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G = M4(2).F5order 320 = 26·5

3rd non-split extension by M4(2) of F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.4C42, M4(2).3F5, C4.9(C4xF5), D5:C8.4C4, C4.F5.1C4, D10.7(C4:C4), (C4xD5).118D4, D5.(C8.C4), C4.Dic5.1C4, (C22xD5).8Q8, C5:2(C4.C42), D5:M4(2).2C2, (D5xM4(2)).5C2, (C5xM4(2)).3C4, C22.11(C4:F5), C4.41(C22:F5), C20.39(C22:C4), (C2xDic5).100D4, Dic5.33(C22:C4), C2.16(D10.3Q8), C10.15(C2.C42), (C2xD5:C8).2C2, (C2xC4).70(C2xF5), (C2xC10).4(C4:C4), (C2xC20).37(C2xC4), (C4xD5).15(C2xC4), (C2xC4xD5).188C22, SmallGroup(320,239)

Series: Derived Chief Lower central Upper central

C1C20 — M4(2).F5
C1C5C10Dic5C4xD5C2xC4xD5D5:M4(2) — M4(2).F5
C5C10C20 — M4(2).F5
C1C4C2xC4M4(2)

Generators and relations for M4(2).F5
 G = < a,b,c,d | a8=b2=c5=1, d4=a4, bab=a5, ac=ca, dad-1=a-1b, bc=cb, dbd-1=a4b, dcd-1=c3 >

Subgroups: 322 in 90 conjugacy classes, 36 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, C23, D5, D5, C10, C10, C2xC8, M4(2), M4(2), C22xC4, Dic5, C20, D10, D10, C2xC10, C22xC8, C2xM4(2), C5:2C8, C40, C5:C8, C4xD5, C2xDic5, C2xC20, C22xD5, C4.C42, C8xD5, C8:D5, C4.Dic5, C5xM4(2), D5:C8, D5:C8, C4.F5, C2xC5:C8, C22.F5, C2xC4xD5, D5xM4(2), C2xD5:C8, D5:M4(2), M4(2).F5
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C42, C22:C4, C4:C4, F5, C2.C42, C8.C4, C2xF5, C4.C42, C4xF5, C4:F5, C22:F5, D10.3Q8, M4(2).F5

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F 5 8A8B8C···8J8K···8P10A10B20A20B20C40A40B40C40D
order1222224444445888···88···8101020202040404040
size1125510112551044410···1020···20484488888

38 irreducible representations

dim111111112222444448
type++++++-+++
imageC1C2C2C2C4C4C4C4D4D4Q8C8.C4F5C2xF5C4xF5C22:F5C4:F5M4(2).F5
kernelM4(2).F5D5xM4(2)C2xD5:C8D5:M4(2)C4.Dic5C5xM4(2)D5:C8C4.F5C4xD5C2xDic5C22xD5D5M4(2)C2xC4C4C4C22C1
# reps111122442118112222

Matrix representation of M4(2).F5 in GL6(F41)

2840000
17130000
003414027
00071427
00271470
002701434
,
4000000
1410000
001000
000100
000010
000001
,
100000
010000
0000040
0010040
0001040
0000140
,
39290000
320000
00370410
00010356
003163510
00311040

G:=sub<GL(6,GF(41))| [28,17,0,0,0,0,4,13,0,0,0,0,0,0,34,0,27,27,0,0,14,7,14,0,0,0,0,14,7,14,0,0,27,27,0,34],[40,14,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[39,3,0,0,0,0,29,2,0,0,0,0,0,0,37,0,31,31,0,0,0,10,6,10,0,0,4,35,35,4,0,0,10,6,10,0] >;

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

M_4(2).F_5
% in TeX

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

G:=SmallGroup(320,239);
// by ID

G=gap.SmallGroup(320,239);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,136,1684,851,102,6278,3156]);
// 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,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^3>;
// generators/relations

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