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G = C242D5order 160 = 25·5

1st semidirect product of C24 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C242D5, C23.25D10, C53C22≀C2, (C2×C10)⋊8D4, (C23×C10)⋊3C2, C10.63(C2×D4), C223(C5⋊D4), C23.D513C2, (C2×C10).61C23, (C2×Dic5)⋊3C22, (C22×D5)⋊2C22, C22.66(C22×D5), (C22×C10).42C22, (C2×C5⋊D4)⋊8C2, C2.26(C2×C5⋊D4), SmallGroup(160,174)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C242D5
C1C5C10C2×C10C22×D5C2×C5⋊D4 — C242D5
C5C2×C10 — C242D5
C1C22C24

Generators and relations for C242D5
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 368 in 130 conjugacy classes, 41 normal (8 characteristic)
C1, C2 [×3], C2 [×7], C4 [×3], C22, C22 [×6], C22 [×17], C5, C2×C4 [×3], D4 [×6], C23 [×3], C23 [×7], D5, C10 [×3], C10 [×6], C22⋊C4 [×3], C2×D4 [×3], C24, Dic5 [×3], D10 [×3], C2×C10, C2×C10 [×6], C2×C10 [×14], C22≀C2, C2×Dic5 [×3], C5⋊D4 [×6], C22×D5, C22×C10 [×3], C22×C10 [×6], C23.D5 [×3], C2×C5⋊D4 [×3], C23×C10, C242D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×6], C22×D5, C2×C5⋊D4 [×3], C242D5

Smallest permutation representation of C242D5
On 40 points
Generators in S40
(1 19)(2 20)(3 16)(4 17)(5 18)(6 11)(7 12)(8 13)(9 14)(10 15)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 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 25)(2 24)(3 23)(4 22)(5 21)(6 28)(7 27)(8 26)(9 30)(10 29)(11 33)(12 32)(13 31)(14 35)(15 34)(16 38)(17 37)(18 36)(19 40)(20 39)

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

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

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

C242D5 is a maximal subgroup of
C242F5  C242D10  C24.27D10  C24.30D10  C24.31D10  C24.56D10  D5×C22≀C2  C243D10  C24.34D10  C24.35D10  C245D10  C24.36D10  C24.72D10  D4×C5⋊D4  C248D10  C24.41D10  C24.42D10  (C22×D5)⋊A4  C15⋊C22≀C2  (C2×C10)⋊11D12  C245D15  C242D15  C244D15
C242D5 is a maximal quotient of
C24.62D10  C24.65D10  (C2×C10)⋊8D8  (C5×D4).31D4  C24.20D10  C24.21D10  (C5×Q8)⋊13D4  (C2×C10)⋊8Q16  C10.C22≀C2  (C22×D5)⋊Q8  (C5×D4)⋊14D4  (C5×D4).32D4  2+ 1+4⋊D5  2+ 1+4.D5  2+ 1+4.2D5  2+ 1+42D5  2- 1+42D5  2- 1+4.2D5  C25.2D5  C15⋊C22≀C2  (C2×C10)⋊11D12  C245D15

46 conjugacy classes

class 1 2A2B2C2D···2I2J4A4B4C5A5B10A···10AD
order12222···224445510···10
size11112···220202020222···2

46 irreducible representations

dim11112222
type+++++++
imageC1C2C2C2D4D5D10C5⋊D4
kernelC242D5C23.D5C2×C5⋊D4C23×C10C2×C10C24C23C22
# reps133162624

Matrix representation of C242D5 in GL4(𝔽41) generated by

40000
04000
0010
00040
,
1000
04000
00400
0001
,
40000
04000
00400
00040
,
1000
0100
00400
00040
,
16000
01800
00100
00037
,
01800
16000
00037
00100
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[16,0,0,0,0,18,0,0,0,0,10,0,0,0,0,37],[0,16,0,0,18,0,0,0,0,0,0,10,0,0,37,0] >;

C242D5 in GAP, Magma, Sage, TeX

C_2^4\rtimes_2D_5
% in TeX

G:=Group("C2^4:2D5");
// GroupNames label

G:=SmallGroup(160,174);
// by ID

G=gap.SmallGroup(160,174);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,218,4613]);
// Polycyclic

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

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