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G = M4(2).1F5order 320 = 26·5

1st non-split extension by M4(2) of F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).1F5, C8.5(C2×F5), C20.7(C4⋊C4), C40.17(C2×C4), (C4×D5).31D4, C8⋊D5.1C4, C4.22(C4⋊F5), D10.Q86C2, C40.C45C2, (C4×D5).10Q8, C5⋊(M4(2).C4), D10.15(C2×Q8), C4.Dic5.3C4, D10.11(C4⋊C4), C4.42(C22×F5), C20.82(C22×C4), Dic5.34(C2×D4), D5⋊M4(2).4C2, (C4×D5).82C23, (C8×D5).30C22, (D5×M4(2)).2C2, (C5×M4(2)).1C4, C22.14(C4⋊F5), (C22×D5).11Q8, C4.F5.10C22, Dic5.11(C4⋊C4), (C2×Dic5).115D4, C2.21(C2×C4⋊F5), C10.18(C2×C4⋊C4), (C2×C4).32(C2×F5), (C2×C10).7(C4⋊C4), (C2×C4.F5).5C2, (C2×C20).48(C2×C4), C52C8.15(C2×C4), (C4×D5).38(C2×C4), (C2×C4×D5).195C22, SmallGroup(320,1067)

Series: Derived Chief Lower central Upper central

C1C20 — M4(2).1F5
C1C5C10Dic5C4×D5C4.F5C2×C4.F5 — M4(2).1F5
C5C10C20 — M4(2).1F5
C1C2C2×C4M4(2)

Generators and relations for M4(2).1F5
 G = < a,b,c,d | a8=b2=c5=1, d4=a4, bab=a5, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 346 in 102 conjugacy classes, 48 normal (32 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×3], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×5], C23, D5 [×2], C10, C10, C2×C8 [×4], M4(2), M4(2) [×9], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10, C2×C10, C8.C4 [×4], C2×M4(2) [×3], C52C8 [×2], C40 [×2], C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, M4(2).C4, C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C5×M4(2), D5⋊C8, C4.F5 [×2], C4.F5 [×2], C4.F5, C2×C5⋊C8, C22.F5, C2×C4×D5, C40.C4 [×2], D10.Q8 [×2], D5×M4(2), C2×C4.F5, D5⋊M4(2), M4(2).1F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C2×F5 [×3], M4(2).C4, C4⋊F5 [×2], C22×F5, C2×C4⋊F5, M4(2).1F5

Smallest permutation representation of M4(2).1F5
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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(58 62)(60 64)(65 69)(67 71)(73 77)(75 79)
(1 72 43 61 53)(2 65 44 62 54)(3 66 45 63 55)(4 67 46 64 56)(5 68 47 57 49)(6 69 48 58 50)(7 70 41 59 51)(8 71 42 60 52)(9 20 34 76 28)(10 21 35 77 29)(11 22 36 78 30)(12 23 37 79 31)(13 24 38 80 32)(14 17 39 73 25)(15 18 40 74 26)(16 19 33 75 27)
(1 40 3 38 5 36 7 34)(2 39 4 37 6 35 8 33)(9 72 26 55 13 68 30 51)(10 71 27 54 14 67 31 50)(11 70 28 53 15 66 32 49)(12 69 29 52 16 65 25 56)(17 64 79 48 21 60 75 44)(18 63 80 47 22 59 76 43)(19 62 73 46 23 58 77 42)(20 61 74 45 24 57 78 41)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E 5 8A8B8C···8L10A10B20A20B20C40A40B40C40D
order12222444445888···8101020202040404040
size112101022551044420···20484488888

32 irreducible representations

dim11111111122224444448
type+++++++-+-+++
imageC1C2C2C2C2C2C4C4C4D4Q8D4Q8F5C2×F5C2×F5M4(2).C4C4⋊F5C4⋊F5M4(2).1F5
kernelM4(2).1F5C40.C4D10.Q8D5×M4(2)C2×C4.F5D5⋊M4(2)C8⋊D5C4.Dic5C5×M4(2)C4×D5C4×D5C2×Dic5C22×D5M4(2)C8C2×C4C5C4C22C1
# reps12211142211111212222

Matrix representation of M4(2).1F5 in GL8(𝔽41)

34027270000
1471400000
0147140000
27270340000
00000100
00009000
000000032
00000010
,
10000000
01000000
00100000
00010000
00001000
000004000
00000010
000000040
,
404040400000
10000000
01000000
00100000
00001000
00000100
00000010
00000001
,
03223320000
320990000
990320000
32233200000
00000010
00000001
00009000
00000900

G:=sub<GL(8,GF(41))| [34,14,0,27,0,0,0,0,0,7,14,27,0,0,0,0,27,14,7,0,0,0,0,0,27,0,14,34,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40],[40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,32,9,32,0,0,0,0,32,0,9,23,0,0,0,0,23,9,0,32,0,0,0,0,32,9,32,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

M4(2).1F5 in GAP, Magma, Sage, TeX

M_4(2)._1F_5
% in TeX

G:=Group("M4(2).1F5");
// GroupNames label

G:=SmallGroup(320,1067);
// by ID

G=gap.SmallGroup(320,1067);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,100,136,1684,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^5=1,d^4=a^4,b*a*b=a^5,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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