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G = D408C4order 320 = 26·5

2nd semidirect product of D40 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D408C4, C40.86D4, Dic208C4, M5(2)⋊6D5, C20.7SD16, C22.3D40, C8.6(C4×D5), C406C41C2, (C2×C10).2D8, C54(D82C4), C40.44(C2×C4), (C2×C8).48D10, (C2×C4).11D20, (C2×C20).101D4, C8.43(C5⋊D4), D407C2.7C2, C4.12(C40⋊C2), (C5×M5(2))⋊10C2, (C2×C40).52C22, C20.91(C22⋊C4), C10.34(D4⋊C4), C4.20(D10⋊C4), C2.11(D205C4), SmallGroup(320,76)

Series: Derived Chief Lower central Upper central

C1C40 — D408C4
C1C5C10C20C40C2×C40D407C2 — D408C4
C5C10C20C40 — D408C4
C1C2C2×C4C2×C8M5(2)

Generators and relations for D408C4
 G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a19, cbc-1=a23b >

Subgroups: 334 in 58 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, D10, C2×C10, C4.Q8, M5(2), C4○D8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, D82C4, C80, C40⋊C2, D40, Dic20, C4⋊Dic5, C2×C40, C4○D20, C406C4, C5×M5(2), D407C2, D408C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, D10, D4⋊C4, C4×D5, D20, C5⋊D4, D82C4, C40⋊C2, D40, D10⋊C4, D205C4, D408C4

Smallest permutation representation of D408C4
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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 80)(39 79)(40 78)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 70 61 50)(42 49 62 69)(43 68 63 48)(44 47 64 67)(45 66 65 46)(51 60 71 80)(52 79 72 59)(53 58 73 78)(54 77 74 57)(55 56 75 76)

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

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

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

56 conjugacy classes

class 1 2A2B2C4A4B4C4D4E5A5B8A8B8C10A10B10C10D16A16B16C16D20A20B20C20D20E20F40A···40H40I40J40K40L80A···80P
order12224444455888101010101616161620202020202040···404040404080···80
size112402240404022224224444442222442···244444···4

56 irreducible representations

dim1111112222222222244
type+++++++++++
imageC1C2C2C2C4C4D4D4D5SD16D8D10C4×D5C5⋊D4D20C40⋊C2D40D82C4D408C4
kernelD408C4C406C4C5×M5(2)D407C2D40Dic20C40C2×C20M5(2)C20C2×C10C2×C8C8C8C2×C4C4C22C5C1
# reps1111221122224448828

Matrix representation of D408C4 in GL4(𝔽241) generated by

31200108187
41109197227
001694
00200173
,
159110161
159110110117
15318814625
9017813167
,
1901570
5151938
00173234
004168
G:=sub<GL(4,GF(241))| [31,41,0,0,200,109,0,0,108,197,169,200,187,227,4,173],[159,159,153,90,110,110,188,178,16,110,146,131,1,117,25,67],[190,51,0,0,1,51,0,0,57,9,173,41,0,38,234,68] >;

D408C4 in GAP, Magma, Sage, TeX

D_{40}\rtimes_8C_4
% in TeX

G:=Group("D40:8C4");
// GroupNames label

G:=SmallGroup(320,76);
// by ID

G=gap.SmallGroup(320,76);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,92,422,387,268,570,136,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^23*b>;
// generators/relations

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