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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, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C8.C4, C22×C8, C2×M4(2), C52C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C8.C4, C8×D5, C2×C52C8, C2×C40, D5⋊C8, C4.F5, C22.F5, C2×C4×D5, C40.C4, D10.Q8, D5×C2×C8, D5⋊M4(2), (C8×D5).C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, F5, C8.C4, C2×C4⋊C4, C2×F5, C2×C8.C4, C4⋊F5, 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 29 23 76 33)(10 30 24 77 34)(11 31 17 78 35)(12 32 18 79 36)(13 25 19 80 37)(14 26 20 73 38)(15 27 21 74 39)(16 28 22 75 40)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 76)(10 77)(11 78)(12 79)(13 80)(14 73)(15 74)(16 75)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(57 72)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(64 71)
(1 17 7 19 5 21 3 23)(2 20 8 22 6 24 4 18)(9 68 35 55 13 72 39 51)(10 71 36 50 14 67 40 54)(11 66 37 53 15 70 33 49)(12 69 38 56 16 65 34 52)(25 57 74 42 29 61 78 46)(26 60 75 45 30 64 79 41)(27 63 76 48 31 59 80 44)(28 58 77 43 32 62 73 47)

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

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

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

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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