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## G = D8⋊4Dic5order 320 = 26·5

### 4th semidirect product of D8 and Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D8⋊4Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — D4.Dic5 — D8⋊4Dic5
 Lower central C5 — C10 — C20 — D8⋊4Dic5
 Upper central C1 — C4 — C2×C4 — C4○D8

Generators and relations for D84Dic5
G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=a-1, ac=ca, dad-1=a5, cbc-1=a4b, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 278 in 104 conjugacy classes, 53 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C10, C10, C42, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C8⋊C4, C4≀C2, C8.C4, C8○D4, C4○D8, C52C8, C52C8, C40, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C8.26D4, C2×C52C8, C2×C52C8, C4.Dic5, C4.Dic5, C4×Dic5, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C408C4, C40.6C4, D42Dic5, D4.Dic5, C5×C4○D8, D84Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, Dic5, D10, C4×D4, C2×Dic5, C22×D5, C8.26D4, D4×D5, D42D5, C22×Dic5, D4×Dic5, D84Dic5

Smallest permutation representation of D84Dic5
On 80 points
Generators in S80
(1 29 14 23 9 31 19 38)(2 30 15 24 10 32 20 39)(3 26 11 25 6 33 16 40)(4 27 12 21 7 34 17 36)(5 28 13 22 8 35 18 37)(41 53 63 77 46 58 68 72)(42 54 64 78 47 59 69 73)(43 55 65 79 48 60 70 74)(44 56 66 80 49 51 61 75)(45 57 67 71 50 52 62 76)
(1 60)(2 56)(3 52)(4 58)(5 54)(6 57)(7 53)(8 59)(9 55)(10 51)(11 71)(12 77)(13 73)(14 79)(15 75)(16 76)(17 72)(18 78)(19 74)(20 80)(21 63)(22 69)(23 65)(24 61)(25 67)(26 50)(27 46)(28 42)(29 48)(30 44)(31 43)(32 49)(33 45)(34 41)(35 47)(36 68)(37 64)(38 70)(39 66)(40 62)
(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)
(2 5)(3 4)(6 7)(8 10)(11 12)(13 15)(16 17)(18 20)(21 40)(22 39)(23 38)(24 37)(25 36)(26 34)(27 33)(28 32)(29 31)(30 35)(41 62 46 67)(42 61 47 66)(43 70 48 65)(44 69 49 64)(45 68 50 63)(51 73 56 78)(52 72 57 77)(53 71 58 76)(54 80 59 75)(55 79 60 74)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 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 5 5 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 20 20 2 2 4 4 10 10 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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - - - + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 D5 C4○D4 D10 Dic5 Dic5 Dic5 D10 C8.26D4 D4×D5 D4⋊2D5 D8⋊4Dic5 kernel D8⋊4Dic5 C40⋊8C4 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 2 2 2 8

Matrix representation of D84Dic5 in GL4(𝔽41) generated by

 10 38 0 0 3 31 0 0 0 0 33 27 0 0 14 8
,
 0 0 10 38 0 0 3 31 33 27 0 0 14 8 0 0
,
 0 40 0 0 1 7 0 0 0 0 0 1 0 0 40 34
,
 19 32 0 0 22 22 0 0 0 0 7 1 0 0 34 34
G:=sub<GL(4,GF(41))| [10,3,0,0,38,31,0,0,0,0,33,14,0,0,27,8],[0,0,33,14,0,0,27,8,10,3,0,0,38,31,0,0],[0,1,0,0,40,7,0,0,0,0,0,40,0,0,1,34],[19,22,0,0,32,22,0,0,0,0,7,34,0,0,1,34] >;

D84Dic5 in GAP, Magma, Sage, TeX

D_8\rtimes_4{\rm Dic}_5
% in TeX

G:=Group("D8:4Dic5");
// GroupNames label

G:=SmallGroup(320,824);
// by ID

G=gap.SmallGroup(320,824);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,136,851,438,102,12550]);
// 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,d*a*d^-1=a^5,c*b*c^-1=a^4*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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