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## G = C2×D4⋊F5order 320 = 26·5

### Direct product of C2 and D4⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D4⋊F5
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — C4.F5 — C2×C4.F5 — C2×D4⋊F5
 Lower central C5 — C10 — C20 — C2×D4⋊F5
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for C2×D4⋊F5
G = < a,b,c,d,e | a2=b4=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b-1c, ede-1=d3 >

Subgroups: 682 in 170 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×8], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×16], D4 [×2], D4 [×5], Q8 [×3], C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C42 [×3], C2×C8, M4(2) [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C4○D4 [×6], Dic5 [×2], Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C4≀C2 [×4], C2×C42, C2×M4(2), C2×C4○D4, C5⋊C8 [×2], Dic10 [×2], Dic10, C4×D5 [×4], C2×Dic5, C2×Dic5 [×5], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C5×D4, C2×F5 [×6], C22×D5, C22×C10, C2×C4≀C2, C4.F5 [×2], C4.F5, C4×F5 [×2], C4×F5, C2×C5⋊C8, C2×Dic10, C2×C4×D5, D42D5 [×4], D42D5 [×2], C22×Dic5, C2×C5⋊D4, D4×C10, C22×F5, D4⋊F5 [×4], C2×C4.F5, C2×C4×F5, C2×D42D5, C2×D4⋊F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C4≀C2 [×2], C2×C22⋊C4, C2×F5 [×3], C2×C4≀C2, C22⋊F5 [×2], C22×F5, D4⋊F5 [×2], C2×C22⋊F5, C2×D4⋊F5

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 8 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D4 C4≀C2 F5 C2×F5 C2×F5 C22⋊F5 C22⋊F5 D4⋊F5 kernel C2×D4⋊F5 D4⋊F5 C2×C4.F5 C2×C4×F5 C2×D4⋊2D5 C2×Dic10 D4⋊2D5 D4×C10 C4×D5 C2×Dic5 C22×D5 C10 C2×D4 C2×C4 D4 C4 C22 C2 # reps 1 4 1 1 1 2 4 2 2 1 1 8 1 1 2 2 2 2

Matrix representation of C2×D4⋊F5 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 9 0 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 32 0 0 0 0 9 0 0 0 0 0 0 0 19 0 38 38 0 0 3 22 3 0 0 0 0 3 22 3 0 0 38 38 0 19
,
 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
,
 40 0 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 40 0 0 0 0 1 1 1 1

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,19,3,0,38,0,0,0,22,3,38,0,0,38,3,22,0,0,0,38,0,3,19],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40],[40,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,1,0,0,0,0,40,1,0,0,0,0,0,1,0,0,0,40,0,1] >;

C2×D4⋊F5 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes F_5
% in TeX

G:=Group("C2xD4:F5");
// GroupNames label

G:=SmallGroup(320,1106);
// by ID

G=gap.SmallGroup(320,1106);
# by ID

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

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

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