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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(C2xF5), C20.7(C4:C4), C40.17(C2xC4), (C4xD5).31D4, C8:D5.1C4, C4.22(C4:F5), D10.Q8:6C2, C40.C4:5C2, (C4xD5).10Q8, C5:(M4(2).C4), D10.15(C2xQ8), C4.Dic5.3C4, D10.11(C4:C4), C4.42(C22xF5), C20.82(C22xC4), Dic5.34(C2xD4), D5:M4(2).4C2, (C4xD5).82C23, (C8xD5).30C22, (D5xM4(2)).2C2, (C5xM4(2)).1C4, C22.14(C4:F5), (C22xD5).11Q8, C4.F5.10C22, Dic5.11(C4:C4), (C2xDic5).115D4, C2.21(C2xC4:F5), C10.18(C2xC4:C4), (C2xC4).32(C2xF5), (C2xC10).7(C4:C4), (C2xC4.F5).5C2, (C2xC20).48(C2xC4), C5:2C8.15(C2xC4), (C4xD5).38(C2xC4), (C2xC4xD5).195C22, SmallGroup(320,1067)

Series: Derived Chief Lower central Upper central

C1C20 — M4(2).1F5
C1C5C10Dic5C4xD5C4.F5C2xC4.F5 — M4(2).1F5
C5C10C20 — M4(2).1F5
C1C2C2xC4M4(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, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, C23, D5, C10, C10, C2xC8, M4(2), M4(2), C22xC4, Dic5, C20, D10, D10, C2xC10, C8.C4, C2xM4(2), C5:2C8, C40, C5:C8, C4xD5, C2xDic5, C2xC20, C22xD5, M4(2).C4, C8xD5, C8:D5, C4.Dic5, C5xM4(2), D5:C8, C4.F5, C4.F5, C4.F5, C2xC5:C8, C22.F5, C2xC4xD5, C40.C4, D10.Q8, D5xM4(2), C2xC4.F5, D5:M4(2), M4(2).1F5
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C23, C4:C4, C22xC4, C2xD4, C2xQ8, F5, C2xC4:C4, C2xF5, M4(2).C4, C4:F5, C22xF5, C2xC4: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)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)(66 70)(68 72)(73 77)(75 79)
(1 69 63 46 53)(2 70 64 47 54)(3 71 57 48 55)(4 72 58 41 56)(5 65 59 42 49)(6 66 60 43 50)(7 67 61 44 51)(8 68 62 45 52)(9 40 24 73 25)(10 33 17 74 26)(11 34 18 75 27)(12 35 19 76 28)(13 36 20 77 29)(14 37 21 78 30)(15 38 22 79 31)(16 39 23 80 32)
(1 17 3 23 5 21 7 19)(2 24 4 22 6 20 8 18)(9 72 31 50 13 68 27 54)(10 71 32 49 14 67 28 53)(11 70 25 56 15 66 29 52)(12 69 26 55 16 65 30 51)(33 48 80 59 37 44 76 63)(34 47 73 58 38 43 77 62)(35 46 74 57 39 42 78 61)(36 45 75 64 40 41 79 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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)(66,70)(68,72)(73,77)(75,79), (1,69,63,46,53)(2,70,64,47,54)(3,71,57,48,55)(4,72,58,41,56)(5,65,59,42,49)(6,66,60,43,50)(7,67,61,44,51)(8,68,62,45,52)(9,40,24,73,25)(10,33,17,74,26)(11,34,18,75,27)(12,35,19,76,28)(13,36,20,77,29)(14,37,21,78,30)(15,38,22,79,31)(16,39,23,80,32), (1,17,3,23,5,21,7,19)(2,24,4,22,6,20,8,18)(9,72,31,50,13,68,27,54)(10,71,32,49,14,67,28,53)(11,70,25,56,15,66,29,52)(12,69,26,55,16,65,30,51)(33,48,80,59,37,44,76,63)(34,47,73,58,38,43,77,62)(35,46,74,57,39,42,78,61)(36,45,75,64,40,41,79,60)>;

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111111122224444448
type+++++++-+-+++
imageC1C2C2C2C2C2C4C4C4D4Q8D4Q8F5C2xF5C2xF5M4(2).C4C4:F5C4:F5M4(2).1F5
kernelM4(2).1F5C40.C4D10.Q8D5xM4(2)C2xC4.F5D5:M4(2)C8:D5C4.Dic5C5xM4(2)C4xD5C4xD5C2xDic5C22xD5M4(2)C8C2xC4C5C4C22C1
# 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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