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## G = C2×D5×D8order 320 = 26·5

### Direct product of C2, D5 and D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D5×D8
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×D4×D5 — C2×D5×D8
 Lower central C5 — C10 — C20 — C2×D5×D8
 Upper central C1 — C22 — C2×C4 — C2×D8

Generators and relations for C2×D5×D8
G = < a,b,c,d,e | a2=b5=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1694 in 338 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×2], C4 [×2], C22, C22 [×38], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], D4 [×4], D4 [×16], C23 [×21], D5 [×4], D5 [×4], C10, C10 [×2], C10 [×4], C2×C8, C2×C8 [×5], D8 [×4], D8 [×12], C22×C4, C2×D4 [×2], C2×D4 [×16], C24 [×2], Dic5 [×2], C20 [×2], D10 [×6], D10 [×24], C2×C10, C2×C10 [×8], C22×C8, C2×D8, C2×D8 [×11], C22×D4 [×2], C52C8 [×2], C40 [×2], C4×D5 [×4], D20 [×4], D20 [×2], C2×Dic5, C5⋊D4 [×8], C2×C20, C5×D4 [×4], C5×D4 [×2], C22×D5, C22×D5 [×18], C22×C10 [×2], C22×D8, C8×D5 [×4], D40 [×4], C2×C52C8, D4⋊D5 [×8], C2×C40, C5×D8 [×4], C2×C4×D5, C2×D20 [×2], D4×D5 [×8], D4×D5 [×4], C2×C5⋊D4 [×2], D4×C10 [×2], C23×D5 [×2], D5×C2×C8, C2×D40, D5×D8 [×8], C2×D4⋊D5 [×2], C10×D8, C2×D4×D5 [×2], C2×D5×D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, D8 [×4], C2×D4 [×6], C24, D10 [×7], C2×D8 [×6], C22×D4, C22×D5 [×7], C22×D8, D4×D5 [×2], C23×D5, D5×D8 [×2], C2×D4×D5, C2×D5×D8

Smallest permutation representation of C2×D5×D8
On 80 points
Generators in S80
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 57)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 54)(18 55)(19 56)(20 49)(21 50)(22 51)(23 52)(24 53)(25 77)(26 78)(27 79)(28 80)(29 73)(30 74)(31 75)(32 76)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)
(1 36 71 25 20)(2 37 72 26 21)(3 38 65 27 22)(4 39 66 28 23)(5 40 67 29 24)(6 33 68 30 17)(7 34 69 31 18)(8 35 70 32 19)(9 79 51 60 48)(10 80 52 61 41)(11 73 53 62 42)(12 74 54 63 43)(13 75 55 64 44)(14 76 56 57 45)(15 77 49 58 46)(16 78 50 59 47)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(41 80)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)
(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)
(2 8)(3 7)(4 6)(9 13)(10 12)(14 16)(17 23)(18 22)(19 21)(26 32)(27 31)(28 30)(33 39)(34 38)(35 37)(41 43)(44 48)(45 47)(50 56)(51 55)(52 54)(57 59)(60 64)(61 63)(65 69)(66 68)(70 72)(74 80)(75 79)(76 78)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 10 ··· 10 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 4 4 5 5 5 5 20 20 20 20 2 2 10 10 2 2 2 2 2 2 10 10 10 10 2 ··· 2 8 ··· 8 4 4 4 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D8 D10 D10 D10 D4×D5 D4×D5 D5×D8 kernel C2×D5×D8 D5×C2×C8 C2×D40 D5×D8 C2×D4⋊D5 C10×D8 C2×D4×D5 C4×D5 C2×Dic5 C22×D5 C2×D8 D10 C2×C8 D8 C2×D4 C4 C22 C2 # reps 1 1 1 8 2 1 2 2 1 1 2 8 2 8 4 2 2 8

Matrix representation of C2×D5×D8 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 40 34 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 17 24 0 0 29 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 40
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,1,34,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,17,29,0,0,24,0],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,40] >;

C2×D5×D8 in GAP, Magma, Sage, TeX

C_2\times D_5\times D_8
% in TeX

G:=Group("C2xD5xD8");
// GroupNames label

G:=SmallGroup(320,1426);
// by ID

G=gap.SmallGroup(320,1426);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,185,438,235,102,12550]);
// Polycyclic

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

׿
×
𝔽