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G = C40.50D4order 320 = 26·5

50th non-split extension by C40 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.50D4, M4(2).37D10, C8○D47D5, D4⋊D510C4, Q8⋊D510C4, D4.8(C4×D5), Q8.8(C4×D5), C57(C8.26D4), D4.D510C4, C408C429C2, C5⋊Q1610C4, C4○D4.35D10, D20.33(C2×C4), C10.112(C4×D4), (C2×C8).191D10, C20.448(C2×D4), C8.47(C5⋊D4), D207C414C2, D42Dic53C2, C20.65(C22×C4), D20.3C414C2, C20.53D414C2, (C2×C40).237C22, (C2×C20).425C23, Dic10.34(C2×C4), D4.8D10.3C2, C4○D20.45C22, C22.4(C4○D20), (C4×Dic5).47C22, C4.Dic5.45C22, (C5×M4(2)).40C22, C4.30(C2×C4×D5), (C5×C8○D4)⋊7C2, C52C8.6(C2×C4), C2.27(C4×C5⋊D4), (C5×D4).29(C2×C4), C4.139(C2×C5⋊D4), (C5×Q8).30(C2×C4), (C2×C10).10(C4○D4), (C5×C4○D4).40C22, (C2×C4).515(C22×D5), (C2×C52C8).144C22, SmallGroup(320,772)

Series: Derived Chief Lower central Upper central

C1C20 — C40.50D4
C1C5C10C20C2×C20C4○D20D4.8D10 — C40.50D4
C5C10C20 — C40.50D4
C1C4C2×C8C8○D4

Generators and relations for C40.50D4
 G = < a,b,c | a40=c2=1, b4=a20, bab-1=a29, cac=a9, cbc=a20b3 >

Subgroups: 326 in 104 conjugacy classes, 47 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×3], C22, C22 [×2], C5, C8 [×2], C8 [×4], C2×C4, C2×C4 [×3], D4, D4 [×3], Q8, Q8, D5, C10, C10 [×2], C42, C2×C8, C2×C8 [×3], M4(2), M4(2) [×3], D8, SD16 [×2], Q16, C4○D4, C4○D4, Dic5 [×2], C20 [×2], C20, D10, C2×C10, C2×C10, C8⋊C4, C4≀C2 [×2], C8.C4, C8○D4, C8○D4, C4○D8, C52C8 [×2], C52C8, C40 [×2], C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C8.26D4, C8×D5, C8⋊D5, C2×C52C8, C4.Dic5, C4×Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C4○D20, C5×C4○D4, C408C4, C20.53D4, D207C4, D42Dic5, D20.3C4, D4.8D10, C5×C8○D4, C40.50D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C8.26D4, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, C40.50D4

Smallest permutation representation of C40.50D4
On 80 points
Generators in S80
(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 16 11 26 21 36 31 6)(2 5 12 15 22 25 32 35)(3 34 13 4 23 14 33 24)(7 30 17 40 27 10 37 20)(8 19 18 29 28 39 38 9)(41 66 71 56 61 46 51 76)(42 55 72 45 62 75 52 65)(43 44 73 74 63 64 53 54)(47 80 77 70 67 60 57 50)(48 69 78 59 68 49 58 79)
(1 46)(2 55)(3 64)(4 73)(5 42)(6 51)(7 60)(8 69)(9 78)(10 47)(11 56)(12 65)(13 74)(14 43)(15 52)(16 61)(17 70)(18 79)(19 48)(20 57)(21 66)(22 75)(23 44)(24 53)(25 62)(26 71)(27 80)(28 49)(29 58)(30 67)(31 76)(32 45)(33 54)(34 63)(35 72)(36 41)(37 50)(38 59)(39 68)(40 77)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A8B8C8D8E8F8G8H8I8J10A10B10C···10H20A20B20C20D20E···20J40A···40H40I···40T
order122224444444558888888888101010···102020202020···2040···4040···40
size11242011242020202222224420202020224···422224···42···24···4

62 irreducible representations

dim111111111111222222222244
type+++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4D4D5C4○D4D10D10D10C5⋊D4C4×D5C4×D5C4○D20C8.26D4C40.50D4
kernelC40.50D4C408C4C20.53D4D207C4D42Dic5D20.3C4D4.8D10C5×C8○D4D4⋊D5D4.D5Q8⋊D5C5⋊Q16C40C8○D4C2×C10C2×C8M4(2)C4○D4C8D4Q8C22C5C1
# reps111111112222222222844828

Matrix representation of C40.50D4 in GL4(𝔽41) generated by

343400
73500
0077
00346
,
131900
232800
00357
0026
,
00357
0026
131900
232800
G:=sub<GL(4,GF(41))| [34,7,0,0,34,35,0,0,0,0,7,34,0,0,7,6],[13,23,0,0,19,28,0,0,0,0,35,2,0,0,7,6],[0,0,13,23,0,0,19,28,35,2,0,0,7,6,0,0] >;

C40.50D4 in GAP, Magma, Sage, TeX

C_{40}._{50}D_4
% in TeX

G:=Group("C40.50D4");
// GroupNames label

G:=SmallGroup(320,772);
// by ID

G=gap.SmallGroup(320,772);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,387,58,136,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=c^2=1,b^4=a^20,b*a*b^-1=a^29,c*a*c=a^9,c*b*c=a^20*b^3>;
// generators/relations

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