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## G = C2.(D4×F5)  order 320 = 26·5

### 25th central stem extension by C2 of D4×F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2.(D4×F5)
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C2×C4×F5 — C2.(D4×F5)
 Lower central C5 — C2×C10 — C2.(D4×F5)
 Upper central C1 — C22 — C2×D4

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

Subgroups: 1258 in 258 conjugacy classes, 64 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×26], C5, C2×C4, C2×C4 [×19], D4 [×8], C23 [×2], C23 [×15], D5 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×5], C2×D4, C2×D4 [×7], C24 [×2], Dic5 [×2], C20 [×2], F5 [×6], D10 [×2], D10 [×4], D10 [×14], C2×C10, C2×C10 [×6], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, C4×D5 [×4], D20 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4 [×2], C2×F5 [×4], C2×F5 [×10], C22×D5, C22×D5 [×2], C22×D5 [×12], C22×C10 [×2], C24.3C22, C4×F5 [×2], C4⋊F5 [×2], C22⋊F5 [×8], C2×C4×D5, C2×D20, D4×D5 [×4], C2×C5⋊D4 [×2], D4×C10, C22×F5 [×4], C23×D5 [×2], C2×C4×F5, C2×C4⋊F5, C2×C22⋊F5 [×4], C2×D4×D5, C2.(D4×F5)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], F5, C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C2×F5 [×3], C24.3C22, 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 14)(2 15)(3 11)(4 12)(5 13)(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 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(1 24)(2 25)(3 21)(4 22)(5 23)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(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 42)(2 44 5 45)(3 41 4 43)(6 46 7 48)(8 50 10 49)(9 47)(11 51 12 53)(13 55 15 54)(14 52)(16 56 17 58)(18 60 20 59)(19 57)(21 61 22 63)(23 65 25 64)(24 62)(26 66 27 68)(28 70 30 69)(29 67)(31 71 32 73)(33 75 35 74)(34 72)(36 76 37 78)(38 80 40 79)(39 77)

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 8 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 C4○D4 F5 C2×F5 C2×F5 C22⋊F5 D4×F5 kernel C2.(D4×F5) C2×C4×F5 C2×C4⋊F5 C2×C22⋊F5 C2×D4×D5 C2×D20 C2×C5⋊D4 D4×C10 C4×D5 C2×F5 D10 C2×D4 C2×C4 C23 C4 C2 # reps 1 1 1 4 1 2 4 2 4 4 4 1 1 2 4 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
,
 28 5 0 0 0 0 7 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 28 5 0 0 0 0 32 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 6 0 0 0 0 0 0 40 6 0 0 0 0 35 35
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 6 40 0 0

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],[28,7,0,0,0,0,5,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,32,0,0,0,0,5,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,35,0,0,0,0,6,35],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;

C2.(D4×F5) in GAP, Magma, Sage, TeX

C_2.(D_4\times F_5)
% in TeX

G:=Group("C2.(D4xF5)");
// GroupNames label

G:=SmallGroup(320,1118);
// by ID

G=gap.SmallGroup(320,1118);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,6278,1595]);
// Polycyclic

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

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