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G = (D4×C10)⋊C4  order 320 = 26·5

6th semidirect product of D4×C10 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (D4×D5)⋊7C4, (C2×D4)⋊4F5, (C2×D20)⋊9C4, (D4×C10)⋊6C4, C4⋊F5⋊2C22, D4.7(C2×F5), D20⋊C4⋊3C2, D5⋊C8⋊1C22, (C4×D5).33D4, D20.7(C2×C4), D10.94(C2×D4), D5⋊M4(2)⋊1C2, Dic5.4(C2×D4), C5⋊(C23.37D4), C4.14(C22×F5), C20.14(C22×C4), D5.4(C8⋊C22), (D4×D5).13C22, (C4×D5).36C23, C4.10(C22⋊F5), C20.10(C22⋊C4), (C2×Dic5).118D4, (C22×D5).146D4, D10.42(C22⋊C4), C22.26(C22⋊F5), D10.C23⋊1C2, Dic5.10(C22⋊C4), (C2×D4×D5).14C2, (C5×D4).7(C2×C4), (C2×C4).33(C2×F5), (C2×C20).50(C2×C4), (C4×D5).20(C2×C4), C2.15(C2×C22⋊F5), C10.14(C2×C22⋊C4), (C2×C4×D5).197C22, (C2×C10).54(C22⋊C4), SmallGroup(320,1105)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — (D4×C10)⋊C4
Chief series
C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — D10.C23 — (D4×C10)⋊C4
Lower central
C5 — C10 — C20 — (D4×C10)⋊C4
Upper central
C1 — C2 — C2×C4 — C2×D4

Generators and relations for (D4×C10)⋊C4
 G = < a,b,c,d | a10=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a3b2, cbc=dbd-1=b-1, dcd-1=b-1c >

Subgroups: 1018 in 190 conjugacy classes, 50 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, D5, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, Dic5, C20, F5, D10, D10, C2×C10, C2×C10, D4⋊C4, C42⋊C2, C2×M4(2), C22×D4, C5⋊C8, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×D4, C2×F5, C22×D5, C22×D5, C22×C10, C23.37D4, D5⋊C8, C4.F5, C4×F5, C4⋊F5, C22.F5, C22⋊F5, C2×C4×D5, C2×D20, D4×D5, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, D20⋊C4, D5⋊M4(2), D10.C23, C2×D4×D5, (D4×C10)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, F5, C2×C22⋊C4, C8⋊C22, C2×F5, C23.37D4, C22⋊F5, C22×F5, C2×C22⋊F5, (D4×C10)⋊C4

Smallest permutation representation of (D4×C10)⋊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 8 19 12)(2 9 20 13)(3 10 16 14)(4 6 17 15)(5 7 18 11)(21 39 26 34)(22 40 27 35)(23 31 28 36)(24 32 29 37)(25 33 30 38)

(1 19)(2 20)(3 16)(4 17)(5 18)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 31)(29 32)(30 33)

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H 5 8A8B8C8D10A10B10C10D10E10F10G20A20B
order12222222224444444458888101010101010102020
size112445510202022101020202020420202020444888888

32 irreducible representations

dim111111112224444448
type+++++++++++++++
imageC1C2C2C2C2C4C4C4D4D4D4F5C8⋊C22C2×F5C2×F5C22⋊F5C22⋊F5(D4×C10)⋊C4
kernel(D4×C10)⋊C4D20⋊C4D5⋊M4(2)D10.C23C2×D4×D5C2×D20D4×D5D4×C10C4×D5C2×Dic5C22×D5C2×D4D5C2×C4D4C4C22C1
# reps141112422111212222

Matrix representation of (D4×C10)⋊C4 ►in GL8(𝔽41)

035000000
77000000
00110000
0033340000
000040000
000004000
00000010
000033501
,
400000000
040000000
004000000
000400000
00001900
0000184000
000029120
0000436440
,
400000000
040000000
00100000
00010000
00001900
000004000
00000010
0000836440
,
004000000
000400000
76000000
3334000000
00000010
000018401816
000040000
000033501

G:=sub<GL(8,GF(41))| [0,7,0,0,0,0,0,0,35,7,0,0,0,0,0,0,0,0,1,33,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,40,0,0,33,0,0,0,0,0,40,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,18,2,4,0,0,0,0,9,40,9,36,0,0,0,0,0,0,1,4,0,0,0,0,0,0,20,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,9,40,0,36,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,40],[0,0,7,33,0,0,0,0,0,0,6,34,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,18,40,33,0,0,0,0,0,40,0,5,0,0,0,0,1,18,0,0,0,0,0,0,0,16,0,1] >;
 

(D4×C10)⋊C4 in GAP, Magma, Sage, TeX 

(D_4\times C_{10})\rtimes C_4
 
% in TeX
 
G:=Group("(D4xC10):C4");
 
// GroupNames label
 
G:=SmallGroup(320,1105);
 
// by ID
 
G=gap.SmallGroup(320,1105);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,1684,438,102,6278,1595]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^10=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^3*b^2,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
 
// generators/relations
 

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