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G = C20.10M4(2)  order 320 = 26·5

4th non-split extension by C20 of M4(2) acting via M4(2)/C4=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.10M4(2), (C2×C8).2F5, (C2×C40).7C4, C52C8.19D4, C51(C23.C8), C20.C85C2, C4.7(C4.F5), (C2×Dic5).2C8, (C22×D5).2C8, C10.4(C22⋊C8), C22.2(D5⋊C8), C4.36(C22⋊F5), C2.5(D10⋊C8), C20.34(C22⋊C4), (C2×C4×D5).7C4, (C2×C10).7(C2×C8), (C2×C8⋊D5).6C2, (C2×C4).123(C2×F5), (C2×C20).140(C2×C4), (C2×C52C8).213C22, SmallGroup(320,229)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20.10M4(2)
C1C5C10C20C52C8C2×C52C8C20.C8 — C20.10M4(2)
C5C10C2×C10 — C20.10M4(2)
C1C4C2×C4C2×C8

Generators and relations for C20.10M4(2)
 G = < a,b,c | a20=c2=1, b8=a10, bab-1=a7, cac=a9, cbc=a5b5 >

Subgroups: 226 in 58 conjugacy classes, 24 normal (20 characteristic)
C1, C2, C2 [×2], C4 [×2], C4, C22, C22 [×2], C5, C8 [×3], C2×C4, C2×C4 [×3], C23, D5, C10, C10, C16 [×2], C2×C8, C2×C8, M4(2) [×2], C22×C4, Dic5, C20 [×2], D10 [×2], C2×C10, M5(2) [×2], C2×M4(2), C52C8 [×2], C40, C4×D5 [×2], C2×Dic5, C2×C20, C22×D5, C23.C8, C5⋊C16 [×2], C8⋊D5 [×2], C2×C52C8, C2×C40, C2×C4×D5, C20.C8 [×2], C2×C8⋊D5, C20.10M4(2)
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C2×F5, C23.C8, D5⋊C8, C4.F5, C22⋊F5, D10⋊C8, C20.10M4(2)

Smallest permutation representation of C20.10M4(2)
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 67 24 48 6 62 29 43 11 77 34 58 16 72 39 53)(2 70 33 55 7 65 38 50 12 80 23 45 17 75 28 60)(3 73 22 42 8 68 27 57 13 63 32 52 18 78 37 47)(4 76 31 49 9 71 36 44 14 66 21 59 19 61 26 54)(5 79 40 56 10 74 25 51 15 69 30 46 20 64 35 41)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 27)(22 36)(23 25)(24 34)(26 32)(28 30)(29 39)(31 37)(33 35)(38 40)(41 45)(42 54)(44 52)(46 50)(47 59)(49 57)(51 55)(56 60)(61 63)(62 72)(64 70)(65 79)(66 68)(67 77)(69 75)(71 73)(74 80)(76 78)

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

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

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

38 conjugacy classes

class 1 2A2B2C4A4B4C4D 5 8A8B8C8D8E8F10A10B10C16A···16H20A20B20C20D40A···40H
order12224444588888810101016···162020202040···40
size11220112204441010101044420···2044444···4

38 irreducible representations

dim1111111224444444
type+++++++
imageC1C2C2C4C4C8C8D4M4(2)F5C2×F5C23.C8C4.F5C22⋊F5D5⋊C8C20.10M4(2)
kernelC20.10M4(2)C20.C8C2×C8⋊D5C2×C40C2×C4×D5C2×Dic5C22×D5C52C8C20C2×C8C2×C4C5C4C4C22C1
# reps1212244221122228

Matrix representation of C20.10M4(2) in GL4(𝔽241) generated by

466400
2236400
000110
0019546
,
001891
0018952
259000
5021600
,
189100
1895200
001891
0018952
G:=sub<GL(4,GF(241))| [46,223,0,0,64,64,0,0,0,0,0,195,0,0,110,46],[0,0,25,50,0,0,90,216,189,189,0,0,1,52,0,0],[189,189,0,0,1,52,0,0,0,0,189,189,0,0,1,52] >;

C20.10M4(2) in GAP, Magma, Sage, TeX

C_{20}._{10}M_4(2)
% in TeX

G:=Group("C20.10M4(2)");
// GroupNames label

G:=SmallGroup(320,229);
// by ID

G=gap.SmallGroup(320,229);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,387,100,1123,102,6278,3156]);
// Polycyclic

G:=Group<a,b,c|a^20=c^2=1,b^8=a^10,b*a*b^-1=a^7,c*a*c=a^9,c*b*c=a^5*b^5>;
// generators/relations

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