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G = D10.12D4order 160 = 25·5

1st non-split extension by D10 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.12D4, C23.4D10, C2.8(D4×D5), C4⋊Dic54C2, C22⋊C43D5, (C2×C4).6D10, C10.19(C2×D4), C23.D54C2, D10⋊C45C2, C10.8(C4○D4), C2.10(C4○D20), C10.D410C2, C2.8(D42D5), (C2×C20).52C22, (C2×C10).24C23, C51(C22.D4), (C2×Dic5).6C22, C22.42(C22×D5), (C22×C10).13C22, (C22×D5).20C22, (C2×C4×D5)⋊10C2, (C5×C22⋊C4)⋊5C2, (C2×C5⋊D4).3C2, SmallGroup(160,104)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D10.12D4
C1C5C10C2×C10C22×D5C2×C4×D5 — D10.12D4
C5C2×C10 — D10.12D4
C1C22C22⋊C4

Generators and relations for D10.12D4
 G = < a,b,c,d | a10=b2=c4=1, d2=a5, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=a5c-1 >

Subgroups: 256 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2 [×3], C2 [×3], C4 [×5], C22, C22 [×7], C5, C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, D5 [×2], C10 [×3], C10, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C22.D4, C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, D10.12D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C22×D5, C4○D20, D4×D5, D42D5, D10.12D4

Smallest permutation representation of D10.12D4
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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 71)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(31 45)(32 44)(33 43)(34 42)(35 41)(36 50)(37 49)(38 48)(39 47)(40 46)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)
(1 11 25 72)(2 12 26 73)(3 13 27 74)(4 14 28 75)(5 15 29 76)(6 16 30 77)(7 17 21 78)(8 18 22 79)(9 19 23 80)(10 20 24 71)(31 64 44 51)(32 65 45 52)(33 66 46 53)(34 67 47 54)(35 68 48 55)(36 69 49 56)(37 70 50 57)(38 61 41 58)(39 62 42 59)(40 63 43 60)
(1 57 6 52)(2 58 7 53)(3 59 8 54)(4 60 9 55)(5 51 10 56)(11 45 16 50)(12 46 17 41)(13 47 18 42)(14 48 19 43)(15 49 20 44)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)

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

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

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

D10.12D4 is a maximal subgroup of
C24.27D10  C24.30D10  C24.31D10  C42.93D10  C42.94D10  C42.95D10  C42.99D10  C4212D10  D2024D4  C4216D10  C42.229D10  C42.114D10  C42.116D10  C42.118D10  C42.119D10  C243D10  C24.33D10  C24.35D10  C24.36D10  C10.342+ 1+4  C10.372+ 1+4  C4⋊C421D10  C10.732- 1+4  D2020D4  C10.442+ 1+4  C10.472+ 1+4  C10.742- 1+4  C10.162- 1+4  D2022D4  C10.202- 1+4  C10.212- 1+4  C10.242- 1+4  C10.582+ 1+4  C10.262- 1+4  D5×C22.D4  C10.822- 1+4  C10.1222+ 1+4  C10.622+ 1+4  C10.632+ 1+4  C10.642+ 1+4  C10.842- 1+4  C10.662+ 1+4  C10.852- 1+4  C10.682+ 1+4  C10.692+ 1+4  C42.137D10  C42.141D10  D2010D4  C4220D10  C4221D10  C42.234D10  C42.143D10  C42.145D10  C4223D10  C4224D10  C42.189D10  C42.161D10  C42.162D10  C42.165D10  D30.35D4  D10.17D12  D30.7D4  D6⋊C4⋊D5  C23.17(S3×D5)  D30.16D4  D30.28D4
D10.12D4 is a maximal quotient of
C10.51(C4×D4)  C4⋊Dic515C4  C2.(C20⋊Q8)  (C2×Dic5).Q8  C22.58(D4×D5)  D103(C4⋊C4)  (C2×C4).21D20  C10.(C4⋊D4)  D10.12D8  D10.16SD16  C406C4⋊C2  C405C4⋊C2  D10.11SD16  D10.7Q16  (C2×C8).D10  D101C8.C2  C24.4D10  C24.7D10  C24.8D10  C24.9D10  C24.12D10  C24.14D10  C24.16D10  D30.35D4  D10.17D12  D30.7D4  D6⋊C4⋊D5  C23.17(S3×D5)  D30.16D4  D30.28D4

34 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B10A···10F10G10H10I10J20A···20H
order122222244444445510···101010101020···20
size11114101022410102020222···244444···4

34 irreducible representations

dim1111111122222244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2D4D5C4○D4D10D10C4○D20D4×D5D42D5
kernelD10.12D4C10.D4C4⋊Dic5D10⋊C4C23.D5C5×C22⋊C4C2×C4×D5C2×C5⋊D4D10C22⋊C4C10C2×C4C23C2C2C2
# reps1111111122442822

Matrix representation of D10.12D4 in GL4(𝔽41) generated by

343400
7100
00400
00040
,
343400
1700
0010
00040
,
303200
91100
00032
0090
,
32000
03200
0090
00032
G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,40,0,0,0,0,40],[34,1,0,0,34,7,0,0,0,0,1,0,0,0,0,40],[30,9,0,0,32,11,0,0,0,0,0,9,0,0,32,0],[32,0,0,0,0,32,0,0,0,0,9,0,0,0,0,32] >;

D10.12D4 in GAP, Magma, Sage, TeX

D_{10}._{12}D_4
% in TeX

G:=Group("D10.12D4");
// GroupNames label

G:=SmallGroup(160,104);
// by ID

G=gap.SmallGroup(160,104);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,55,218,188,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^4=1,d^2=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=a^5*c^-1>;
// generators/relations

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