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G = C2×C8×F5order 320 = 26·5

Direct product of C2×C8 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×C8×F5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4×F5 — C2×C4×F5 — C2×C8×F5
 Lower central C5 — C2×C8×F5
 Upper central C1 — C2×C8

Generators and relations for C2×C8×F5
G = < a,b,c,d | a2=b8=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 442 in 162 conjugacy classes, 92 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×6], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×17], C23, D5 [×4], C10, C10 [×2], C42 [×4], C2×C8, C2×C8 [×11], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×8], D10 [×2], D10 [×4], C2×C10, C4×C8 [×4], C2×C42, C22×C8 [×2], C52C8 [×2], C40 [×2], C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×12], C22×D5, C2×C4×C8, C8×D5 [×4], C2×C52C8, C2×C40, D5⋊C8 [×4], C4×F5 [×4], C2×C5⋊C8 [×2], C2×C4×D5, C22×F5 [×2], C8×F5 [×4], D5×C2×C8, C2×D5⋊C8, C2×C4×F5, C2×C8×F5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], C22×C4 [×3], F5, C4×C8 [×4], C2×C42, C22×C8 [×2], C2×F5 [×3], C2×C4×C8, C4×F5 [×2], C22×F5, C8×F5 [×2], C2×C4×F5, C2×C8×F5

Smallest permutation representation of C2×C8×F5
On 80 points
Generators in S80
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 71)(10 72)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 73)(48 74)
(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 65 42 51 59)(2 66 43 52 60)(3 67 44 53 61)(4 68 45 54 62)(5 69 46 55 63)(6 70 47 56 64)(7 71 48 49 57)(8 72 41 50 58)(9 74 25 33 17)(10 75 26 34 18)(11 76 27 35 19)(12 77 28 36 20)(13 78 29 37 21)(14 79 30 38 22)(15 80 31 39 23)(16 73 32 40 24)
(1 5)(2 6)(3 7)(4 8)(9 78 33 29)(10 79 34 30)(11 80 35 31)(12 73 36 32)(13 74 37 25)(14 75 38 26)(15 76 39 27)(16 77 40 28)(17 21)(18 22)(19 23)(20 24)(41 62 50 68)(42 63 51 69)(43 64 52 70)(44 57 53 71)(45 58 54 72)(46 59 55 65)(47 60 56 66)(48 61 49 67)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4X 5 8A ··· 8H 8I ··· 8AF 10A 10B 10C 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 5 8 ··· 8 8 ··· 8 10 10 10 20 20 20 20 40 ··· 40 size 1 1 1 1 5 5 5 5 1 1 1 1 5 ··· 5 4 1 ··· 1 5 ··· 5 4 4 4 4 4 4 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 C4 C4 C4 C8 F5 C2×F5 C2×F5 C4×F5 C4×F5 C8×F5 kernel C2×C8×F5 C8×F5 D5×C2×C8 C2×D5⋊C8 C2×C4×F5 C8×D5 C2×C5⋊2C8 C2×C40 D5⋊C8 C4×F5 C2×C5⋊C8 C22×F5 C2×F5 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 4 4 4 4 32 1 2 1 2 2 8

Matrix representation of C2×C8×F5 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 9 0 0 0 0 0 27 0 0 0 0 0 27 0 0 0 0 0 27 0 0 0 0 0 27
,
 1 0 0 0 0 0 40 40 40 40 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 9 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 40 0 0 0 1 1 1 1

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[9,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,27],[1,0,0,0,0,0,40,1,0,0,0,40,0,1,0,0,40,0,0,1,0,40,0,0,0],[9,0,0,0,0,0,40,0,0,1,0,0,0,40,1,0,0,0,0,1,0,0,40,0,1] >;

C2×C8×F5 in GAP, Magma, Sage, TeX

C_2\times C_8\times F_5
% in TeX

G:=Group("C2xC8xF5");
// GroupNames label

G:=SmallGroup(320,1054);
// by ID

G=gap.SmallGroup(320,1054);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,100,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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