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G = D4010C4order 320 = 26·5

4th semidirect product of D40 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4010C4, C8.27D20, C40.49D4, Dic2010C4, C42.23D10, C8.8(C4×D5), C8⋊C45D5, C40⋊C26C4, C54(C8.26D4), C40.46(C2×C4), C4.78(C2×D20), C2.17(C4×D20), C10.44(C4×D4), D204C41C2, D20.29(C2×C4), C20.298(C2×D4), (C2×C8).161D10, C40.6C412C2, D407C2.8C2, (C4×C20).17C22, D20.3C412C2, C20.167(C22×C4), (C2×C20).792C23, (C2×C40).229C22, Dic10.30(C2×C4), C4○D20.36C22, C22.21(C4○D20), C4.Dic5.34C22, C4.66(C2×C4×D5), (C5×C8⋊C4)⋊1C2, (C2×C10).63(C4○D4), (C2×C4).682(C22×D5), SmallGroup(320,344)

Series: Derived Chief Lower central Upper central

C1C20 — D4010C4
C1C5C10C20C2×C20C4○D20D407C2 — D4010C4
C5C10C20 — D4010C4
C1C4C2×C8C8⋊C4

Generators and relations for D4010C4
 G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a21, cbc-1=a30b >

Subgroups: 374 in 104 conjugacy classes, 47 normal (29 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×3], C22, C22 [×2], C5, C8 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×3], D4 [×4], Q8 [×2], D5 [×2], C10, C10, C42, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×2], C20 [×2], C20, D10 [×2], C2×C10, C8⋊C4, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C52C8 [×2], C40 [×2], C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C8.26D4, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40, Dic20, C4.Dic5 [×2], C4×C20, C2×C40 [×2], C4○D20 [×2], D204C4 [×2], C40.6C4, C5×C8⋊C4, D20.3C4 [×2], D407C2, D4010C4
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], D20 [×2], C22×D5, C8.26D4, C2×C4×D5, C2×D20, C4○D20, C4×D20, D4010C4

Smallest permutation representation of D4010C4
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 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 80)(9 79)(10 78)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 70)(19 69)(20 68)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 60)(29 59)(30 58)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)
(1 31 21 11)(2 12 22 32)(3 33 23 13)(4 14 24 34)(5 35 25 15)(6 16 26 36)(7 37 27 17)(8 18 28 38)(9 39 29 19)(10 20 30 40)(42 62)(44 64)(46 66)(48 68)(50 70)(52 72)(54 74)(56 76)(58 78)(60 80)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A8B8C8D8E8F8G8H8I8J10A···10F20A···20H20I···20P40A···40P
order12222444444455888888888810···1020···2020···2040···40
size112202011244202022222244202020202···22···24···44···4

62 irreducible representations

dim1111111112222222244
type+++++++++++
imageC1C2C2C2C2C2C4C4C4D4D5C4○D4D10D10C4×D5D20C4○D20C8.26D4D4010C4
kernelD4010C4D204C4C40.6C4C5×C8⋊C4D20.3C4D407C2C40⋊C2D40Dic20C40C8⋊C4C2×C10C42C2×C8C8C8C22C5C1
# reps1211214222222488828

Matrix representation of D4010C4 in GL6(𝔽41)

3600000
080000
00021921
00230040
000095
0000232
,
0330000
500000
001000
0000232
00121016
00230040
,
900000
0320000
009005
0003200
0000409
000001

G:=sub<GL(6,GF(41))| [36,0,0,0,0,0,0,8,0,0,0,0,0,0,0,23,0,0,0,0,21,0,0,0,0,0,9,0,9,2,0,0,21,40,5,32],[0,5,0,0,0,0,33,0,0,0,0,0,0,0,1,0,1,23,0,0,0,0,21,0,0,0,0,2,0,0,0,0,0,32,16,40],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,5,0,9,1] >;

D4010C4 in GAP, Magma, Sage, TeX

D_{40}\rtimes_{10}C_4
% in TeX

G:=Group("D40:10C4");
// GroupNames label

G:=SmallGroup(320,344);
// by ID

G=gap.SmallGroup(320,344);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,387,58,136,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^21,c*b*c^-1=a^30*b>;
// generators/relations

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