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G = C423Dic5order 320 = 26·5

3rd semidirect product of C42 and Dic5 acting via Dic5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C423Dic5, (C4×C20)⋊13C4, (D4×C10)⋊14C4, (C2×D4)⋊2Dic5, C41D4.2D5, C54(C42⋊C4), (C2×D4).10D10, C23⋊Dic58C2, (C22×C10).17D4, C23.8(C5⋊D4), C10.46(C23⋊C4), (D4×C10).173C22, C2.10(C23⋊Dic5), C22.16(C23.D5), (C5×C41D4).7C2, (C2×C4).3(C2×Dic5), (C2×C20).183(C2×C4), (C2×C10).167(C22⋊C4), SmallGroup(320,103)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C423Dic5
C1C5C10C2×C10C22×C10D4×C10C23⋊Dic5 — C423Dic5
C5C10C2×C10C2×C20 — C423Dic5
C1C2C22C2×D4C41D4

Generators and relations for C423Dic5
 G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, cac-1=a-1, dad-1=a-1b, cbc-1=b-1, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 366 in 86 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2 [×4], C4 [×5], C22, C22 [×7], C5, C2×C4, C2×C4 [×3], D4 [×6], C23 [×2], C23, C10, C10 [×4], C42, C22⋊C4 [×2], C2×D4 [×2], C2×D4 [×2], Dic5 [×2], C20 [×3], C2×C10, C2×C10 [×7], C23⋊C4 [×2], C41D4, C2×Dic5 [×2], C2×C20, C2×C20, C5×D4 [×6], C22×C10 [×2], C22×C10, C42⋊C4, C23.D5 [×2], C4×C20, D4×C10 [×2], D4×C10 [×2], C23⋊Dic5 [×2], C5×C41D4, C423Dic5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, Dic5 [×2], D10, C23⋊C4, C2×Dic5, C5⋊D4 [×2], C42⋊C4, C23.D5, C23⋊Dic5, C423Dic5

Smallest permutation representation of C423Dic5
On 40 points
Generators in S40
(1 30 25 6)(2 7 26 21)(3 22 27 8)(4 9 28 23)(5 24 29 10)
(1 30 25 6)(2 7 26 21)(3 22 27 8)(4 9 28 23)(5 24 29 10)(11 16 39 34)(12 35 40 17)(13 18 31 36)(14 37 32 19)(15 20 33 38)
(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 39 6 34)(2 38 7 33)(3 37 8 32)(4 36 9 31)(5 35 10 40)(11 30 16 25)(12 29 17 24)(13 28 18 23)(14 27 19 22)(15 26 20 21)

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

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

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

41 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G5A5B10A···10F10G···10N20A···20L
order12222244444445510···1010···1020···20
size11244844440404040222···28···84···4

41 irreducible representations

dim111112222224444
type+++++--+++
imageC1C2C2C4C4D4D5Dic5Dic5D10C5⋊D4C23⋊C4C42⋊C4C23⋊Dic5C423Dic5
kernelC423Dic5C23⋊Dic5C5×C41D4C4×C20D4×C10C22×C10C41D4C42C2×D4C2×D4C23C10C5C2C1
# reps121222222281248

Matrix representation of C423Dic5 in GL4(𝔽41) generated by

0100
40000
0010
0001
,
0100
40000
0001
00400
,
01600
16000
00018
00180
,
0010
0001
0100
1000
G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,1,0,0,0,0,0,0,40,0,0,1,0],[0,16,0,0,16,0,0,0,0,0,0,18,0,0,18,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;

C423Dic5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_3{\rm Dic}_5
% in TeX

G:=Group("C4^2:3Dic5");
// GroupNames label

G:=SmallGroup(320,103);
// by ID

G=gap.SmallGroup(320,103);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,1571,570,297,136,1684,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c^5,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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