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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, D20.7(C2×C4), (C4×D5).33D4, 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), (C4×D5).36C23, (D4×D5).13C22, 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

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

Generators and relations
 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 >

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

(11 18)(12 19)(13 20)(14 16)(15 17)(21 37)(22 38)(23 39)(24 40)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)

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

Matrix representation ►G ⊆ 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] >;
 

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

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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