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G = C242D10order 320 = 26·5

1st semidirect product of C24 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C242D10, C52C2≀C22, (C2×C20)⋊2D4, C22≀C21D5, C22⋊C43D10, (C2×Dic5)⋊2D4, (C22×D5)⋊2D4, (C22×C10)⋊3D4, D46D103C2, (C2×D4).33D10, C242D51C2, C231(C5⋊D4), C23⋊Dic55C2, C22.33(D4×D5), C10.43C22≀C2, (C23×C10)⋊7C22, C23.D54C22, (D4×C10).49C22, C23.1D105C2, C2.11(C23⋊D10), C23.74(C22×D5), (C22×C10).113C23, (C2×C4)⋊1(C5⋊D4), (C5×C22≀C2)⋊1C2, (C2×C10).30(C2×D4), (C2×C5⋊D4).5C22, C22.29(C2×C5⋊D4), (C5×C22⋊C4)⋊34C22, SmallGroup(320,659)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C242D10
C1C5C10C2×C10C22×C10C2×C5⋊D4D46D10 — C242D10
C5C10C22×C10 — C242D10
C1C2C23C22≀C2

Generators and relations for C242D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=f2=1, ab=ba, eae-1=ac=ca, ad=da, faf=abcd, bc=cb, ebe-1=fbf=bd=db, fcf=cd=dc, ce=ec, de=ed, df=fd, fef=e-1 >

Subgroups: 926 in 198 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2 [×8], C4 [×6], C22, C22 [×2], C22 [×18], C5, C2×C4, C2×C4 [×8], D4 [×15], Q8, C23 [×2], C23 [×8], D5 [×2], C10, C10 [×6], C22⋊C4, C22⋊C4 [×5], C2×D4, C2×D4 [×8], C4○D4 [×3], C24, Dic5 [×4], C20 [×2], D10 [×5], C2×C10, C2×C10 [×2], C2×C10 [×13], C23⋊C4 [×3], C22≀C2, C22≀C2 [×2], 2+ 1+4, Dic10, C4×D5 [×2], D20, C2×Dic5 [×2], C2×Dic5 [×3], C5⋊D4 [×10], C2×C20, C2×C20, C5×D4 [×4], C22×D5 [×2], C22×D5, C22×C10 [×2], C22×C10 [×5], C2≀C22, C23.D5 [×2], C23.D5 [×2], C5×C22⋊C4, C5×C22⋊C4, C4○D20, D4×D5 [×2], D42D5 [×2], C2×C5⋊D4 [×2], C2×C5⋊D4 [×3], D4×C10, D4×C10, C23×C10, C23.1D10 [×2], C23⋊Dic5, C242D5 [×2], C5×C22≀C2, D46D10, C242D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×2], C22×D5, C2≀C22, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, C242D10

Smallest permutation representation of C242D10
On 40 points
Generators in S40
(1 33)(2 27)(3 35)(4 29)(5 37)(6 21)(7 39)(8 23)(9 31)(10 25)(11 36)(12 30)(13 38)(14 22)(15 40)(16 24)(17 32)(18 26)(19 34)(20 28)
(1 26)(2 19)(3 28)(4 11)(5 30)(6 13)(7 22)(8 15)(9 24)(10 17)(12 37)(14 39)(16 31)(18 33)(20 35)(21 38)(23 40)(25 32)(27 34)(29 36)
(1 18)(2 19)(3 20)(4 11)(5 12)(6 13)(7 14)(8 15)(9 16)(10 17)(21 38)(22 39)(23 40)(24 31)(25 32)(26 33)(27 34)(28 35)(29 36)(30 37)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 29)(12 30)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)
(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 5)(2 4)(6 10)(7 9)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 30)(19 29)(20 28)(31 39)(32 38)(33 37)(34 36)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F5A5B10A···10F10G···10R10S10T20A···20F
order12222222224444445510···1010···10101020···20
size1122244420204820204040222···24···4888···8

44 irreducible representations

dim1111112222222222444
type++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D4D5D10D10D10C5⋊D4C5⋊D4C2≀C22D4×D5C242D10
kernelC242D10C23.1D10C23⋊Dic5C242D5C5×C22≀C2D46D10C2×Dic5C2×C20C22×D5C22×C10C22≀C2C22⋊C4C2×D4C24C2×C4C23C5C22C1
# reps1212112121222244248

Matrix representation of C242D10 in GL4(𝔽41) generated by

40000
04000
002440
00117
,
244000
11700
00171
004024
,
17100
402400
00171
004024
,
40000
04000
00400
00040
,
00407
00347
40700
34700
,
40000
34100
00400
00341
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,24,1,0,0,40,17],[24,1,0,0,40,17,0,0,0,0,17,40,0,0,1,24],[17,40,0,0,1,24,0,0,0,0,17,40,0,0,1,24],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,0,40,34,0,0,7,7,40,34,0,0,7,7,0,0],[40,34,0,0,0,1,0,0,0,0,40,34,0,0,0,1] >;

C242D10 in GAP, Magma, Sage, TeX

C_2^4\rtimes_2D_{10}
% in TeX

G:=Group("C2^4:2D10");
// GroupNames label

G:=SmallGroup(320,659);
// by ID

G=gap.SmallGroup(320,659);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,570,1684,12550]);
// Polycyclic

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

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