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G = C40.44D4order 320 = 26·5

44th non-split extension by C40 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.44D4, D20.26D4, Dic10.26D4, C4.66(D4xD5), (C2xSD16):3D5, (C2xC8).92D10, (C10xSD16):1C2, (C2xD4).78D10, C20.D4:8C2, C20.180(C2xD4), C5:5(D4.3D4), C8.32(C5:D4), (C2xQ8).59D10, C40.6C4:10C2, D20.3C4:4C2, C20.C23:4C2, C20.10D4:7C2, (C2xC40).48C22, C2.21(C20:2D4), (C2xC20).454C23, D4.D10.2C2, C4oD20.47C22, (Q8xC10).83C22, C10.118(C4:D4), (D4xC10).103C22, C4.Dic5.20C22, C22.21(D4:2D5), C4.84(C2xC5:D4), (C2xC4).127(C22xD5), (C2xC10).159(C4oD4), SmallGroup(320,804)

Series: Derived Chief Lower central Upper central

C1C2xC20 — C40.44D4
C1C5C10C20C2xC20C4oD20D20.3C4 — C40.44D4
C5C10C2xC20 — C40.44D4
C1C2C2xC4C2xSD16

Generators and relations for C40.44D4
 G = < a,b,c | a40=c2=1, b4=a20, bab-1=a19, cac=a9, cbc=a20b3 >

Subgroups: 366 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, D4, Q8, C23, D5, C10, C10, C2xC8, C2xC8, M4(2), D8, SD16, Q16, C2xD4, C2xQ8, C4oD4, Dic5, C20, C20, D10, C2xC10, C2xC10, C4.D4, C4.10D4, C8.C4, C8oD4, C2xSD16, C8:C22, C8.C22, C5:2C8, C40, Dic10, C4xD5, D20, C5:D4, C2xC20, C2xC20, C5xD4, C5xQ8, C22xC10, D4.3D4, C8xD5, C8:D5, C4.Dic5, D4:D5, D4.D5, Q8:D5, C5:Q16, C2xC40, C5xSD16, C4oD20, D4xC10, Q8xC10, C40.6C4, C20.D4, C20.10D4, D20.3C4, D4.D10, C20.C23, C10xSD16, C40.44D4
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C4oD4, D10, C4:D4, C5:D4, C22xD5, D4.3D4, D4xD5, D4:2D5, C2xC5:D4, C20:2D4, C40.44D4

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D5A5B8A8B8C8D8E8F8G10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222444455888888810···1010101010202020202020202040···40
size1128202282022224202040402···28888444488884···4

44 irreducible representations

dim111111112222222224444
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D4D5C4oD4D10D10D10C5:D4D4.3D4D4xD5D4:2D5C40.44D4
kernelC40.44D4C40.6C4C20.D4C20.10D4D20.3C4D4.D10C20.C23C10xSD16C40Dic10D20C2xSD16C2xC10C2xC8C2xD4C2xQ8C8C5C4C22C1
# reps111111112112222282228

Matrix representation of C40.44D4 in GL4(F41) generated by

0253030
40383021
00023
004028
,
34171627
389332
13391616
0383423
,
738316
318319
2803413
0383423
G:=sub<GL(4,GF(41))| [0,40,0,0,25,38,0,0,30,30,0,40,30,21,23,28],[34,38,13,0,17,9,39,38,16,33,16,34,27,2,16,23],[7,3,28,0,38,18,0,38,3,31,34,34,16,9,13,23] >;

C40.44D4 in GAP, Magma, Sage, TeX

C_{40}._{44}D_4
% in TeX

G:=Group("C40.44D4");
// GroupNames label

G:=SmallGroup(320,804);
// by ID

G=gap.SmallGroup(320,804);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,555,1123,297,136,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=c^2=1,b^4=a^20,b*a*b^-1=a^19,c*a*c=a^9,c*b*c=a^20*b^3>;
// generators/relations

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