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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 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C10 [×3], C10 [×2], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊Q8, Dic10 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×2], C22×C10, C10.D4 [×2], C4⋊Dic5, C23.D5 [×2], C2×Dic10, C22×C20, C20.48D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, Dic10 [×2], C5⋊D4 [×2], 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 25 77 50)(2 24 78 49)(3 23 79 48)(4 22 80 47)(5 21 61 46)(6 40 62 45)(7 39 63 44)(8 38 64 43)(9 37 65 42)(10 36 66 41)(11 35 67 60)(12 34 68 59)(13 33 69 58)(14 32 70 57)(15 31 71 56)(16 30 72 55)(17 29 73 54)(18 28 74 53)(19 27 75 52)(20 26 76 51)
(1 60 11 50)(2 59 12 49)(3 58 13 48)(4 57 14 47)(5 56 15 46)(6 55 16 45)(7 54 17 44)(8 53 18 43)(9 52 19 42)(10 51 20 41)(21 61 31 71)(22 80 32 70)(23 79 33 69)(24 78 34 68)(25 77 35 67)(26 76 36 66)(27 75 37 65)(28 74 38 64)(29 73 39 63)(30 72 40 62)

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

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

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

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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