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## G = D4⋊F5⋊C2order 320 = 26·5

### 6th semidirect product of D4⋊F5 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D4⋊F5⋊C2
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — C4.F5 — C2×C4.F5 — D4⋊F5⋊C2
 Lower central C5 — C10 — C20 — D4⋊F5⋊C2
 Upper central C1 — C2 — C2×C4 — C4○D4

Generators and relations for D4⋊F5⋊C2
G = < a,b,c,d,e | a4=b2=c5=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=a-1b, ebe=a2b, dcd-1=c3, ce=ec, ede=a2d >

Subgroups: 682 in 154 conjugacy classes, 48 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, F5, D10, D10, C2×C10, C2×C10, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C5⋊C8, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×F5, C22×D5, C22×D5, C42⋊C22, C4.F5, C4.F5, C4×F5, C4⋊F5, C2×C5⋊C8, C22⋊F5, C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, D4⋊F5, Q82F5, C2×C4.F5, D10.C23, D5×C4○D4, D4⋊F5⋊C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, F5, C2×C22⋊C4, C2×F5, C42⋊C22, C22⋊F5, C22×F5, C2×C22⋊F5, D4⋊F5⋊C2

Smallest permutation representation of D4⋊F5⋊C2
On 80 points
Generators in S80
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(1 39)(2 40)(3 36)(4 37)(5 38)(6 31)(7 32)(8 33)(9 34)(10 35)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 66)(52 67)(53 68)(54 69)(55 70)(56 61)(57 62)(58 63)(59 64)(60 65)
(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 49 9 44)(2 46 8 42)(3 48 7 45)(4 50 6 43)(5 47 10 41)(11 58 17 55)(12 60 16 53)(13 57 20 51)(14 59 19 54)(15 56 18 52)(21 78 22 80)(23 77 25 76)(24 79)(26 73 27 75)(28 72 30 71)(29 74)(31 63 32 65)(33 62 35 61)(34 64)(36 68 37 70)(38 67 40 66)(39 69)
(1 44)(2 45)(3 41)(4 42)(5 43)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G ··· 4K 5 8A 8B 8C 8D 10A 10B 10C 10D 20A 20B 20C 20D 20E order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 5 8 8 8 8 10 10 10 10 20 20 20 20 20 size 1 1 2 4 10 10 20 2 2 4 5 5 10 20 ··· 20 4 20 20 20 20 4 8 8 8 4 4 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D4 D4 F5 C2×F5 C2×F5 C2×F5 C42⋊C22 C22⋊F5 C22⋊F5 D4⋊F5⋊C2 kernel D4⋊F5⋊C2 D4⋊F5 Q8⋊2F5 C2×C4.F5 D10.C23 D5×C4○D4 C4○D20 D4×D5 Q8×D5 C5×C4○D4 C4×D5 C2×Dic5 C22×D5 C4○D4 C2×C4 D4 Q8 C5 C4 C22 C1 # reps 1 2 2 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2 2

Matrix representation of D4⋊F5⋊C2 in GL8(𝔽41)

 32 0 0 0 0 0 0 0 24 9 0 0 0 0 0 0 0 0 32 0 0 0 0 0 37 0 27 9 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 3 33 0 0 0 0 0 0 1 38 0 0 0 0 0 0 29 4 35 37 0 0 0 0 10 15 19 6 0 0 0 0 0 0 0 0 19 0 38 38 0 0 0 0 3 22 3 0 0 0 0 0 0 3 22 3 0 0 0 0 38 38 0 19
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 40 40 40
,
 40 0 39 0 0 0 0 0 13 0 23 9 0 0 0 0 1 0 1 0 0 0 0 0 25 32 14 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 0 1 1 1 1
,
 40 0 39 0 0 0 0 0 33 0 11 1 0 0 0 0 0 0 1 0 0 0 0 0 33 1 14 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40

G:=sub<GL(8,GF(41))| [32,24,0,37,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,27,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[3,1,29,10,0,0,0,0,33,38,4,15,0,0,0,0,0,0,35,19,0,0,0,0,0,0,37,6,0,0,0,0,0,0,0,0,19,3,0,38,0,0,0,0,0,22,3,38,0,0,0,0,38,3,22,0,0,0,0,0,38,0,3,19],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,1,40],[40,13,1,25,0,0,0,0,0,0,0,32,0,0,0,0,39,23,1,14,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,1,0,0,0,0,0,40,0,1],[40,33,0,33,0,0,0,0,0,0,0,1,0,0,0,0,39,11,1,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;

D4⋊F5⋊C2 in GAP, Magma, Sage, TeX

D_4\rtimes F_5\rtimes C_2
% in TeX

G:=Group("D4:F5:C2");
// GroupNames label

G:=SmallGroup(320,1133);
// by ID

G=gap.SmallGroup(320,1133);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,136,1684,438,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=c^5=d^4=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=a^-1*b,e*b*e=a^2*b,d*c*d^-1=c^3,c*e=e*c,e*d*e=a^2*d>;
// generators/relations

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