Copied to
clipboard

G = M4(2).31D10order 320 = 26·5

4th non-split extension by M4(2) of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).31D10, C4.66(C2×D20), (C2×C4).50D20, C20.418(C2×D4), (C2×C20).174D4, C23.21(C4×D5), (C2×M4(2))⋊12D5, C20.46D415C2, C4.12D2015C2, (C10×M4(2))⋊20C2, (C2×C20).417C23, (C22×C4).140D10, C4.56(D10⋊C4), C20.114(C22⋊C4), (C2×D20).259C22, C4.Dic5.42C22, C22.7(D10⋊C4), (C22×C20).189C22, (C5×M4(2)).34C22, (C2×Dic10).286C22, C55(M4(2).8C22), (C2×C4×D5).4C4, C22.21(C2×C4×D5), (C2×C4).160(C4×D5), (C2×C5⋊D4).22C4, C4.111(C2×C5⋊D4), (C2×C20).282(C2×C4), (C2×C4○D20).12C2, (C2×C4).77(C5⋊D4), (C2×C4.Dic5)⋊16C2, (C2×Dic5).5(C2×C4), (C22×D5).6(C2×C4), C2.31(C2×D10⋊C4), C10.100(C2×C22⋊C4), (C2×C4).121(C22×D5), (C2×C10).116(C22×C4), (C22×C10).139(C2×C4), (C2×C10).130(C22⋊C4), SmallGroup(320,759)

Series: Derived Chief Lower central Upper central

C1C2×C10 — M4(2).31D10
C1C5C10C20C2×C20C2×D20C2×C4○D20 — M4(2).31D10
C5C10C2×C10 — M4(2).31D10
C1C4C22×C4C2×M4(2)

Generators and relations for M4(2).31D10
 G = < a,b,c,d | a8=b2=1, c10=d2=a4, bab=a5, ac=ca, dad-1=ab, bc=cb, bd=db, dcd-1=c9 >

Subgroups: 574 in 150 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×2], C22 [×3], C22 [×5], C5, C8 [×4], C2×C4 [×6], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C23 [×2], D5 [×2], C10, C10 [×3], C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×4], D10 [×4], C2×C10 [×3], C2×C10, C4.D4 [×2], C4.10D4 [×2], C2×M4(2), C2×M4(2), C2×C4○D4, C52C8 [×2], C40 [×2], Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×6], C22×D5 [×2], C22×C10, M4(2).8C22, C2×C52C8, C4.Dic5 [×2], C4.Dic5, C2×C40, C5×M4(2) [×2], C5×M4(2), C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, C20.46D4 [×2], C4.12D20 [×2], C2×C4.Dic5, C10×M4(2), C2×C4○D20, M4(2).31D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, M4(2).8C22, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, M4(2).31D10

Smallest permutation representation of M4(2).31D10
On 80 points
Generators in S80
(1 59 73 27 11 49 63 37)(2 60 74 28 12 50 64 38)(3 41 75 29 13 51 65 39)(4 42 76 30 14 52 66 40)(5 43 77 31 15 53 67 21)(6 44 78 32 16 54 68 22)(7 45 79 33 17 55 69 23)(8 46 80 34 18 56 70 24)(9 47 61 35 19 57 71 25)(10 48 62 36 20 58 72 26)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 22 31 32)(23 40 33 30)(24 29 34 39)(25 38 35 28)(26 27 36 37)(41 56 51 46)(42 45 52 55)(43 54 53 44)(47 50 57 60)(48 59 58 49)(61 64 71 74)(62 73 72 63)(65 80 75 70)(66 69 76 79)(67 78 77 68)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222224444444558888888810···101010101020···202020202040···40
size112222020112222020224444202020202···244442···244444···4

62 irreducible representations

dim111111112222222244
type+++++++++++
imageC1C2C2C2C2C2C4C4D4D5D10D10C4×D5D20C5⋊D4C4×D5M4(2).8C22M4(2).31D10
kernelM4(2).31D10C20.46D4C4.12D20C2×C4.Dic5C10×M4(2)C2×C4○D20C2×C4×D5C2×C5⋊D4C2×C20C2×M4(2)M4(2)C22×C4C2×C4C2×C4C2×C4C23C5C1
# reps122111444242488428

Matrix representation of M4(2).31D10 in GL4(𝔽41) generated by

0090
0009
183500
62300
,
1000
0100
00400
00040
,
282800
133200
002828
001332
,
282800
321300
002828
003213
G:=sub<GL(4,GF(41))| [0,0,18,6,0,0,35,23,9,0,0,0,0,9,0,0],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[28,13,0,0,28,32,0,0,0,0,28,13,0,0,28,32],[28,32,0,0,28,13,0,0,0,0,28,32,0,0,28,13] >;

M4(2).31D10 in GAP, Magma, Sage, TeX

M_4(2)._{31}D_{10}
% in TeX

G:=Group("M4(2).31D10");
// GroupNames label

G:=SmallGroup(320,759);
// by ID

G=gap.SmallGroup(320,759);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,422,58,1123,136,438,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^10=d^2=a^4,b*a*b=a^5,a*c=c*a,d*a*d^-1=a*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations

׿
×
𝔽