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G = D20.3C4order 160 = 25·5

The non-split extension by D20 of C4 acting through Inn(D20)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.3C4, C8.18D10, C40.23C22, C20.37C23, Dic10.3C4, (C2×C8)⋊7D5, (C8×D5)⋊6C2, C53(C8○D4), (C2×C40)⋊10C2, C8⋊D57C2, C4.10(C4×D5), C5⋊D4.3C4, C20.41(C2×C4), C4○D20.6C2, D10.3(C2×C4), (C2×C4).78D10, C22.2(C4×D5), C4.Dic511C2, Dic5.5(C2×C4), C4.37(C22×D5), (C2×C20).98C22, C10.27(C22×C4), C52C8.11C22, (C4×D5).23C22, C2.15(C2×C4×D5), (C2×C10).36(C2×C4), SmallGroup(160,122)

Series: Derived Chief Lower central Upper central

C1C10 — D20.3C4
C1C5C10C20C4×D5C4○D20 — D20.3C4
C5C10 — D20.3C4
C1C8C2×C8

Generators and relations for D20.3C4
 G = < a,b,c | a20=b2=1, c4=a10, bab=a-1, ac=ca, bc=cb >

Subgroups: 160 in 62 conjugacy classes, 37 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C2×C8, C2×C8, M4(2), C4○D4, Dic5, C20, D10, C2×C10, C8○D4, C52C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C8×D5, C8⋊D5, C4.Dic5, C2×C40, C4○D20, D20.3C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, D10, C8○D4, C4×D5, C22×D5, C2×C4×D5, D20.3C4

Smallest permutation representation of D20.3C4
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)
(1 67 57 26 11 77 47 36)(2 68 58 27 12 78 48 37)(3 69 59 28 13 79 49 38)(4 70 60 29 14 80 50 39)(5 71 41 30 15 61 51 40)(6 72 42 31 16 62 52 21)(7 73 43 32 17 63 53 22)(8 74 44 33 18 64 54 23)(9 75 45 34 19 65 55 24)(10 76 46 35 20 66 56 25)

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

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

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

D20.3C4 is a maximal subgroup of
D20.3C8  D20.4C8  D4017C4  D4010C4  C8.20D20  C8.21D20  C8.24D20  D20.6C8  D20.5C8  C40.93D4  C40.50D4  C40.23D4  C40.44D4  C40.29D4  C40.47C23  D5×C8○D4  C20.72C24  D813D10  D20.29D4  D20.30D4  D815D10  D811D10  D20.47D4  C40.54D6  C40.55D6  D20.3Dic3  D60.4C4  D60.6C4
D20.3C4 is a maximal quotient of
C8×Dic10  C4011Q8  C8×D20  C86D20  D10.5C42  C42.243D10  C408C4⋊C2  C22⋊C8⋊D5  D104M4(2)  Dic52M4(2)  Dic5.5M4(2)  D105M4(2)  C42.30D10  C42.31D10  C20.42C42  C20.65(C4⋊C4)  C8×C5⋊D4  (C22×C8)⋊D5  C4032D4  C40.54D6  C40.55D6  D20.3Dic3  D60.4C4  D60.6C4

52 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B8A8B8C8D8E8F8G8H8I8J10A···10F20A···20H40A···40P
order122224444455888888888810···1020···2040···40
size1121010112101022111122101010102···22···22···2

52 irreducible representations

dim1111111112222222
type+++++++++
imageC1C2C2C2C2C2C4C4C4D5D10D10C8○D4C4×D5C4×D5D20.3C4
kernelD20.3C4C8×D5C8⋊D5C4.Dic5C2×C40C4○D20Dic10D20C5⋊D4C2×C8C8C2×C4C5C4C22C1
# reps12211122424244416

Matrix representation of D20.3C4 in GL2(𝔽41) generated by

1439
1630
,
11
040
,
140
014
G:=sub<GL(2,GF(41))| [14,16,39,30],[1,0,1,40],[14,0,0,14] >;

D20.3C4 in GAP, Magma, Sage, TeX

D_{20}._3C_4
% in TeX

G:=Group("D20.3C4");
// GroupNames label

G:=SmallGroup(160,122);
// by ID

G=gap.SmallGroup(160,122);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,50,69,4613]);
// Polycyclic

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

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