Copied to
clipboard

## G = C5×C23.9D4order 320 = 26·5

### Direct product of C5 and C23.9D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C23.9D4
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C10 — C10×C22⋊C4 — C5×C23.9D4
 Lower central C1 — C2 — C22 — C5×C23.9D4
 Upper central C1 — C2×C10 — C23×C10 — C5×C23.9D4

Generators and relations for C5×C23.9D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=1, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=bde-1 >

Subgroups: 298 in 142 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×6], C22 [×3], C22 [×4], C22 [×10], C5, C2×C4 [×12], C23 [×3], C23 [×4], C23 [×2], C10, C10 [×2], C10 [×6], C22⋊C4 [×4], C22⋊C4 [×4], C22×C4 [×2], C22×C4 [×2], C24, C20 [×6], C2×C10 [×3], C2×C10 [×4], C2×C10 [×10], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C20 [×12], C22×C10 [×3], C22×C10 [×4], C22×C10 [×2], C23.9D4, C5×C22⋊C4 [×4], C5×C22⋊C4 [×4], C22×C20 [×2], C22×C20 [×2], C23×C10, C10×C22⋊C4, C10×C22⋊C4 [×2], C5×C23.9D4
Quotients: C1, C2 [×3], C4 [×6], C22, C5, C2×C4 [×3], D4 [×3], Q8, C10 [×3], C42, C22⋊C4 [×3], C4⋊C4 [×3], C20 [×6], C2×C10, C2.C42, C23⋊C4 [×2], C2×C20 [×3], C5×D4 [×3], C5×Q8, C23.9D4, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C5×C2.C42, C5×C23⋊C4 [×2], C5×C23.9D4

Smallest permutation representation of C5×C23.9D4
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 42)(2 43)(3 44)(4 45)(5 41)(6 60)(7 56)(8 57)(9 58)(10 59)(11 53)(12 54)(13 55)(14 51)(15 52)(16 61)(17 62)(18 63)(19 64)(20 65)(21 75)(22 71)(23 72)(24 73)(25 74)(26 69)(27 70)(28 66)(29 67)(30 68)(31 76)(32 77)(33 78)(34 79)(35 80)(36 46)(37 47)(38 48)(39 49)(40 50)
(1 42)(2 43)(3 44)(4 45)(5 41)(6 19)(7 20)(8 16)(9 17)(10 18)(11 53)(12 54)(13 55)(14 51)(15 52)(21 34)(22 35)(23 31)(24 32)(25 33)(26 69)(27 70)(28 66)(29 67)(30 68)(36 46)(37 47)(38 48)(39 49)(40 50)(56 65)(57 61)(58 62)(59 63)(60 64)(71 80)(72 76)(73 77)(74 78)(75 79)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 64)(7 65)(8 61)(9 62)(10 63)(16 57)(17 58)(18 59)(19 60)(20 56)(21 79)(22 80)(23 76)(24 77)(25 78)(26 50)(27 46)(28 47)(29 48)(30 49)(31 72)(32 73)(33 74)(34 75)(35 71)(36 70)(37 66)(38 67)(39 68)(40 69)(41 53)(42 54)(43 55)(44 51)(45 52)
(1 24 66 7)(2 25 67 8)(3 21 68 9)(4 22 69 10)(5 23 70 6)(11 76 36 64)(12 77 37 65)(13 78 38 61)(14 79 39 62)(15 80 40 63)(16 55 33 48)(17 51 34 49)(18 52 35 50)(19 53 31 46)(20 54 32 47)(26 59 45 71)(27 60 41 72)(28 56 42 73)(29 57 43 74)(30 58 44 75)
(1 28 12 47)(2 29 13 48)(3 30 14 49)(4 26 15 50)(5 27 11 46)(16 57)(17 58)(18 59)(19 60)(20 56)(31 72)(32 73)(33 74)(34 75)(35 71)(36 53 70 41)(37 54 66 42)(38 55 67 43)(39 51 68 44)(40 52 69 45)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A ··· 4L 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AJ 20A ··· 20AV order 1 2 2 2 2 ··· 2 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 4 ··· 4 1 1 1 1 1 ··· 1 2 ··· 2 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + - + image C1 C2 C4 C4 C5 C10 C20 C20 D4 Q8 C5×D4 C5×Q8 C23⋊C4 C5×C23⋊C4 kernel C5×C23.9D4 C10×C22⋊C4 C5×C22⋊C4 C22×C20 C23.9D4 C2×C22⋊C4 C22⋊C4 C22×C4 C22×C10 C22×C10 C23 C23 C10 C2 # reps 1 3 8 4 4 12 32 16 3 1 12 4 2 8

Matrix representation of C5×C23.9D4 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 39 1 8 33 0 0 0 0 0 1 0 0 0 0 1 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 8 33 0 0 39 1 8 33 0 0 0 0 0 40 0 0 0 0 40 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 40 0 0 0 0 2 40 0 0 0 0 0 0 33 0 11 30 0 0 0 0 0 1 0 0 39 1 8 33 0 0 0 1 0 0
,
 32 0 0 0 0 0 23 9 0 0 0 0 0 0 40 1 8 0 0 0 39 1 8 33 0 0 0 0 1 0 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,39,0,0,0,0,0,1,0,0,0,0,0,8,0,1,0,0,0,33,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,39,0,0,0,0,0,1,0,0,0,0,8,8,0,40,0,0,33,33,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,2,0,0,0,0,40,40,0,0,0,0,0,0,33,0,39,0,0,0,0,0,1,1,0,0,11,0,8,0,0,0,30,1,33,0],[32,23,0,0,0,0,0,9,0,0,0,0,0,0,40,39,0,0,0,0,1,1,0,0,0,0,8,8,1,0,0,0,0,33,0,40] >;

C5×C23.9D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3._9D_4
% in TeX

G:=Group("C5xC2^3.9D4");
// GroupNames label

G:=SmallGroup(320,147);
// by ID

G=gap.SmallGroup(320,147);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,5043,3511]);
// Polycyclic

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

׿
×
𝔽