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G = D5⋊(C4.D4)  order 320 = 26·5

The semidirect product of D5 and C4.D4 acting via C4.D4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5⋊(C4.D4), (C4×D5).38D4, (C2×D4).10F5, C23.F54C2, C23.4(C2×F5), (C2×D20).13C4, (D4×C10).11C4, D5⋊M4(2)⋊3C2, Dic5.6(C2×D4), (C23×D5).5C4, C4.15(C22⋊F5), C20.15(C22⋊C4), C22.F51C22, D10.44(C22⋊C4), C22.14(C22×F5), (C2×Dic5).174C23, (C2×D4×D5).15C2, C51(C2×C4.D4), (C2×C4).38(C2×F5), (C2×C20).57(C2×C4), C2.23(C2×C22⋊F5), C10.22(C2×C22⋊C4), (C22×D5).9(C2×C4), (C2×C4×D5).202C22, (C22×C10).29(C2×C4), (C2×C10).81(C22×C4), (C2×C5⋊D4).90C22, SmallGroup(320,1116)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D5⋊(C4.D4)
C1C5C10Dic5C2×Dic5C22.F5D5⋊M4(2) — D5⋊(C4.D4)
C5C10C2×C10 — D5⋊(C4.D4)
C1C2C2×C4C2×D4

Generators and relations for D5⋊(C4.D4)
 G = < a,b,c,d,e | a5=b2=c4=1, d4=c2, e2=c, bab=cac-1=a-1, dad-1=eae-1=a3, cbc-1=a3b, dbd-1=ebe-1=a2b, dcd-1=c-1, ce=ec, ede-1=c-1d3 >

Subgroups: 970 in 186 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2 [×8], C4 [×2], C4 [×2], C22, C22 [×20], C5, C8 [×4], C2×C4, C2×C4 [×5], D4 [×8], C23 [×2], C23 [×11], D5 [×2], D5 [×3], C10, C10 [×3], C2×C8 [×2], M4(2) [×6], C22×C4, C2×D4, C2×D4 [×7], C24 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10 [×14], C2×C10, C2×C10 [×4], C4.D4 [×4], C2×M4(2) [×2], C22×D4, C5⋊C8 [×4], C4×D5 [×4], D20 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×D5 [×2], C22×D5 [×8], C22×C10 [×2], C2×C4.D4, D5⋊C8 [×2], C4.F5 [×2], C22.F5 [×4], C2×C4×D5, C2×D20, D4×D5 [×4], C2×C5⋊D4 [×2], D4×C10, C23×D5 [×2], C23.F5 [×4], D5⋊M4(2) [×2], C2×D4×D5, D5⋊(C4.D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C4.D4 [×2], C2×C22⋊C4, C2×F5 [×3], C2×C4.D4, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, D5⋊(C4.D4)

Smallest permutation representation of D5⋊(C4.D4)
On 40 points
Generators in S40
(1 26 9 22 35)(2 23 27 36 10)(3 37 24 11 28)(4 12 38 29 17)(5 30 13 18 39)(6 19 31 40 14)(7 33 20 15 32)(8 16 34 25 21)
(1 39)(2 14)(3 32)(4 21)(5 35)(6 10)(7 28)(8 17)(9 13)(11 33)(12 25)(15 37)(16 29)(18 26)(19 36)(20 24)(22 30)(23 40)(27 31)(34 38)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)(33 35 37 39)(34 40 38 36)
(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 6 3 4 5 2 7 8)(9 14 11 12 13 10 15 16)(17 18 23 20 21 22 19 24)(25 26 31 28 29 30 27 32)(33 34 35 40 37 38 39 36)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D 5 8A···8H10A10B10C10D10E10F10G20A20B
order1222222222444458···8101010101010102020
size1124455102020221010420···20444888888

32 irreducible representations

dim11111112444448
type+++++++++++
imageC1C2C2C2C4C4C4D4F5C4.D4C2×F5C2×F5C22⋊F5D5⋊(C4.D4)
kernelD5⋊(C4.D4)C23.F5D5⋊M4(2)C2×D4×D5C2×D20D4×C10C23×D5C4×D5C2×D4D5C2×C4C23C4C1
# reps14212244121242

Matrix representation of D5⋊(C4.D4) in GL8(ℤ)

-1-1-1-10000
10000000
01000000
00100000
00001000
00000100
00000010
00000001
,
-1-1-1-10000
00010000
00100000
01000000
0000-1000
00000-100
000000-10
0000000-1
,
-10000000
11110000
000-10000
00-100000
00000100
0000-1000
00000001
000000-10
,
10110000
11010000
-10-100000
-100-10000
00000010
0000000-1
00000100
00001000
,
10110000
11010000
-10-100000
-100-10000
00000010
00000001
00000100
0000-1000

G:=sub<GL(8,Integers())| [-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,1,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0],[1,1,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0],[1,1,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

D5⋊(C4.D4) in GAP, Magma, Sage, TeX

D_5\rtimes (C_4.D_4)
% in TeX

G:=Group("D5:(C4.D4)");
// GroupNames label

G:=SmallGroup(320,1116);
// by ID

G=gap.SmallGroup(320,1116);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,297,136,1684,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^4=1,d^4=c^2,e^2=c,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=e*a*e^-1=a^3,c*b*c^-1=a^3*b,d*b*d^-1=e*b*e^-1=a^2*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=c^-1*d^3>;
// generators/relations

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