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G = (D4×C10)⋊C4order 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⋊F52C22, D4.7(C2×F5), D20⋊C43C2, D5⋊C81C22, (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.C231C2, 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

C1C20 — (D4×C10)⋊C4
C1C5C10D10C4×D5C4⋊F5D10.C23 — (D4×C10)⋊C4
C5C10C20 — (D4×C10)⋊C4
C1C2C2×C4C2×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 [×8], C4 [×2], C4 [×4], C22, C22 [×20], C5, C8 [×2], C2×C4, C2×C4 [×7], D4 [×2], D4 [×8], C23 [×11], D5 [×2], D5 [×3], C10, C10 [×3], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C2×D4, C2×D4 [×8], C24, Dic5 [×2], C20 [×2], F5 [×2], D10 [×2], D10 [×14], C2×C10, C2×C10 [×4], D4⋊C4 [×4], C42⋊C2, C2×M4(2), C22×D4, C5⋊C8 [×2], C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4 [×2], C5×D4, C2×F5 [×2], C22×D5, C22×D5 [×9], C22×C10, C23.37D4, D5⋊C8 [×2], C4.F5, C4×F5, C4⋊F5 [×2], C22.F5, C22⋊F5, C2×C4×D5, C2×D20, D4×D5 [×4], D4×D5 [×2], C2×C5⋊D4, D4×C10, C23×D5, D20⋊C4 [×4], D5⋊M4(2), D10.C23, C2×D4×D5, (D4×C10)⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C8⋊C22 [×2], C2×F5 [×3], C23.37D4, C22⋊F5 [×2], 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 19 14 9)(2 20 15 10)(3 16 11 6)(4 17 12 7)(5 18 13 8)(21 39 26 34)(22 40 27 35)(23 31 28 36)(24 32 29 37)(25 33 30 38)
(1 14)(2 15)(3 11)(4 12)(5 13)(21 39)(22 40)(23 31)(24 32)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)
(1 26 13 29)(2 28 12 27)(3 30 11 25)(4 22 15 23)(5 24 14 21)(6 38 16 33)(7 40 20 31)(8 32 19 39)(9 34 18 37)(10 36 17 35)

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

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

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

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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