Copied to
clipboard

G = C2×Q8×D5order 160 = 25·5

Direct product of C2, Q8 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×Q8×D5
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C2×C4×D5 — C2×Q8×D5
 Lower central C5 — C10 — C2×Q8×D5
 Upper central C1 — C22 — C2×Q8

Generators and relations for C2×Q8×D5
G = < a,b,c,d,e | a2=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 392 in 156 conjugacy classes, 97 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, Q8, Q8, C23, D5, C10, C10, C22×C4, C2×Q8, C2×Q8, Dic5, C20, D10, C2×C10, C22×Q8, Dic10, C4×D5, C2×Dic5, C2×C20, C5×Q8, C22×D5, C2×Dic10, C2×C4×D5, Q8×D5, Q8×C10, C2×Q8×D5
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C24, D10, C22×Q8, C22×D5, Q8×D5, C23×D5, C2×Q8×D5

Smallest permutation representation of C2×Q8×D5
On 80 points
Generators in S80
(1 44)(2 45)(3 41)(4 42)(5 43)(6 46)(7 47)(8 48)(9 49)(10 50)(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)
(1 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(1 59 9 54)(2 60 10 55)(3 56 6 51)(4 57 7 52)(5 58 8 53)(11 41 16 46)(12 42 17 47)(13 43 18 48)(14 44 19 49)(15 45 20 50)(21 71 26 76)(22 72 27 77)(23 73 28 78)(24 74 29 79)(25 75 30 80)(31 66 36 61)(32 67 37 62)(33 68 38 63)(34 69 39 64)(35 70 40 65)
(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 43)(2 42)(3 41)(4 45)(5 44)(6 46)(7 50)(8 49)(9 48)(10 47)(11 51)(12 55)(13 54)(14 53)(15 52)(16 56)(17 60)(18 59)(19 58)(20 57)(21 61)(22 65)(23 64)(24 63)(25 62)(26 66)(27 70)(28 69)(29 68)(30 67)(31 71)(32 75)(33 74)(34 73)(35 72)(36 76)(37 80)(38 79)(39 78)(40 77)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4F 4G ··· 4L 5A 5B 10A ··· 10F 20A ··· 20L order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 5 5 5 5 2 ··· 2 10 ··· 10 2 2 2 ··· 2 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 type + + + + + - + + + - image C1 C2 C2 C2 C2 Q8 D5 D10 D10 Q8×D5 kernel C2×Q8×D5 C2×Dic10 C2×C4×D5 Q8×D5 Q8×C10 D10 C2×Q8 C2×C4 Q8 C2 # reps 1 3 3 8 1 4 2 6 8 4

Matrix representation of C2×Q8×D5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 9 23 0 0 0 32
,
 40 0 0 0 0 40 0 0 0 0 9 0 0 0 9 32
,
 0 1 0 0 40 6 0 0 0 0 1 0 0 0 0 1
,
 0 40 0 0 40 0 0 0 0 0 40 0 0 0 0 40
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,23,32],[40,0,0,0,0,40,0,0,0,0,9,9,0,0,0,32],[0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,40,0,0,0,0,0,40,0,0,0,0,40] >;

C2×Q8×D5 in GAP, Magma, Sage, TeX

C_2\times Q_8\times D_5
% in TeX

G:=Group("C2xQ8xD5");
// GroupNames label

G:=SmallGroup(160,220);
// by ID

G=gap.SmallGroup(160,220);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,86,159,69,4613]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^5=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=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