Copied to
clipboard

## G = C5×C2.C25order 320 = 26·5

### Direct product of C5 and C2.C25

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×C2.C25
 Chief series C1 — C2 — C10 — C2×C10 — C5×D4 — D4×C10 — C5×2+ 1+4 — C5×C2.C25
 Lower central C1 — C2 — C5×C2.C25
 Upper central C1 — C20 — C5×C2.C25

Generators and relations for C5×C2.C25
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=e2=f2=1, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, dcd=fcf=bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, ce=ec, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 930 in 810 conjugacy classes, 750 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, D4, Q8, C23, C10, C10, C22×C4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C20, C5×D4, C5×Q8, C22×C10, C2.C25, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C10×C4○D4, C5×2+ 1+4, C5×2- 1+4, C5×C2.C25
Quotients: C1, C2, C22, C5, C23, C10, C24, C2×C10, C25, C22×C10, C2.C25, C23×C10, C24×C10, C5×C2.C25

Smallest permutation representation of C5×C2.C25
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 13)(2 14)(3 15)(4 11)(5 12)(6 80)(7 76)(8 77)(9 78)(10 79)(16 22)(17 23)(18 24)(19 25)(20 21)(26 32)(27 33)(28 34)(29 35)(30 31)(36 42)(37 43)(38 44)(39 45)(40 41)(46 52)(47 53)(48 54)(49 55)(50 51)(56 62)(57 63)(58 64)(59 65)(60 61)(66 72)(67 73)(68 74)(69 75)(70 71)
(1 47)(2 48)(3 49)(4 50)(5 46)(6 41)(7 42)(8 43)(9 44)(10 45)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(26 32)(27 33)(28 34)(29 35)(30 31)(36 42)(37 43)(38 44)(39 45)(40 41)(46 52)(47 53)(48 54)(49 55)(50 51)(56 62)(57 63)(58 64)(59 65)(60 61)
(1 27)(2 28)(3 29)(4 30)(5 26)(6 61)(7 62)(8 63)(9 64)(10 65)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(6 80)(7 76)(8 77)(9 78)(10 79)(46 52)(47 53)(48 54)(49 55)(50 51)(56 62)(57 63)(58 64)(59 65)(60 61)(66 72)(67 73)(68 74)(69 75)(70 71)
(1 23 13 17)(2 24 14 18)(3 25 15 19)(4 21 11 20)(5 22 12 16)(6 71 80 70)(7 72 76 66)(8 73 77 67)(9 74 78 68)(10 75 79 69)(26 42 32 36)(27 43 33 37)(28 44 34 38)(29 45 35 39)(30 41 31 40)(46 62 52 56)(47 63 53 57)(48 64 54 58)(49 65 55 59)(50 61 51 60)

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

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

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

170 conjugacy classes

 class 1 2A 2B ··· 2P 4A 4B 4C ··· 4Q 5A 5B 5C 5D 10A 10B 10C 10D 10E ··· 10BL 20A ··· 20H 20I ··· 20BP order 1 2 2 ··· 2 4 4 4 ··· 4 5 5 5 5 10 10 10 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 2 ··· 2 1 1 2 ··· 2 1 1 1 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2

170 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 type + + + + image C1 C2 C2 C2 C5 C10 C10 C10 C2.C25 C5×C2.C25 kernel C5×C2.C25 C10×C4○D4 C5×2+ 1+4 C5×2- 1+4 C2.C25 C2×C4○D4 2+ 1+4 2- 1+4 C5 C1 # reps 1 15 10 6 4 60 40 24 2 8

Matrix representation of C5×C2.C25 in GL4(𝔽41) generated by

 10 0 0 0 0 10 0 0 0 0 10 0 0 0 0 10
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 0 0 1 0 40 40 40 39 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 40 0 0 0 0 40 0 0 1 1 1
,
 0 1 0 0 1 0 0 0 40 40 40 39 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 40 0 40 40 0 40
,
 32 0 0 0 0 32 0 0 0 0 32 0 0 0 0 32
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,40,1,0,0,40,0,0,1,40,0,0,0,39,0,1],[1,0,0,0,0,40,0,1,0,0,40,1,0,0,0,1],[0,1,40,0,1,0,40,0,0,0,40,0,0,0,39,1],[1,0,0,40,0,1,0,40,0,0,40,0,0,0,0,40],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32] >;

C5×C2.C25 in GAP, Magma, Sage, TeX

C_5\times C_2.C_2^5
% in TeX

G:=Group("C5xC2.C2^5");
// GroupNames label

G:=SmallGroup(320,1634);
// by ID

G=gap.SmallGroup(320,1634);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-5,-2,2269,1731,4707,382]);
// Polycyclic

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

׿
×
𝔽