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## G = M4(2).Dic5order 320 = 26·5

### 1st non-split extension by M4(2) of Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — M4(2).Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4.Dic5 — C2×C4.Dic5 — M4(2).Dic5
 Lower central C5 — C10 — C20 — M4(2).Dic5
 Upper central C1 — C4 — C22×C4 — C2×M4(2)

Generators and relations for M4(2).Dic5
G = < a,b,c,d | a8=b2=1, c10=a4, d2=c5, bab=a5, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c9 >

Subgroups: 238 in 102 conjugacy classes, 67 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, C10, C10, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C8.C4, C2×M4(2), C2×M4(2), C52C8, C40, C2×C20, C2×C20, C22×C10, M4(2).C4, C2×C52C8, C4.Dic5, C4.Dic5, C2×C40, C5×M4(2), C22×C20, C40.6C4, C2×C4.Dic5, C10×M4(2), M4(2).Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, D10, C2×C4⋊C4, Dic10, D20, C2×Dic5, C22×D5, M4(2).C4, C4⋊Dic5, C2×Dic10, C2×D20, C22×Dic5, C2×C4⋊Dic5, M4(2).Dic5

Smallest permutation representation of M4(2).Dic5
On 80 points
Generators in S80
(1 26 6 31 11 36 16 21)(2 27 7 32 12 37 17 22)(3 28 8 33 13 38 18 23)(4 29 9 34 14 39 19 24)(5 30 10 35 15 40 20 25)(41 66 56 61 51 76 46 71)(42 67 57 62 52 77 47 72)(43 68 58 63 53 78 48 73)(44 69 59 64 54 79 49 74)(45 70 60 65 55 80 50 75)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 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)
(1 46 6 51 11 56 16 41)(2 55 7 60 12 45 17 50)(3 44 8 49 13 54 18 59)(4 53 9 58 14 43 19 48)(5 42 10 47 15 52 20 57)(21 71 26 76 31 61 36 66)(22 80 27 65 32 70 37 75)(23 69 28 74 33 79 38 64)(24 78 29 63 34 68 39 73)(25 67 30 72 35 77 40 62)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 8E ··· 8L 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 4 4 4 4 4 5 5 8 8 8 8 8 ··· 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 2 2 2 1 1 2 2 2 2 2 4 4 4 4 20 ··· 20 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - - + + - + - + - image C1 C2 C2 C2 C4 D4 Q8 Q8 D5 D10 Dic5 D10 Dic10 D20 Dic10 M4(2).C4 M4(2).Dic5 kernel M4(2).Dic5 C40.6C4 C2×C4.Dic5 C10×M4(2) C5×M4(2) C2×C20 C2×C20 C22×C10 C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C2×C4 C23 C5 C1 # reps 1 4 2 1 8 2 1 1 2 4 8 2 4 8 4 2 8

Matrix representation of M4(2).Dic5 in GL4(𝔽41) generated by

 0 1 0 0 32 0 0 0 36 29 0 9 15 0 1 0
,
 1 0 0 0 0 40 0 0 0 4 1 0 37 0 0 40
,
 33 0 0 0 0 33 0 0 20 6 36 0 6 21 0 36
,
 33 14 7 0 14 8 0 7 26 0 8 27 0 26 27 33
G:=sub<GL(4,GF(41))| [0,32,36,15,1,0,29,0,0,0,0,1,0,0,9,0],[1,0,0,37,0,40,4,0,0,0,1,0,0,0,0,40],[33,0,20,6,0,33,6,21,0,0,36,0,0,0,0,36],[33,14,26,0,14,8,0,26,7,0,8,27,0,7,27,33] >;

M4(2).Dic5 in GAP, Magma, Sage, TeX

M_4(2).{\rm Dic}_5
% in TeX

G:=Group("M4(2).Dic5");
// GroupNames label

G:=SmallGroup(320,752);
// by ID

G=gap.SmallGroup(320,752);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,100,136,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^10=a^4,d^2=c^5,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^9>;
// generators/relations

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