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## G = Dic5.5D4order 160 = 25·5

### 1st non-split extension by Dic5 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic5.5D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — D10⋊C4 — Dic5.5D4
 Lower central C5 — C2×C10 — Dic5.5D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for Dic5.5D4
G = < a,b,c,d | a10=c4=1, b2=d2=a5, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=a5c-1 >

Subgroups: 256 in 76 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C2×D4, C2×Q8, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C4.4D4, Dic10, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4×Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×Dic10, C2×C5⋊D4, Dic5.5D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C22×D5, C4○D20, D4×D5, D42D5, Dic5.5D4

Smallest permutation representation of Dic5.5D4
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 34 6 39)(2 33 7 38)(3 32 8 37)(4 31 9 36)(5 40 10 35)(11 57 16 52)(12 56 17 51)(13 55 18 60)(14 54 19 59)(15 53 20 58)(21 49 26 44)(22 48 27 43)(23 47 28 42)(24 46 29 41)(25 45 30 50)(61 76 66 71)(62 75 67 80)(63 74 68 79)(64 73 69 78)(65 72 70 77)
(1 59 22 63)(2 60 23 64)(3 51 24 65)(4 52 25 66)(5 53 26 67)(6 54 27 68)(7 55 28 69)(8 56 29 70)(9 57 30 61)(10 58 21 62)(11 45 71 31)(12 46 72 32)(13 47 73 33)(14 48 74 34)(15 49 75 35)(16 50 76 36)(17 41 77 37)(18 42 78 38)(19 43 79 39)(20 44 80 40)
(1 54 6 59)(2 53 7 58)(3 52 8 57)(4 51 9 56)(5 60 10 55)(11 32 16 37)(12 31 17 36)(13 40 18 35)(14 39 19 34)(15 38 20 33)(21 69 26 64)(22 68 27 63)(23 67 28 62)(24 66 29 61)(25 65 30 70)(41 71 46 76)(42 80 47 75)(43 79 48 74)(44 78 49 73)(45 77 50 72)

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

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

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

34 conjugacy classes

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

34 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 C4○D20 D4×D5 D4⋊2D5 kernel Dic5.5D4 C4×Dic5 D10⋊C4 C23.D5 C5×C22⋊C4 C2×Dic10 C2×C5⋊D4 Dic5 C22⋊C4 C10 C2×C4 C23 C2 C2 C2 # reps 1 1 2 1 1 1 1 2 2 4 4 2 8 2 2

Matrix representation of Dic5.5D4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 1 0 0 0 0 5 35 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 20 0 0 0 0 4 1 0 0 0 0 0 0 6 1 0 0 0 0 6 35 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 5 0 0 0 0 16 40
,
 9 0 0 0 0 0 5 32 0 0 0 0 0 0 35 40 0 0 0 0 35 6 0 0 0 0 0 0 40 36 0 0 0 0 0 1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,4,0,0,0,0,20,1,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,16,0,0,0,0,5,40],[9,5,0,0,0,0,0,32,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,40,0,0,0,0,0,36,1] >;

Dic5.5D4 in GAP, Magma, Sage, TeX

{\rm Dic}_5._5D_4
% in TeX

G:=Group("Dic5.5D4");
// GroupNames label

G:=SmallGroup(160,106);
// by ID

G=gap.SmallGroup(160,106);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,55,506,188,4613]);
// Polycyclic

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

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