Copied to
clipboard

## G = C5×Q8×D5order 400 = 24·52

### Direct product of C5, Q8 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×Q8×D5
 Chief series C1 — C5 — C10 — C5×C10 — D5×C10 — D5×C20 — C5×Q8×D5
 Lower central C5 — C10 — C5×Q8×D5
 Upper central C1 — C10 — C5×Q8

Generators and relations for C5×Q8×D5
G = < a,b,c,d,e | a5=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: 204 in 88 conjugacy classes, 50 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, Q8, Q8, D5, C10, C10, C2×Q8, Dic5, C20, C20, D10, C2×C10, C52, Dic10, C4×D5, C2×C20, C5×Q8, C5×Q8, C5×D5, C5×C10, Q8×D5, Q8×C10, C5×Dic5, C5×C20, D5×C10, C5×Dic10, D5×C20, Q8×C52, C5×Q8×D5
Quotients: C1, C2, C22, C5, Q8, C23, D5, C10, C2×Q8, D10, C2×C10, C5×Q8, C22×D5, C22×C10, C5×D5, Q8×D5, Q8×C10, D5×C10, D5×C2×C10, C5×Q8×D5

Smallest permutation representation of C5×Q8×D5
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 18 8 13)(2 19 9 14)(3 20 10 15)(4 16 6 11)(5 17 7 12)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(1 28 8 23)(2 29 9 24)(3 30 10 25)(4 26 6 21)(5 27 7 22)(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 3 5 2 4)(6 8 10 7 9)(11 13 15 12 14)(16 18 20 17 19)(21 23 25 22 24)(26 28 30 27 29)(31 33 35 32 34)(36 38 40 37 39)(41 44 42 45 43)(46 49 47 50 48)(51 54 52 55 53)(56 59 57 60 58)(61 64 62 65 63)(66 69 67 70 68)(71 74 72 75 73)(76 79 77 80 78)
(1 45)(2 41)(3 42)(4 43)(5 44)(6 48)(7 49)(8 50)(9 46)(10 47)(11 53)(12 54)(13 55)(14 51)(15 52)(16 58)(17 59)(18 60)(19 56)(20 57)(21 63)(22 64)(23 65)(24 61)(25 62)(26 68)(27 69)(28 70)(29 66)(30 67)(31 73)(32 74)(33 75)(34 71)(35 72)(36 78)(37 79)(38 80)(39 76)(40 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), (1,18,8,13)(2,19,9,14)(3,20,10,15)(4,16,6,11)(5,17,7,12)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (1,28,8,23)(2,29,9,24)(3,30,10,25)(4,26,6,21)(5,27,7,22)(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), (1,3,5,2,4)(6,8,10,7,9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)(36,38,40,37,39)(41,44,42,45,43)(46,49,47,50,48)(51,54,52,55,53)(56,59,57,60,58)(61,64,62,65,63)(66,69,67,70,68)(71,74,72,75,73)(76,79,77,80,78), (1,45)(2,41)(3,42)(4,43)(5,44)(6,48)(7,49)(8,50)(9,46)(10,47)(11,53)(12,54)(13,55)(14,51)(15,52)(16,58)(17,59)(18,60)(19,56)(20,57)(21,63)(22,64)(23,65)(24,61)(25,62)(26,68)(27,69)(28,70)(29,66)(30,67)(31,73)(32,74)(33,75)(34,71)(35,72)(36,78)(37,79)(38,80)(39,76)(40,77)>;

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

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

100 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 5E ··· 5N 10A 10B 10C 10D 10E ··· 10N 10O ··· 10V 20A ··· 20L 20M ··· 20AP 20AQ ··· 20BB order 1 2 2 2 4 4 4 4 4 4 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 size 1 1 5 5 2 2 2 10 10 10 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 5 ··· 5 2 ··· 2 4 ··· 4 10 ··· 10

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + - + + - image C1 C2 C2 C2 C5 C10 C10 C10 Q8 D5 D10 C5×Q8 C5×D5 D5×C10 Q8×D5 C5×Q8×D5 kernel C5×Q8×D5 C5×Dic10 D5×C20 Q8×C52 Q8×D5 Dic10 C4×D5 C5×Q8 C5×D5 C5×Q8 C20 D5 Q8 C4 C5 C1 # reps 1 3 3 1 4 12 12 4 2 2 6 8 8 24 2 8

Matrix representation of C5×Q8×D5 in GL4(𝔽41) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 0 40 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 1 30 0 0 30 40
,
 18 0 0 0 17 16 0 0 0 0 1 0 0 0 0 1
,
 25 26 0 0 17 16 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,1,30,0,0,30,40],[18,17,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[25,17,0,0,26,16,0,0,0,0,1,0,0,0,0,1] >;

C5×Q8×D5 in GAP, Magma, Sage, TeX

C_5\times Q_8\times D_5
% in TeX

G:=Group("C5xQ8xD5");
// GroupNames label

G:=SmallGroup(400,187);
// by ID

G=gap.SmallGroup(400,187);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,247,404,194,11525]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=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