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G = C20.48D4order 160 = 25·5

5th non-split extension by C20 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.48D4, C222Dic10, C23.20D10, (C2×C10)⋊3Q8, C4⋊Dic58C2, C54(C22⋊Q8), C10.8(C2×Q8), (C2×C4).83D10, C10.39(C2×D4), (C22×C4).5D5, (C2×Dic10)⋊6C2, C10.D42C2, C4.23(C5⋊D4), (C22×C20).6C2, C2.9(C2×Dic10), C23.D5.4C2, C10.15(C4○D4), C2.17(C4○D20), (C2×C20).91C22, (C2×C10).42C23, C22.54(C22×D5), (C22×C10).34C22, (C2×Dic5).14C22, C2.5(C2×C5⋊D4), SmallGroup(160,145)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20.48D4
C1C5C10C2×C10C2×Dic5C2×Dic10 — C20.48D4
C5C2×C10 — C20.48D4
C1C22C22×C4

Generators and relations for C20.48D4
 G = < a,b,c | a20=b4=1, c2=a10, bab-1=cac-1=a-1, cbc-1=a10b-1 >

Subgroups: 192 in 74 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C10.D4, C4⋊Dic5, C23.D5, C2×Dic10, C22×C20, C20.48D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, Dic10, C5⋊D4, C22×D5, C2×Dic10, C4○D20, C2×C5⋊D4, C20.48D4

Smallest permutation representation of C20.48D4
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 50 40 76)(2 49 21 75)(3 48 22 74)(4 47 23 73)(5 46 24 72)(6 45 25 71)(7 44 26 70)(8 43 27 69)(9 42 28 68)(10 41 29 67)(11 60 30 66)(12 59 31 65)(13 58 32 64)(14 57 33 63)(15 56 34 62)(16 55 35 61)(17 54 36 80)(18 53 37 79)(19 52 38 78)(20 51 39 77)
(1 66 11 76)(2 65 12 75)(3 64 13 74)(4 63 14 73)(5 62 15 72)(6 61 16 71)(7 80 17 70)(8 79 18 69)(9 78 19 68)(10 77 20 67)(21 59 31 49)(22 58 32 48)(23 57 33 47)(24 56 34 46)(25 55 35 45)(26 54 36 44)(27 53 37 43)(28 52 38 42)(29 51 39 41)(30 50 40 60)

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

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

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

C20.48D4 is a maximal subgroup of
C4⋊Dic5⋊C4  C23.30D20  C23.34D20  C23.35D20  C23.10D20  D20.31D4  D20.32D4  C22⋊Dic20  C4⋊C4.230D10  C4⋊C4.231D10  C4⋊C4.233D10  C52C823D4  C4.(D4×D5)  (C2×C10)⋊Q16  C5⋊(C8.D4)  C4030D4  C40.82D4  C402D4  C40.4D4  (C5×D4).31D4  (C2×C10)⋊8Q16  (C5×D4).32D4  C42.274D10  C42.277D10  C232Dic10  C24.30D10  C24.31D10  C10.12- 1+4  C10.102+ 1+4  C10.52- 1+4  C10.62- 1+4  C42.89D10  C42.94D10  C42.98D10  C42.99D10  D4×Dic10  D45Dic10  C42.105D10  C42.106D10  D46Dic10  D2023D4  D2024D4  Dic1023D4  C4216D10  C42.115D10  C42.118D10  C4⋊C4.178D10  C10.352+ 1+4  C10.362+ 1+4  C4⋊C421D10  C10.392+ 1+4  C10.462+ 1+4  C10.742- 1+4  (Q8×Dic5)⋊C2  C10.502+ 1+4  C10.152- 1+4  D5×C22⋊Q8  C10.162- 1+4  C10.512+ 1+4  C10.582+ 1+4  C10.812- 1+4  C10.632+ 1+4  C10.842- 1+4  C10.692+ 1+4  C24.72D10  C24.42D10  Q8×C5⋊D4  C10.1042- 1+4  C10.1052- 1+4  C10.1072- 1+4  C10.1472+ 1+4  D6⋊Dic10  C60.45D4  C60.46D4  (C2×C10)⋊8Dic6  C60.205D4  C20.1S4
C20.48D4 is a maximal quotient of
C207(C4⋊C4)  (C2×C20)⋊10Q8  C10.92(C4×D4)  C23⋊Dic10  C24.6D10  C24.7D10  (C2×C4)⋊Dic10  (C2×C20).54D4  (C2×C20).55D4  C20.50D8  C20.38SD16  D4.3Dic10  C20.48SD16  C20.23Q16  Q8.3Dic10  C24.62D10  C24.64D10  D6⋊Dic10  C60.45D4  C60.46D4  (C2×C10)⋊8Dic6  C60.205D4

46 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B10A···10N20A···20P
order122222444444445510···1020···20
size111122222220202020222···22···2

46 irreducible representations

dim111111222222222
type+++++++-+++-
imageC1C2C2C2C2C2D4Q8D5C4○D4D10D10C5⋊D4Dic10C4○D20
kernelC20.48D4C10.D4C4⋊Dic5C23.D5C2×Dic10C22×C20C20C2×C10C22×C4C10C2×C4C23C4C22C2
# reps121211222242888

Matrix representation of C20.48D4 in GL4(𝔽41) generated by

39000
02000
001817
00016
,
0100
1000
00117
002440
,
0100
40000
00400
00171
G:=sub<GL(4,GF(41))| [39,0,0,0,0,20,0,0,0,0,18,0,0,0,17,16],[0,1,0,0,1,0,0,0,0,0,1,24,0,0,17,40],[0,40,0,0,1,0,0,0,0,0,40,17,0,0,0,1] >;

C20.48D4 in GAP, Magma, Sage, TeX

C_{20}._{48}D_4
% in TeX

G:=Group("C20.48D4");
// GroupNames label

G:=SmallGroup(160,145);
// by ID

G=gap.SmallGroup(160,145);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,217,103,218,4613]);
// Polycyclic

G:=Group<a,b,c|a^20=b^4=1,c^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^10*b^-1>;
// generators/relations

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