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## G = (C2×C8)⋊F5order 320 = 26·5

### 1st semidirect product of C2×C8 and F5 acting via F5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×C8)⋊F5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C2×C4×D5 — D10.C23 — (C2×C8)⋊F5
 Lower central C5 — C10 — C2×C10 — (C2×C8)⋊F5
 Upper central C1 — C4 — C2×C4 — C2×C8

Generators and relations for (C2×C8)⋊F5
G = < a,b,c,d | a2=b8=c5=d4=1, ab=ba, ac=ca, dad-1=ab4, bc=cb, dbd-1=ab5, dcd-1=c3 >

Subgroups: 370 in 90 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×4], C22, C22 [×3], C5, C8 [×4], C2×C4, C2×C4 [×7], C23, D5 [×2], C10, C10, C42, C22⋊C4 [×2], C4⋊C4, C2×C8, C2×C8 [×2], M4(2) [×5], C22×C4, Dic5 [×2], C20 [×2], F5 [×2], D10 [×2], D10, C2×C10, C42⋊C2, C2×M4(2) [×2], C52C8, C40, C5⋊C8 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×2], C22×D5, M4(2)⋊4C4, C8⋊D5 [×2], C2×C52C8, C2×C40, D5⋊C8, C4.F5, C4×F5, C4⋊F5, C22.F5 [×2], C22⋊F5 [×2], C2×C4×D5, C2×C8⋊D5, D5⋊M4(2), D10.C23, (C2×C8)⋊F5
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], F5, C2.C42, C2×F5, M4(2)⋊4C4, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, (C2×C8)⋊F5

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 type + + + + + - + + + image C1 C2 C2 C2 C4 C4 C4 C4 D4 Q8 F5 C2×F5 M4(2)⋊4C4 C4⋊F5 C22⋊F5 C4×F5 (C2×C8)⋊F5 kernel (C2×C8)⋊F5 C2×C8⋊D5 D5⋊M4(2) D10.C23 C2×C5⋊2C8 C2×C40 C22.F5 C22⋊F5 C4×D5 C4×D5 C2×C8 C2×C4 C5 C4 C4 C22 C1 # reps 1 1 1 1 2 2 4 4 3 1 1 1 2 2 2 2 8

Matrix representation of (C2×C8)⋊F5 in GL4(𝔽41) generated by

 19 0 38 38 3 22 3 0 0 3 22 3 38 38 0 19
,
 9 18 21 38 3 12 21 24 17 20 29 38 3 20 23 32
,
 40 40 40 40 1 0 0 0 0 1 0 0 0 0 1 0
,
 9 0 0 0 0 0 0 9 0 9 0 0 32 32 32 32
G:=sub<GL(4,GF(41))| [19,3,0,38,0,22,3,38,38,3,22,0,38,0,3,19],[9,3,17,3,18,12,20,20,21,21,29,23,38,24,38,32],[40,1,0,0,40,0,1,0,40,0,0,1,40,0,0,0],[9,0,0,32,0,0,9,32,0,0,0,32,0,9,0,32] >;

(C2×C8)⋊F5 in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes F_5
% in TeX

G:=Group("(C2xC8):F5");
// GroupNames label

G:=SmallGroup(320,232);
// by ID

G=gap.SmallGroup(320,232);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,136,1684,6278,3156]);
// Polycyclic

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

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