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## G = M4(2)⋊4Dic5order 320 = 26·5

### 4th semidirect product of M4(2) and Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — M4(2)⋊4Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C22×C20 — C2×C4.Dic5 — M4(2)⋊4Dic5
 Lower central C5 — C10 — C2×C10 — M4(2)⋊4Dic5
 Upper central C1 — C4 — C22×C4 — C2×M4(2)

Generators and relations for M4(2)⋊4Dic5
G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=cac-1=a5, dad-1=a5b, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 262 in 90 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, C2×C10, C2×C10, C42⋊C2, C2×M4(2), C2×M4(2), C52C8, C40, C2×Dic5, C2×C20, C22×C10, M4(2)⋊4C4, C2×C52C8, C4.Dic5, C4×Dic5, C4⋊Dic5, C23.D5, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C2×C4.Dic5, C23.21D10, C10×M4(2), M4(2)⋊4Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, M4(2)⋊4C4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C10.10C42, M4(2)⋊4Dic5

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 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 4 4 4 4 5 5 8 8 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 20 20 20 20 2 2 4 4 4 4 20 20 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 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - + - + - + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D5 Dic5 D10 Dic10 C4×D5 D20 C5⋊D4 C4×D5 M4(2)⋊4C4 M4(2)⋊4Dic5 kernel M4(2)⋊4Dic5 C2×C4.Dic5 C23.21D10 C10×M4(2) C2×C5⋊2C8 C23.D5 C5×M4(2) C2×C20 C2×C20 C2×M4(2) M4(2) C22×C4 C2×C4 C2×C4 C2×C4 C2×C4 C23 C5 C1 # reps 1 1 1 1 4 4 4 3 1 2 4 2 4 4 4 8 4 2 8

Matrix representation of M4(2)⋊4Dic5 in GL4(𝔽41) generated by

 0 0 18 6 0 0 35 23 9 0 0 0 0 9 0 0
,
 18 6 0 0 35 23 0 0 0 0 23 35 0 0 6 18
,
 6 1 0 0 40 0 0 0 0 0 35 40 0 0 1 0
,
 39 16 0 0 28 2 0 0 0 0 32 28 0 0 0 9
G:=sub<GL(4,GF(41))| [0,0,9,0,0,0,0,9,18,35,0,0,6,23,0,0],[18,35,0,0,6,23,0,0,0,0,23,6,0,0,35,18],[6,40,0,0,1,0,0,0,0,0,35,1,0,0,40,0],[39,28,0,0,16,2,0,0,0,0,32,0,0,0,28,9] >;

M4(2)⋊4Dic5 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_4{\rm Dic}_5
% in TeX

G:=Group("M4(2):4Dic5");
// GroupNames label

G:=SmallGroup(320,117);
// by ID

G=gap.SmallGroup(320,117);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,1123,136,851,12550]);
// Polycyclic

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

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