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## G = C52⋊4SD16order 400 = 24·52

### 3rd semidirect product of C52 and SD16 acting via SD16/C4=C22

Aliases: C524SD16, Dic101D5, C10.14D20, C20.13D10, C4.3D52, C52C83D5, C51(Q8⋊D5), C52(C40⋊C2), (C5×C10).10D4, (C5×Dic10)⋊2C2, C20⋊D5.2C2, C10.3(C5⋊D4), (C5×C20).5C22, C2.6(C5⋊D20), (C5×C52C8)⋊3C2, SmallGroup(400,68)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C20 — C52⋊4SD16
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — C5×Dic10 — C52⋊4SD16
 Lower central C52 — C5×C10 — C5×C20 — C52⋊4SD16
 Upper central C1 — C2 — C4

Generators and relations for C524SD16
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=dad=a-1, bc=cb, dbd=b-1, dcd=c3 >

Subgroups: 500 in 56 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C8, D4, Q8, D5, C10, C10, SD16, Dic5, C20, C20, D10, C52, C52C8, C40, Dic10, D20, C5×Q8, C5⋊D5, C5×C10, C40⋊C2, Q8⋊D5, C5×Dic5, C5×C20, C2×C5⋊D5, C5×C52C8, C5×Dic10, C20⋊D5, C524SD16
Quotients: C1, C2, C22, D4, D5, SD16, D10, D20, C5⋊D4, C40⋊C2, Q8⋊D5, D52, C5⋊D20, C524SD16

Smallest permutation representation of C524SD16
On 40 points
Generators in S40
(1 37 20 14 31)(2 32 15 21 38)(3 39 22 16 25)(4 26 9 23 40)(5 33 24 10 27)(6 28 11 17 34)(7 35 18 12 29)(8 30 13 19 36)
(1 31 14 20 37)(2 32 15 21 38)(3 25 16 22 39)(4 26 9 23 40)(5 27 10 24 33)(6 28 11 17 34)(7 29 12 18 35)(8 30 13 19 36)
(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 3)(2 6)(5 7)(9 23)(10 18)(11 21)(12 24)(13 19)(14 22)(15 17)(16 20)(25 37)(26 40)(27 35)(28 38)(29 33)(30 36)(31 39)(32 34)

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

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

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

49 conjugacy classes

 class 1 2A 2B 4A 4B 5A 5B 5C 5D 5E 5F 5G 5H 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 20A 20B 20C 20D 20E ··· 20N 20O 20P 20Q 20R 40A ··· 40H order 1 2 2 4 4 5 5 5 5 5 5 5 5 8 8 10 10 10 10 10 10 10 10 20 20 20 20 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 100 2 20 2 2 2 2 4 4 4 4 10 10 2 2 2 2 4 4 4 4 2 2 2 2 4 ··· 4 20 20 20 20 10 ··· 10

49 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 D4 D5 D5 SD16 D10 D20 C5⋊D4 C40⋊C2 Q8⋊D5 D52 C5⋊D20 C52⋊4SD16 kernel C52⋊4SD16 C5×C5⋊2C8 C5×Dic10 C20⋊D5 C5×C10 C5⋊2C8 Dic10 C52 C20 C10 C10 C5 C5 C4 C2 C1 # reps 1 1 1 1 1 2 2 2 4 4 4 8 2 4 4 8

Matrix representation of C524SD16 in GL4(𝔽41) generated by

 0 40 0 0 1 34 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 40 0 0 1 34
,
 0 40 0 0 40 0 0 0 0 0 14 12 0 0 29 16
,
 0 1 0 0 1 0 0 0 0 0 32 11 0 0 30 9
G:=sub<GL(4,GF(41))| [0,1,0,0,40,34,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,34],[0,40,0,0,40,0,0,0,0,0,14,29,0,0,12,16],[0,1,0,0,1,0,0,0,0,0,32,30,0,0,11,9] >;

C524SD16 in GAP, Magma, Sage, TeX

C_5^2\rtimes_4{\rm SD}_{16}
% in TeX

G:=Group("C5^2:4SD16");
// GroupNames label

G:=SmallGroup(400,68);
// by ID

G=gap.SmallGroup(400,68);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,73,31,218,50,970,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^3>;
// generators/relations

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