Copied to
clipboard

## G = C10×2+ 1+4order 320 = 26·5

### Direct product of C10 and 2+ 1+4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C10×2+ 1+4
G = < a,b,c,d,e | a10=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 1186 in 898 conjugacy classes, 754 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, D4, Q8, C23, C23, C10, C10, C10, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C20, C2×C10, C2×C10, C2×C10, C22×D4, C2×C4○D4, 2+ 1+4, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C2×2+ 1+4, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C23×C10, D4×C2×C10, C10×C4○D4, C5×2+ 1+4, C10×2+ 1+4
Quotients: C1, C2, C22, C5, C23, C10, C24, C2×C10, 2+ 1+4, C25, C22×C10, C2×2+ 1+4, C23×C10, C5×2+ 1+4, C24×C10, C10×2+ 1+4

Smallest permutation representation of C10×2+ 1+4
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 73 28 14)(2 74 29 15)(3 75 30 16)(4 76 21 17)(5 77 22 18)(6 78 23 19)(7 79 24 20)(8 80 25 11)(9 71 26 12)(10 72 27 13)(31 56 49 64)(32 57 50 65)(33 58 41 66)(34 59 42 67)(35 60 43 68)(36 51 44 69)(37 52 45 70)(38 53 46 61)(39 54 47 62)(40 55 48 63)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 51)(18 52)(19 53)(20 54)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 41)(29 42)(30 43)(61 78)(62 79)(63 80)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)
(1 78 28 19)(2 79 29 20)(3 80 30 11)(4 71 21 12)(5 72 22 13)(6 73 23 14)(7 74 24 15)(8 75 25 16)(9 76 26 17)(10 77 27 18)(31 69 49 51)(32 70 50 52)(33 61 41 53)(34 62 42 54)(35 63 43 55)(36 64 44 56)(37 65 45 57)(38 66 46 58)(39 67 47 59)(40 68 48 60)
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 51)(10 52)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 31)(18 32)(19 33)(20 34)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 61)(29 62)(30 63)(41 78)(42 79)(43 80)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)

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

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

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

170 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2U 4A ··· 4L 5A 5B 5C 5D 10A ··· 10L 10M ··· 10CF 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 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 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 2+ 1+4 C5×2+ 1+4 kernel C10×2+ 1+4 D4×C2×C10 C10×C4○D4 C5×2+ 1+4 C2×2+ 1+4 C22×D4 C2×C4○D4 2+ 1+4 C10 C2 # reps 1 9 6 16 4 36 24 64 2 8

Matrix representation of C10×2+ 1+4 in GL5(𝔽41)

 40 0 0 0 0 0 37 0 0 0 0 0 37 0 0 0 0 0 37 0 0 0 0 0 37
,
 1 0 0 0 0 0 40 0 0 39 0 1 0 40 1 0 0 1 0 40 0 1 0 0 1
,
 40 0 0 0 0 0 1 2 0 0 0 0 40 0 0 0 0 1 0 40 0 0 40 40 0
,
 1 0 0 0 0 0 40 0 0 39 0 0 0 1 1 0 40 40 0 40 0 1 0 0 1
,
 40 0 0 0 0 0 40 0 2 0 0 0 0 40 40 0 0 0 1 0 0 0 40 40 0

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,37,0,0,0,0,0,37,0,0,0,0,0,37,0,0,0,0,0,37],[1,0,0,0,0,0,40,1,0,1,0,0,0,1,0,0,0,40,0,0,0,39,1,40,1],[40,0,0,0,0,0,1,0,0,0,0,2,40,1,40,0,0,0,0,40,0,0,0,40,0],[1,0,0,0,0,0,40,0,40,1,0,0,0,40,0,0,0,1,0,0,0,39,1,40,1],[40,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,2,40,1,40,0,0,40,0,0] >;

C10×2+ 1+4 in GAP, Magma, Sage, TeX

C_{10}\times 2_+^{1+4}
% in TeX

G:=Group("C10xES+(2,2)");
// GroupNames label

G:=SmallGroup(320,1632);
// by ID

G=gap.SmallGroup(320,1632);
# by ID

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

G:=Group<a,b,c,d,e|a^10=b^4=c^2=e^2=1,d^2=b^2,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=b^2*d>;
// generators/relations

׿
×
𝔽