Copied to
clipboard

G = C5×C22.D4order 160 = 25·5

Direct product of C5 and C22.D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C22.D4
 Chief series C1 — C2 — C22 — C2×C10 — C22×C10 — D4×C10 — C5×C22.D4
 Lower central C1 — C22 — C5×C22.D4
 Upper central C1 — C2×C10 — C5×C22.D4

Generators and relations for C5×C22.D4
G = < a,b,c,d,e | a5=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >

Subgroups: 116 in 78 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×5], C22, C22 [×2], C22 [×5], C5, C2×C4, C2×C4 [×4], C2×C4 [×2], D4 [×2], C23 [×2], C10, C10 [×2], C10 [×3], C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, C20 [×5], C2×C10, C2×C10 [×2], C2×C10 [×5], C22.D4, C2×C20, C2×C20 [×4], C2×C20 [×2], C5×D4 [×2], C22×C10 [×2], C5×C22⋊C4, C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C22×C20, D4×C10, C5×C22.D4
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], C2×D4, C4○D4 [×2], C2×C10 [×7], C22.D4, C5×D4 [×2], C22×C10, D4×C10, C5×C4○D4 [×2], C5×C22.D4

Smallest permutation representation of C5×C22.D4
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 61)(2 62)(3 63)(4 64)(5 65)(6 46)(7 47)(8 48)(9 49)(10 50)(11 37)(12 38)(13 39)(14 40)(15 36)(16 55)(17 51)(18 52)(19 53)(20 54)(21 56)(22 57)(23 58)(24 59)(25 60)(26 74)(27 75)(28 71)(29 72)(30 73)(31 68)(32 69)(33 70)(34 66)(35 67)(41 76)(42 77)(43 78)(44 79)(45 80)
(1 35)(2 31)(3 32)(4 33)(5 34)(6 80)(7 76)(8 77)(9 78)(10 79)(11 17)(12 18)(13 19)(14 20)(15 16)(21 27)(22 28)(23 29)(24 30)(25 26)(36 55)(37 51)(38 52)(39 53)(40 54)(41 47)(42 48)(43 49)(44 50)(45 46)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)
(1 55 27 41)(2 51 28 42)(3 52 29 43)(4 53 30 44)(5 54 26 45)(6 65 14 74)(7 61 15 75)(8 62 11 71)(9 63 12 72)(10 64 13 73)(16 56 76 67)(17 57 77 68)(18 58 78 69)(19 59 79 70)(20 60 80 66)(21 47 35 36)(22 48 31 37)(23 49 32 38)(24 50 33 39)(25 46 34 40)
(6 14)(7 15)(8 11)(9 12)(10 13)(16 76)(17 77)(18 78)(19 79)(20 80)(36 41)(37 42)(38 43)(39 44)(40 45)(46 54)(47 55)(48 51)(49 52)(50 53)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 5A 5B 5C 5D 10A ··· 10L 10M ··· 10T 10U 10V 10W 10X 20A ··· 20P 20Q ··· 20AB order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 5 5 5 5 10 ··· 10 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 2 2 2 2 4 4 4 1 1 1 1 1 ··· 1 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

70 irreducible representations

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

Matrix representation of C5×C22.D4 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 32 0 0 9 0
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 0 40 0 0 1 0 0 0 0 0 0 40 0 0 40 0
,
 1 0 0 0 0 40 0 0 0 0 1 0 0 0 0 40
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,9,0,0,32,0],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[0,1,0,0,40,0,0,0,0,0,0,40,0,0,40,0],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;

C5×C22.D4 in GAP, Magma, Sage, TeX

C_5\times C_2^2.D_4
% in TeX

G:=Group("C5xC2^2.D4");
// GroupNames label

G:=SmallGroup(160,184);
// by ID

G=gap.SmallGroup(160,184);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,1514,194]);
// Polycyclic

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

׿
×
𝔽