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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 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, D5 [×2], C10, C10, C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C8○D4, C52C8 [×2], C40 [×2], Dic10, C4×D5 [×2], D20, C5⋊D4 [×2], C2×C20, C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C2×C40, C4○D20, D20.3C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, C22×C4, D10 [×3], C8○D4, C4×D5 [×2], 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 29)(22 28)(23 27)(24 26)(30 40)(31 39)(32 38)(33 37)(34 36)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)
(1 38 75 41 11 28 65 51)(2 39 76 42 12 29 66 52)(3 40 77 43 13 30 67 53)(4 21 78 44 14 31 68 54)(5 22 79 45 15 32 69 55)(6 23 80 46 16 33 70 56)(7 24 61 47 17 34 71 57)(8 25 62 48 18 35 72 58)(9 26 63 49 19 36 73 59)(10 27 64 50 20 37 74 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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73), (1,38,75,41,11,28,65,51)(2,39,76,42,12,29,66,52)(3,40,77,43,13,30,67,53)(4,21,78,44,14,31,68,54)(5,22,79,45,15,32,69,55)(6,23,80,46,16,33,70,56)(7,24,61,47,17,34,71,57)(8,25,62,48,18,35,72,58)(9,26,63,49,19,36,73,59)(10,27,64,50,20,37,74,60)>;

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

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

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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