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## G = D8⋊2Dic5order 320 = 26·5

### 2nd semidirect product of D8 and Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D8⋊2Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C40⋊6C4 — D8⋊2Dic5
 Lower central C5 — C10 — C20 — C40 — D8⋊2Dic5
 Upper central C1 — C2 — C2×C4 — C2×C8 — C4○D8

Generators and relations for D82Dic5
G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=a-1, ac=ca, dad-1=a3, cbc-1=a4b, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 206 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C10, C10, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C4.Q8, M5(2), C4○D8, C40, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, D82C4, C52C16, C4⋊Dic5, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C20.4C8, C406C4, C5×C4○D8, D82Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, Dic5, D10, D4⋊C4, C2×Dic5, C5⋊D4, D82C4, D4⋊D5, D4.D5, C23.D5, D4⋊Dic5, D82Dic5

Smallest permutation representation of D82Dic5
On 80 points
Generators in S80
(1 39 19 29 9 34 14 22)(2 40 20 30 10 35 15 23)(3 36 16 26 6 31 11 24)(4 37 17 27 7 32 12 25)(5 38 18 28 8 33 13 21)(41 61 51 71 46 66 56 76)(42 62 52 72 47 67 57 77)(43 63 53 73 48 68 58 78)(44 64 54 74 49 69 59 79)(45 65 55 75 50 70 60 80)
(1 74)(2 80)(3 76)(4 72)(5 78)(6 71)(7 77)(8 73)(9 79)(10 75)(11 61)(12 67)(13 63)(14 69)(15 65)(16 66)(17 62)(18 68)(19 64)(20 70)(21 43)(22 49)(23 45)(24 41)(25 47)(26 46)(27 42)(28 48)(29 44)(30 50)(31 51)(32 57)(33 53)(34 59)(35 55)(36 56)(37 52)(38 58)(39 54)(40 60)
(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 17)(12 16)(13 20)(14 19)(15 18)(21 35)(22 34)(23 33)(24 32)(25 31)(26 37)(27 36)(28 40)(29 39)(30 38)(41 67 46 62)(42 66 47 61)(43 65 48 70)(44 64 49 69)(45 63 50 68)(51 72 56 77)(52 71 57 76)(53 80 58 75)(54 79 59 74)(55 78 60 73)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 10A 10B 10C 10D 10E 10F 10G 10H 16A 16B 16C 16D 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 40A ··· 40H order 1 2 2 2 4 4 4 4 4 5 5 8 8 8 10 10 10 10 10 10 10 10 16 16 16 16 20 20 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 2 8 2 2 8 40 40 2 2 2 2 4 2 2 4 4 8 8 8 8 20 20 20 20 2 2 2 2 4 4 8 8 8 8 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - - - + image C1 C2 C2 C2 C4 C4 D4 D4 D5 SD16 D8 D10 Dic5 Dic5 C5⋊D4 C5⋊D4 D8⋊2C4 D4.D5 D4⋊D5 D8⋊2Dic5 kernel D8⋊2Dic5 C20.4C8 C40⋊6C4 C5×C4○D8 C5×D8 C5×Q16 C40 C2×C20 C4○D8 C20 C2×C10 C2×C8 D8 Q16 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 2 2 1 1 2 2 2 2 2 2 4 4 2 2 2 8

Matrix representation of D82Dic5 in GL4(𝔽241) generated by

 94 57 0 0 184 109 0 0 169 104 132 57 6 61 184 147
,
 114 104 132 57 192 61 184 147 42 166 231 76 132 218 231 76
,
 0 1 0 0 240 51 0 0 28 134 0 240 52 232 1 190
,
 1 0 0 0 51 240 0 0 83 59 147 184 170 113 210 94
G:=sub<GL(4,GF(241))| [94,184,169,6,57,109,104,61,0,0,132,184,0,0,57,147],[114,192,42,132,104,61,166,218,132,184,231,231,57,147,76,76],[0,240,28,52,1,51,134,232,0,0,0,1,0,0,240,190],[1,51,83,170,0,240,59,113,0,0,147,210,0,0,184,94] >;

D82Dic5 in GAP, Magma, Sage, TeX

D_8\rtimes_2{\rm Dic}_5
% in TeX

G:=Group("D8:2Dic5");
// GroupNames label

G:=SmallGroup(320,124);
// by ID

G=gap.SmallGroup(320,124);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,387,675,794,80,1684,851,102,12550]);
// 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^3,c*b*c^-1=a^4*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

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