Copied to
clipboard

## G = D8⋊5Dic5order 320 = 26·5

### The semidirect product of D8 and Dic5 acting through Inn(D8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D8⋊5Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — D4.Dic5 — D8⋊5Dic5
 Lower central C5 — C10 — C20 — D8⋊5Dic5
 Upper central C1 — C4 — C2×C4 — C4○D8

Generators and relations for D85Dic5
G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=a-1, ac=ca, ad=da, cbc-1=a4b, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 278 in 106 conjugacy classes, 53 normal (31 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C5, C8 [×2], C8 [×4], C2×C4, C2×C4 [×3], D4 [×2], D4 [×2], Q8 [×2], C10, C10 [×3], C42, C2×C8, C2×C8 [×3], M4(2) [×4], D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C4×C8, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C52C8 [×2], C52C8 [×2], C40 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], C8○D8, C2×C52C8, C2×C52C8 [×2], C4.Dic5 [×2], C4.Dic5 [×2], C4×Dic5, C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C5×C4○D4 [×2], C8×Dic5, C40.6C4, D42Dic5 [×2], D4.Dic5 [×2], C5×C4○D8, D85Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C2×Dic5 [×6], C22×D5, C8○D8, D4×D5, D42D5, C22×Dic5, D4×Dic5, D85Dic5

Smallest permutation representation of D85Dic5
On 80 points
Generators in S80
(1 24 14 28 31 38 19 9)(2 25 15 29 32 39 20 10)(3 21 11 30 33 40 16 6)(4 22 12 26 34 36 17 7)(5 23 13 27 35 37 18 8)(41 69 56 74 46 64 51 79)(42 70 57 75 47 65 52 80)(43 61 58 76 48 66 53 71)(44 62 59 77 49 67 54 72)(45 63 60 78 50 68 55 73)
(1 41)(2 47)(3 43)(4 49)(5 45)(6 61)(7 67)(8 63)(9 69)(10 65)(11 53)(12 59)(13 55)(14 51)(15 57)(16 58)(17 54)(18 60)(19 56)(20 52)(21 71)(22 77)(23 73)(24 79)(25 75)(26 62)(27 68)(28 64)(29 70)(30 66)(31 46)(32 42)(33 48)(34 44)(35 50)(36 72)(37 78)(38 74)(39 80)(40 76)
(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 31)(2 35)(3 34)(4 33)(5 32)(6 26)(7 30)(8 29)(9 28)(10 27)(11 17)(12 16)(13 20)(14 19)(15 18)(21 36)(22 40)(23 39)(24 38)(25 37)(41 51 46 56)(42 60 47 55)(43 59 48 54)(44 58 49 53)(45 57 50 52)(61 77 66 72)(62 76 67 71)(63 75 68 80)(64 74 69 79)(65 73 70 78)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 8M 8N 10A 10B 10C 10D 10E 10F 10G 10H 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 40A ··· 40H order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 2 4 4 1 1 2 4 4 10 10 10 10 2 2 2 2 2 2 5 5 5 5 10 10 20 20 20 20 2 2 4 4 8 8 8 8 2 2 2 2 4 4 8 8 8 8 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - - - + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 D5 C4○D4 D10 Dic5 Dic5 Dic5 D10 C8○D8 D4×D5 D4⋊2D5 D8⋊5Dic5 kernel D8⋊5Dic5 C8×Dic5 C40.6C4 D4⋊2Dic5 D4.Dic5 C5×C4○D8 C5×D8 C5×SD16 C5×Q16 C5⋊2C8 C4○D8 C2×C10 C2×C8 D8 SD16 Q16 C4○D4 C5 C4 C22 C1 # reps 1 1 1 2 2 1 2 4 2 2 2 2 2 2 4 2 4 8 2 2 8

Matrix representation of D85Dic5 in GL4(𝔽41) generated by

 14 26 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 3 15 0 0 35 38 0 0 0 0 40 0 0 0 0 40
,
 40 40 0 0 0 1 0 0 0 0 0 1 0 0 40 6
,
 9 5 0 0 0 40 0 0 0 0 1 0 0 0 6 40
G:=sub<GL(4,GF(41))| [14,0,0,0,26,3,0,0,0,0,1,0,0,0,0,1],[3,35,0,0,15,38,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,40,1,0,0,0,0,0,40,0,0,1,6],[9,0,0,0,5,40,0,0,0,0,1,6,0,0,0,40] >;

D85Dic5 in GAP, Magma, Sage, TeX

D_8\rtimes_5{\rm Dic}_5
% in TeX

G:=Group("D8:5Dic5");
// GroupNames label

G:=SmallGroup(320,823);
// by ID

G=gap.SmallGroup(320,823);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,136,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=c^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^4*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽