clipboard

G = C22.D20order 160 = 25·5

3rd non-split extension by C22 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22.4D20, C23.16D10, C4⋊Dic55C2, C22⋊C46D5, C10.6(C2×D4), (C2×C4).9D10, (C2×C10).4D4, C2.8(C2×D20), D10⋊C47C2, (C2×C20).3C22, C10.23(C4○D4), (C2×C10).27C23, (C22×Dic5)⋊2C2, C52(C22.D4), C2.10(D42D5), (C22×D5).5C22, C22.45(C22×D5), (C22×C10).16C22, (C2×Dic5).31C22, (C5×C22⋊C4)⋊4C2, (C2×C5⋊D4).5C2, SmallGroup(160,107)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C22.D20
C1C5C10C2×C10C22×D5C2×C5⋊D4 — C22.D20
C5C2×C10 — C22.D20
C1C22C22⋊C4

Generators and relations for C22.D20
 G = < a,b,c,d | a2=b2=c20=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 256 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×5], C22, C22 [×2], C22 [×5], C5, C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, D5, C10, C10 [×2], C10 [×2], C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22.D4, C2×Dic5, C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C22⋊C4, C22×Dic5, C2×C5⋊D4, C22.D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, D20 [×2], C22×D5, C2×D20, D42D5 [×2], C22.D20

Smallest permutation representation of C22.D20
On 80 points
Generators in S80
(1 52)(2 36)(3 54)(4 38)(5 56)(6 40)(7 58)(8 22)(9 60)(10 24)(11 42)(12 26)(13 44)(14 28)(15 46)(16 30)(17 48)(18 32)(19 50)(20 34)(21 79)(23 61)(25 63)(27 65)(29 67)(31 69)(33 71)(35 73)(37 75)(39 77)(41 62)(43 64)(45 66)(47 68)(49 70)(51 72)(53 74)(55 76)(57 78)(59 80)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 58)(22 59)(23 60)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)
(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 72 73 20)(2 19 74 71)(3 70 75 18)(4 17 76 69)(5 68 77 16)(6 15 78 67)(7 66 79 14)(8 13 80 65)(9 64 61 12)(10 11 62 63)(21 28 58 45)(22 44 59 27)(23 26 60 43)(24 42 41 25)(29 40 46 57)(30 56 47 39)(31 38 48 55)(32 54 49 37)(33 36 50 53)(34 52 51 35)

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

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

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

C22.D20 is a maximal subgroup of
C22⋊C4⋊F5  C23.5D20  C233D20  C24.31D10  C428D10  C42.92D10  C42.96D10  C42.102D10  D45D20  D46D20  C42.118D10  C24.56D10  C243D10  C24.33D10  C10.462+ 1+4  C10.1152+ 1+4  C10.472+ 1+4  C10.482+ 1+4  C22⋊Q825D5  C10.532+ 1+4  C10.772- 1+4  C10.572+ 1+4  C10.792- 1+4  D5×C22.D4  C10.822- 1+4  C10.1222+ 1+4  C10.662+ 1+4  C10.852- 1+4  C10.692+ 1+4  C4220D10  C42.143D10  C42.144D10  C42.145D10  C42.161D10  C42.163D10  C42.164D10  C42.165D10  D6.D20  D6.9D20  C6.(C2×D20)  C6.D4⋊D5  C22.D60
C22.D20 is a maximal quotient of
C2.(C4×D20)  (C2×C20).28D4  C10.(C4⋊Q8)  C10.55(C4×D4)  (C2×C4).21D20  (C2×C20).33D4  C23.34D20  C23.35D20  C23.10D20  C23.38D20  C22.D40  C23.13D20  C23.42D20  C24.47D10  C23.14D20  C23.45D20  C24.16D10  D6.D20  D6.9D20  C6.(C2×D20)  C6.D4⋊D5  C22.D60

34 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B10A···10F10G10H10I10J20A···20H
order122222244444445510···101010101020···20
size11112220441010101020222···244444···4

34 irreducible representations

dim1111112222224
type+++++++++++-
imageC1C2C2C2C2C2D4D5C4○D4D10D10D20D42D5
kernelC22.D20C4⋊Dic5D10⋊C4C5×C22⋊C4C22×Dic5C2×C5⋊D4C2×C10C22⋊C4C10C2×C4C23C22C2
# reps1221112244284

Matrix representation of C22.D20 in GL4(𝔽41) generated by

1000
0100
00211
001120
,
1000
0100
00400
00040
,
163000
27200
00259
004016
,
393000
4200
00259
001716
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,21,11,0,0,1,20],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[16,27,0,0,30,2,0,0,0,0,25,40,0,0,9,16],[39,4,0,0,30,2,0,0,0,0,25,17,0,0,9,16] >;

C22.D20 in GAP, Magma, Sage, TeX

C_2^2.D_{20}
% in TeX

G:=Group("C2^2.D20");
// GroupNames label

G:=SmallGroup(160,107);
// by ID

G=gap.SmallGroup(160,107);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,218,188,122,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^20=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations

׿
×
𝔽