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G = D10⋊(C4⋊C4)  order 320 = 26·5

The semidirect product of D10 and C4⋊C4 acting via C4⋊C4/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D10⋊(C4⋊C4)
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C23×F5 — D10⋊(C4⋊C4)
 Lower central C5 — C2×C10 — D10⋊(C4⋊C4)
 Upper central C1 — C22 — C22⋊C4

Generators and relations for D10⋊(C4⋊C4)
G = < a,b,c,d | a10=b2=c4=d4=1, bab=cac-1=a-1, dad-1=a3, cbc-1=a3b, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 1018 in 234 conjugacy classes, 64 normal (24 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, D5, D5, C10, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C24, Dic5, C20, F5, D10, D10, D10, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C22×D5, C22×D5, C22×C10, C23.8Q8, D10⋊C4, C23.D5, C5×C22⋊C4, C4⋊F5, C22⋊F5, C2×C4×D5, C22×F5, C22×F5, C22×F5, C23×D5, D10.3Q8, D5×C22⋊C4, C2×C4⋊F5, C2×C22⋊F5, C23×F5, D10⋊(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, F5, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C2×F5, C23.8Q8, C4⋊F5, C22×F5, C2×C4⋊F5, D4×F5, D10⋊(C4⋊C4)

Smallest permutation representation of D10⋊(C4⋊C4)
On 40 points
Generators in S40
(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)
(1 12)(2 11)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(30 40)
(1 33 13 23)(2 32 14 22)(3 31 15 21)(4 40 16 30)(5 39 17 29)(6 38 18 28)(7 37 19 27)(8 36 20 26)(9 35 11 25)(10 34 12 24)
(2 8 10 4)(3 5 9 7)(11 19 15 17)(12 16 14 20)(21 39 25 37)(22 36 24 40)(23 33)(26 34 30 32)(27 31 29 35)(28 38)

G:=sub<Sym(40)| (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), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(30,40), (1,33,13,23)(2,32,14,22)(3,31,15,21)(4,40,16,30)(5,39,17,29)(6,38,18,28)(7,37,19,27)(8,36,20,26)(9,35,11,25)(10,34,12,24), (2,8,10,4)(3,5,9,7)(11,19,15,17)(12,16,14,20)(21,39,25,37)(22,36,24,40)(23,33)(26,34,30,32)(27,31,29,35)(28,38)>;

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), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(30,40), (1,33,13,23)(2,32,14,22)(3,31,15,21)(4,40,16,30)(5,39,17,29)(6,38,18,28)(7,37,19,27)(8,36,20,26)(9,35,11,25)(10,34,12,24), (2,8,10,4)(3,5,9,7)(11,19,15,17)(12,16,14,20)(21,39,25,37)(22,36,24,40)(23,33)(26,34,30,32)(27,31,29,35)(28,38) );

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)], [(1,12),(2,11),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(30,40)], [(1,33,13,23),(2,32,14,22),(3,31,15,21),(4,40,16,30),(5,39,17,29),(6,38,18,28),(7,37,19,27),(8,36,20,26),(9,35,11,25),(10,34,12,24)], [(2,8,10,4),(3,5,9,7),(11,19,15,17),(12,16,14,20),(21,39,25,37),(22,36,24,40),(23,33),(26,34,30,32),(27,31,29,35),(28,38)]])

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 8 type + + + + + + + + - + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 D4 Q8 C4○D4 F5 C2×F5 C2×F5 C4⋊F5 D4×F5 kernel D10⋊(C4⋊C4) D10.3Q8 D5×C22⋊C4 C2×C4⋊F5 C2×C22⋊F5 C23×F5 D10⋊C4 C23.D5 C5×C22⋊C4 C2×F5 C22×D5 C22×D5 D10 C22⋊C4 C2×C4 C23 C22 C2 # reps 1 2 1 2 1 1 4 2 2 4 2 2 4 1 2 1 4 2

Matrix representation of D10⋊(C4⋊C4) in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 1 0 0 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40 0
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 1 0 0 0 40 0 1 0 0 0 0 0 1 0 0 0 0 0 1 40
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 27 0 14 34 0 0 27 14 7 0 0 0 0 7 14 27 0 0 34 14 0 27
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40,0,0,1,0,0,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,1,1,1,1,0,0,0,0,0,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,27,27,0,34,0,0,0,14,7,14,0,0,14,7,14,0,0,0,34,0,27,27],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

D10⋊(C4⋊C4) in GAP, Magma, Sage, TeX

D_{10}\rtimes (C_4\rtimes C_4)
% in TeX

G:=Group("D10:(C4:C4)");
// GroupNames label

G:=SmallGroup(320,1037);
// by ID

G=gap.SmallGroup(320,1037);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,100,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^4=d^4=1,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^3,c*b*c^-1=a^3*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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