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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(C4×F5), D5⋊C8.4C4, C4.F5.1C4, D10.7(C4⋊C4), (C4×D5).118D4, D5.(C8.C4), C4.Dic5.1C4, (C22×D5).8Q8, C52(C4.C42), D5⋊M4(2).2C2, (D5×M4(2)).5C2, (C5×M4(2)).3C4, C22.11(C4⋊F5), C4.41(C22⋊F5), C20.39(C22⋊C4), (C2×Dic5).100D4, Dic5.33(C22⋊C4), C2.16(D10.3Q8), C10.15(C2.C42), (C2×D5⋊C8).2C2, (C2×C4).70(C2×F5), (C2×C10).4(C4⋊C4), (C2×C20).37(C2×C4), (C4×D5).15(C2×C4), (C2×C4×D5).188C22, SmallGroup(320,239)

Series: Derived Chief Lower central Upper central

C1C20 — M4(2).F5
C1C5C10Dic5C4×D5C2×C4×D5D5⋊M4(2) — M4(2).F5
C5C10C20 — M4(2).F5
C1C4C2×C4M4(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 [×4], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×6], C2×C4, C2×C4 [×5], C23, D5 [×2], D5, C10, C10, C2×C8 [×6], M4(2), M4(2) [×5], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C22×C8, C2×M4(2) [×2], C52C8, C40, C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C4.C42, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), D5⋊C8 [×2], D5⋊C8 [×2], C4.F5 [×2], C2×C5⋊C8, C22.F5, C2×C4×D5, D5×M4(2), C2×D5⋊C8, D5⋊M4(2), M4(2).F5
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], F5, C2.C42, C8.C4 [×2], C2×F5, C4.C42, C4×F5, 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)(25 29)(27 31)(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 55 61 44)(2 66 56 62 45)(3 67 49 63 46)(4 68 50 64 47)(5 69 51 57 48)(6 70 52 58 41)(7 71 53 59 42)(8 72 54 60 43)(9 25 23 80 33)(10 26 24 73 34)(11 27 17 74 35)(12 28 18 75 36)(13 29 19 76 37)(14 30 20 77 38)(15 31 21 78 39)(16 32 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 57 74 53 29 61 78 49)(26 60 75 56 30 64 79 52)(27 59 76 55 31 63 80 51)(28 62 77 50 32 58 73 54)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111111112222444448
type++++++-+++
imageC1C2C2C2C4C4C4C4D4D4Q8C8.C4F5C2×F5C4×F5C22⋊F5C4⋊F5M4(2).F5
kernelM4(2).F5D5×M4(2)C2×D5⋊C8D5⋊M4(2)C4.Dic5C5×M4(2)D5⋊C8C4.F5C4×D5C2×Dic5C22×D5D5M4(2)C2×C4C4C4C22C1
# 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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