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G = C8.21D20order 320 = 26·5

7th non-split extension by C8 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.21D20, C40.19D4, D20.21D4, Dic10.21D4, M4(2).10D10, (C2×D40)⋊21C2, C8.C46D5, C4.57(C2×D20), (C2×C8).71D10, C4.136(D4×D5), C8⋊D1010C2, C20.137(C2×D4), C52(D4.4D4), D20.3C46C2, C20.46D43C2, C10.50(C4⋊D4), C2.23(C4⋊D20), (C2×C20).313C23, (C2×C40).103C22, C4○D20.40C22, (C2×D20).92C22, C22.7(Q82D5), (C5×M4(2)).7C22, C4.Dic5.38C22, (C5×C8.C4)⋊7C2, (C2×C10).4(C4○D4), (C2×C4).114(C22×D5), SmallGroup(320,524)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C8.21D20
C1C5C10C20C2×C20C4○D20D20.3C4 — C8.21D20
C5C10C2×C20 — C8.21D20
C1C2C2×C4C8.C4

Generators and relations for C8.21D20
 G = < a,b,c | a8=c2=1, b20=a4, bab-1=cac=a-1, cbc=a4b19 >

Subgroups: 606 in 108 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C5, C8 [×2], C8 [×3], C2×C4, C2×C4, D4 [×6], Q8, C23 [×2], D5 [×3], C10, C10, C2×C8, C2×C8, M4(2) [×2], M4(2) [×2], D8 [×4], SD16 [×2], C2×D4 [×2], C4○D4, Dic5, C20 [×2], D10 [×5], C2×C10, C4.D4 [×2], C8.C4, C8○D4, C2×D8, C8⋊C22 [×2], C52C8, C40 [×2], C40 [×2], Dic10, C4×D5, D20, D20 [×4], C5⋊D4, C2×C20, C22×D5 [×2], D4.4D4, C8×D5, C8⋊D5, C40⋊C2 [×2], D40 [×4], C4.Dic5, C2×C40, C5×M4(2) [×2], C2×D20 [×2], C4○D20, C20.46D4 [×2], C5×C8.C4, D20.3C4, C2×D40, C8⋊D10 [×2], C8.21D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, D20 [×2], C22×D5, D4.4D4, C2×D20, D4×D5, Q82D5, C4⋊D20, C8.21D20

Smallest permutation representation of C8.21D20
On 80 points
Generators in S80
(1 44 11 54 21 64 31 74)(2 75 32 65 22 55 12 45)(3 46 13 56 23 66 33 76)(4 77 34 67 24 57 14 47)(5 48 15 58 25 68 35 78)(6 79 36 69 26 59 16 49)(7 50 17 60 27 70 37 80)(8 41 38 71 28 61 18 51)(9 52 19 62 29 72 39 42)(10 43 40 73 30 63 20 53)
(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 58)(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 80)(20 79)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 70)(30 69)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 60)(40 59)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C5A5B8A8B8C8D8E8F8G10A10B10C10D20A20B20C20D20E20F40A···40H40I···40P
order1222224445588888881010101020202020202040···4040···40
size11220404022202222488202022442222444···48···8

44 irreducible representations

dim111111222222224444
type+++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5C4○D4D10D10D20D4.4D4D4×D5Q82D5C8.21D20
kernelC8.21D20C20.46D4C5×C8.C4D20.3C4C2×D40C8⋊D10C40Dic10D20C8.C4C2×C10C2×C8M4(2)C8C5C4C22C1
# reps121112211222482228

Matrix representation of C8.21D20 in GL4(𝔽41) generated by

123300
8500
0058
003312
,
0001
00406
13200
392500
,
8500
123300
003312
0058
G:=sub<GL(4,GF(41))| [12,8,0,0,33,5,0,0,0,0,5,33,0,0,8,12],[0,0,13,39,0,0,2,25,0,40,0,0,1,6,0,0],[8,12,0,0,5,33,0,0,0,0,33,5,0,0,12,8] >;

C8.21D20 in GAP, Magma, Sage, TeX

C_8._{21}D_{20}
% in TeX

G:=Group("C8.21D20");
// GroupNames label

G:=SmallGroup(320,524);
// by ID

G=gap.SmallGroup(320,524);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,226,1123,136,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=c^2=1,b^20=a^4,b*a*b^-1=c*a*c=a^-1,c*b*c=a^4*b^19>;
// generators/relations

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