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## G = (C8×D5).C4order 320 = 26·5

### 6th non-split extension by C8×D5 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — (C8×D5).C4
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — C4.F5 — D5⋊M4(2) — (C8×D5).C4
 Lower central C5 — C10 — C20 — (C8×D5).C4
 Upper central C1 — C4 — C2×C4 — C2×C8

Generators and relations for (C8×D5).C4
G = < a,b,c,d | a8=b5=c2=1, d4=a4, ab=ba, ac=ca, dad-1=a3, cbc=b-1, dbd-1=b3, dcd-1=b2c >

Subgroups: 346 in 106 conjugacy classes, 50 normal (38 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×5], C23, D5 [×2], D5, C10, C10, C2×C8, C2×C8 [×7], M4(2) [×6], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C8.C4 [×4], C22×C8, C2×M4(2) [×2], C52C8 [×2], C40 [×2], C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×C8.C4, C8×D5 [×4], C2×C52C8, C2×C40, D5⋊C8 [×2], C4.F5 [×4], C22.F5 [×2], C2×C4×D5, C40.C4 [×2], D10.Q8 [×2], D5×C2×C8, D5⋊M4(2) [×2], (C8×D5).C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, C8.C4 [×2], C2×C4⋊C4, C2×F5 [×3], C2×C8.C4, C4⋊F5 [×2], C22×F5, C2×C4⋊F5, (C8×D5).C4

Smallest permutation representation of (C8×D5).C4
On 80 points
Generators in S80
(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 68 48 61 49)(2 69 41 62 50)(3 70 42 63 51)(4 71 43 64 52)(5 72 44 57 53)(6 65 45 58 54)(7 66 46 59 55)(8 67 47 60 56)(9 26 74 39 20)(10 27 75 40 21)(11 28 76 33 22)(12 29 77 34 23)(13 30 78 35 24)(14 31 79 36 17)(15 32 80 37 18)(16 25 73 38 19)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 79)(18 80)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(57 72)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(64 71)
(1 30 7 32 5 26 3 28)(2 25 8 27 6 29 4 31)(9 63 76 48 13 59 80 44)(10 58 77 43 14 62 73 47)(11 61 78 46 15 57 74 42)(12 64 79 41 16 60 75 45)(17 69 38 56 21 65 34 52)(18 72 39 51 22 68 35 55)(19 67 40 54 23 71 36 50)(20 70 33 49 24 66 37 53)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + - + - + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 Q8 D4 Q8 C8.C4 F5 C2×F5 C2×F5 C4⋊F5 C4⋊F5 (C8×D5).C4 kernel (C8×D5).C4 C40.C4 D10.Q8 D5×C2×C8 D5⋊M4(2) C8×D5 C2×C5⋊2C8 C2×C40 C4×D5 C4×D5 C2×Dic5 C22×D5 D5 C2×C8 C8 C2×C4 C4 C22 C1 # reps 1 2 2 1 2 4 2 2 1 1 1 1 8 1 2 1 2 2 8

Matrix representation of (C8×D5).C4 in GL4(𝔽41) generated by

 38 0 0 0 0 38 0 0 37 3 14 0 35 0 0 14
,
 0 40 0 0 1 6 0 0 7 13 40 35 20 20 6 35
,
 35 40 0 0 35 6 0 0 24 13 40 0 13 20 6 1
,
 27 31 39 0 20 0 0 39 25 2 14 10 40 37 21 0
G:=sub<GL(4,GF(41))| [38,0,37,35,0,38,3,0,0,0,14,0,0,0,0,14],[0,1,7,20,40,6,13,20,0,0,40,6,0,0,35,35],[35,35,24,13,40,6,13,20,0,0,40,6,0,0,0,1],[27,20,25,40,31,0,2,37,39,0,14,21,0,39,10,0] >;

(C8×D5).C4 in GAP, Magma, Sage, TeX

(C_8\times D_5).C_4
% in TeX

G:=Group("(C8xD5).C4");
// GroupNames label

G:=SmallGroup(320,1062);
// by ID

G=gap.SmallGroup(320,1062);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,100,136,1684,102,6278,1595]);
// Polycyclic

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

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