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G = Dic5.14D4  order 160 = 25·5

1st non-split extension by Dic5 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic5.14D4, C22⋊1Dic10, C23.12D10, (C2×C10)⋊Q8, C2.6(D4×D5), C4⋊Dic5⋊2C2, (C2×C4).5D10, C5⋊1(C22⋊Q8), C10.4(C2×Q8), C10.16(C2×D4), C22⋊C4.1D5, (C2×Dic10)⋊2C2, C10.D4⋊4C2, (C2×C20).1C22, C2.6(C2×Dic10), C23.D5.2C2, C10.21(C4○D4), C2.6(D4⋊2D5), (C2×C10).19C23, (C22×C10).8C22, (C22×Dic5).3C2, (C2×Dic5).5C22, C22.39(C22×D5), (C5×C22⋊C4).1C2, SmallGroup(160,99)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — Dic5.14D4
Chief series
C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — Dic5.14D4
Lower central
C5 — C2×C10 — Dic5.14D4
Upper central
C1 — C22 — C22⋊C4

Generators and relations for Dic5.14D4
 G = < a,b,c,d | a10=c4=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=a5c-1 >

Subgroups: 208 in 74 conjugacy classes, 35 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C22⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, Dic5.14D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, Dic10, C22×D5, C2×Dic10, D4×D5, D4⋊2D5, Dic5.14D4

Smallest permutation representation of Dic5.14D4
►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 62 6 67)(2 61 7 66)(3 70 8 65)(4 69 9 64)(5 68 10 63)(11 31 16 36)(12 40 17 35)(13 39 18 34)(14 38 19 33)(15 37 20 32)(21 52 26 57)(22 51 27 56)(23 60 28 55)(24 59 29 54)(25 58 30 53)(41 79 46 74)(42 78 47 73)(43 77 48 72)(44 76 49 71)(45 75 50 80)

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

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

Dic5.14D4 is a maximal subgroup of
 C23⋊2Dic10  C24.27D10  C24.31D10  C42.88D10  C42.90D10  C42⋊10D10  C42.96D10  D4×Dic10  C42.102D10  D4⋊5Dic10  D4⋊6Dic10  C42⋊12D10  C42⋊17D10  C42.118D10  C24.56D10  C24.32D10  C24.33D10  C24.35D10  C10.682- 1+4  Dic10⋊19D4  C10.352+ 1+4  C10.362+ 1+4  C10.392+ 1+4  C10.732- 1+4  C10.742- 1+4  (Q8×Dic5)⋊C2  C10.502+ 1+4  D5×C22⋊Q8  Dic10⋊21D4  C10.512+ 1+4  C10.1182+ 1+4  C10.522+ 1+4  C10.792- 1+4  C4⋊C4.197D10  C10.802- 1+4  C10.812- 1+4  C10.822- 1+4  C10.1222+ 1+4  C10.632+ 1+4  C10.842- 1+4  C10.852- 1+4  C10.692+ 1+4  C42.137D10  C42.139D10  C42.140D10  C42.141D10  Dic10⋊10D4  C42.144D10  C42.145D10  C42.159D10  C42.160D10  C42.161D10  C42.162D10  C42.164D10  C42.165D10  D6⋊1Dic10  D6⋊2Dic10  D6⋊3Dic10  D6⋊4Dic10  (C2×C30)⋊Q8  Dic15.48D4  C22⋊2Dic30  A4⋊Dic10
Dic5.14D4 is a maximal quotient of
 (C2×C20)⋊Q8  C10.49(C4×D4)  C4⋊Dic5⋊15C4  C10.52(C4×D4)  (C2×Dic5)⋊Q8  C2.(C20⋊Q8)  (C2×C4).Dic10  C10.(C4⋊Q8)  Dic5.14D8  D4⋊Dic10  D4.Dic10  D4.2Dic10  Q8⋊Dic10  Dic5.9Q16  Q8.Dic10  Q8.2Dic10  C24.44D10  C24.46D10  C23⋊Dic10  C24.6D10  C24.7D10  C24.47D10  D6⋊1Dic10  D6⋊2Dic10  D6⋊3Dic10  D6⋊4Dic10  (C2×C30)⋊Q8  Dic15.48D4  C22⋊2Dic30

34 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B10A···10F10G10H10I10J20A···20H
order122222444444445510···101010101020···20
size11112244101010102020222···244444···4

34 irreducible representations

dim1111111222222244
type++++++++-+++-+-
imageC1C2C2C2C2C2C2D4Q8D5C4○D4D10D10Dic10D4×D5D4⋊2D5
kernelDic5.14D4C10.D4C4⋊Dic5C23.D5C5×C22⋊C4C2×Dic10C22×Dic5Dic5C2×C10C22⋊C4C10C2×C4C23C22C2C2
# reps1211111222242822

Matrix representation of Dic5.14D4 ►in GL6(𝔽41)

100000
010000
000100
00403400
0000400
0000040
,
100000
010000
001000
00344000
000090
0000032
,
0400000
100000
0040000
0004000
0000040
0000400
,
100000
0400000
001000
000100
000010
0000040

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,32],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
 

Dic5.14D4 in GAP, Magma, Sage, TeX 

{\rm Dic}_5._{14}D_4
 
% in TeX
 
G:=Group("Dic5.14D4");
 
// GroupNames label
 
G:=SmallGroup(160,99);
 
// by ID
 
G=gap.SmallGroup(160,99);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,218,188,50,4613]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^10=c^4=d^2=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=a^5*c^-1>;
 
// generators/relations
 

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