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## G = M4(2)⋊D10order 320 = 26·5

### 4th semidirect product of M4(2) and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — M4(2)⋊D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D4⋊8D10 — M4(2)⋊D10
 Lower central C5 — C10 — C2×C20 — M4(2)⋊D10
 Upper central C1 — C2 — C2×C4 — C4≀C2

Generators and relations for M4(2)⋊D10
G = < a,b,c,d | a8=b2=c10=d2=1, bab=a5, cac-1=a-1b, dad=ab, cbc-1=a4b, bd=db, dcd=c-1 >

Subgroups: 782 in 152 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×6], D4, D4 [×9], Q8, Q8 [×3], C23 [×4], D5 [×3], C10, C10 [×2], C42, C22⋊C4 [×2], M4(2), M4(2), SD16 [×2], Q16 [×2], C2×D4 [×5], C2×Q8, C4○D4, C4○D4 [×3], Dic5 [×2], C20 [×2], C20 [×2], D10 [×7], C2×C10, C2×C10, C4.D4, C4≀C2, C4≀C2, C4.4D4, C8.C22 [×2], 2+ 1+4, C52C8, C40, Dic10, Dic10 [×2], C4×D5 [×3], D20, D20 [×4], C2×Dic5, C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C22×D5 [×2], C22×D5 [×2], D4.9D4, C40⋊C2, Dic20, C4.Dic5, D10⋊C4 [×2], D4.D5, C5⋊Q16, C4×C20, C5×M4(2), C2×Dic10, C2×D20, C2×D20, C4○D20, C4○D20, D4×D5 [×3], Q82D5, C5×C4○D4, D204C4, C20.46D4, C5×C4≀C2, C4.D20, C8.D10, D4.9D10, D48D10, M4(2)⋊D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, D20 [×2], C22×D5, D4.9D4, C2×D20, D4×D5 [×2], C22⋊D20, M4(2)⋊D10

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

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 5A 5B 8A 8B 10A 10B 10C 10D 10E 10F 20A 20B 20C 20D 20E ··· 20N 20O 20P 40A 40B 40C 40D order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 5 5 8 8 10 10 10 10 10 10 20 20 20 20 20 ··· 20 20 20 40 40 40 40 size 1 1 2 4 20 20 20 2 2 4 4 4 20 40 2 2 8 40 2 2 4 4 8 8 2 2 2 2 4 ··· 4 8 8 8 8 8 8

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4 D5 D10 D10 D10 D20 D20 D4.9D4 D4×D5 D4×D5 M4(2)⋊D10 kernel M4(2)⋊D10 D20⋊4C4 C20.46D4 C5×C4≀C2 C4.D20 C8.D10 D4.9D10 D4⋊8D10 Dic10 D20 C5×D4 C5×Q8 C22×D5 C4≀C2 C42 M4(2) C4○D4 D4 Q8 C5 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 2 2 2 8

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

 25 38 16 3 3 7 38 34 16 3 16 3 38 34 38 34
,
 0 0 39 28 0 0 13 2 2 13 0 0 28 39 0 0
,
 35 35 0 0 6 40 0 0 0 0 6 6 0 0 35 1
,
 35 35 0 0 40 6 0 0 0 0 6 6 0 0 1 35
G:=sub<GL(4,GF(41))| [25,3,16,38,38,7,3,34,16,38,16,38,3,34,3,34],[0,0,2,28,0,0,13,39,39,13,0,0,28,2,0,0],[35,6,0,0,35,40,0,0,0,0,6,35,0,0,6,1],[35,40,0,0,35,6,0,0,0,0,6,1,0,0,6,35] >;

M4(2)⋊D10 in GAP, Magma, Sage, TeX

M_4(2)\rtimes D_{10}
% in TeX

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

G:=SmallGroup(320,452);
// by ID

G=gap.SmallGroup(320,452);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,58,1123,136,851,438,102,12550]);
// Polycyclic

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

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