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G = C52⋊3SD16  order 400 = 24·52

2nd semidirect product of C52 and SD16 acting via SD16/C4=C22

metabelian, supersoluble, monomial

Aliases: D20.1D5, C52⋊3SD16, C20.12D10, C10.13D20, C4.2D52, C5⋊2C8⋊2D5, (C5×C10).9D4, C5⋊3(C40⋊C2), C5⋊1(D4.D5), (C5×D20).2C2, C52⋊4Q8⋊2C2, C10.2(C5⋊D4), (C5×C20).4C22, C2.5(C5⋊D20), (C5×C5⋊2C8)⋊2C2, SmallGroup(400,67)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5×C20 — C52⋊3SD16
Chief series
C1 — C5 — C52 — C5×C10 — C5×C20 — C5×D20 — C52⋊3SD16
Lower central
C52 — C5×C10 — C5×C20 — C52⋊3SD16
Upper central
C1 — C2 — C4

Generators and relations for C52⋊3SD16
 G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd=b-1, dcd=c3 >

C1
C2
20C2
C5
C5
2C5
2C5
C4
10C22
50C4
C10
C10
2C10
2C10
4D5
20C10
C52
5D4
5C8
25Q8
C20
C20
2C20
2D10
2C20
10Dic5
10Dic5
10Dic5
10Dic5
10Dic5
10C2×C10
10Dic5
C5×C10
4C5×D5
25SD16
D20
C5⋊2C8
5C40
5Dic10
5Dic10
5C5×D4
10Dic10
10Dic10
C5×C20
2D5×C10
2C52⋊6C4
5C40⋊C2
5D4.D5
C5×D20
C52⋊4Q8
C5×C5⋊2C8
C52⋊3SD16

Smallest permutation representation of C52⋊3SD16
►On 80 points
Generators in S80
(1 42 49 63 66)(2 67 64 50 43)(3 44 51 57 68)(4 69 58 52 45)(5 46 53 59 70)(6 71 60 54 47)(7 48 55 61 72)(8 65 62 56 41)(9 21 74 39 25)(10 26 40 75 22)(11 23 76 33 27)(12 28 34 77 24)(13 17 78 35 29)(14 30 36 79 18)(15 19 80 37 31)(16 32 38 73 20)

(1 49 66 42 63)(2 50 67 43 64)(3 51 68 44 57)(4 52 69 45 58)(5 53 70 46 59)(6 54 71 47 60)(7 55 72 48 61)(8 56 65 41 62)(9 39 21 25 74)(10 40 22 26 75)(11 33 23 27 76)(12 34 24 28 77)(13 35 17 29 78)(14 36 18 30 79)(15 37 19 31 80)(16 38 20 32 73)

(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)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)

(1 23)(2 18)(3 21)(4 24)(5 19)(6 22)(7 17)(8 20)(9 68)(10 71)(11 66)(12 69)(13 72)(14 67)(15 70)(16 65)(25 57)(26 60)(27 63)(28 58)(29 61)(30 64)(31 59)(32 62)(33 49)(34 52)(35 55)(36 50)(37 53)(38 56)(39 51)(40 54)(41 73)(42 76)(43 79)(44 74)(45 77)(46 80)(47 75)(48 78)
 
G:=sub<Sym(80)| (1,42,49,63,66)(2,67,64,50,43)(3,44,51,57,68)(4,69,58,52,45)(5,46,53,59,70)(6,71,60,54,47)(7,48,55,61,72)(8,65,62,56,41)(9,21,74,39,25)(10,26,40,75,22)(11,23,76,33,27)(12,28,34,77,24)(13,17,78,35,29)(14,30,36,79,18)(15,19,80,37,31)(16,32,38,73,20), (1,49,66,42,63)(2,50,67,43,64)(3,51,68,44,57)(4,52,69,45,58)(5,53,70,46,59)(6,54,71,47,60)(7,55,72,48,61)(8,56,65,41,62)(9,39,21,25,74)(10,40,22,26,75)(11,33,23,27,76)(12,34,24,28,77)(13,35,17,29,78)(14,36,18,30,79)(15,37,19,31,80)(16,38,20,32,73), (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)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,23)(2,18)(3,21)(4,24)(5,19)(6,22)(7,17)(8,20)(9,68)(10,71)(11,66)(12,69)(13,72)(14,67)(15,70)(16,65)(25,57)(26,60)(27,63)(28,58)(29,61)(30,64)(31,59)(32,62)(33,49)(34,52)(35,55)(36,50)(37,53)(38,56)(39,51)(40,54)(41,73)(42,76)(43,79)(44,74)(45,77)(46,80)(47,75)(48,78)>;
 
G:=Group( (1,42,49,63,66)(2,67,64,50,43)(3,44,51,57,68)(4,69,58,52,45)(5,46,53,59,70)(6,71,60,54,47)(7,48,55,61,72)(8,65,62,56,41)(9,21,74,39,25)(10,26,40,75,22)(11,23,76,33,27)(12,28,34,77,24)(13,17,78,35,29)(14,30,36,79,18)(15,19,80,37,31)(16,32,38,73,20), (1,49,66,42,63)(2,50,67,43,64)(3,51,68,44,57)(4,52,69,45,58)(5,53,70,46,59)(6,54,71,47,60)(7,55,72,48,61)(8,56,65,41,62)(9,39,21,25,74)(10,40,22,26,75)(11,33,23,27,76)(12,34,24,28,77)(13,35,17,29,78)(14,36,18,30,79)(15,37,19,31,80)(16,38,20,32,73), (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)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,23)(2,18)(3,21)(4,24)(5,19)(6,22)(7,17)(8,20)(9,68)(10,71)(11,66)(12,69)(13,72)(14,67)(15,70)(16,65)(25,57)(26,60)(27,63)(28,58)(29,61)(30,64)(31,59)(32,62)(33,49)(34,52)(35,55)(36,50)(37,53)(38,56)(39,51)(40,54)(41,73)(42,76)(43,79)(44,74)(45,77)(46,80)(47,75)(48,78) );
 
G=PermutationGroup([[(1,42,49,63,66),(2,67,64,50,43),(3,44,51,57,68),(4,69,58,52,45),(5,46,53,59,70),(6,71,60,54,47),(7,48,55,61,72),(8,65,62,56,41),(9,21,74,39,25),(10,26,40,75,22),(11,23,76,33,27),(12,28,34,77,24),(13,17,78,35,29),(14,30,36,79,18),(15,19,80,37,31),(16,32,38,73,20)], [(1,49,66,42,63),(2,50,67,43,64),(3,51,68,44,57),(4,52,69,45,58),(5,53,70,46,59),(6,54,71,47,60),(7,55,72,48,61),(8,56,65,41,62),(9,39,21,25,74),(10,40,22,26,75),(11,33,23,27,76),(12,34,24,28,77),(13,35,17,29,78),(14,36,18,30,79),(15,37,19,31,80),(16,38,20,32,73)], [(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),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)], [(1,23),(2,18),(3,21),(4,24),(5,19),(6,22),(7,17),(8,20),(9,68),(10,71),(11,66),(12,69),(13,72),(14,67),(15,70),(16,65),(25,57),(26,60),(27,63),(28,58),(29,61),(30,64),(31,59),(32,62),(33,49),(34,52),(35,55),(36,50),(37,53),(38,56),(39,51),(40,54),(41,73),(42,76),(43,79),(44,74),(45,77),(46,80),(47,75),(48,78)]])
 

49 conjugacy classes

class 1 2A2B4A4B5A5B5C5D5E5F5G5H8A8B10A10B10C10D10E10F10G10H10I10J10K10L20A20B20C20D20E···20N40A···40H
order1224455555555881010101010101010101010102020202020···2040···40
size11202100222244441010222244442020202022224···410···10

49 irreducible representations

dim1111222222224444
type+++++++++-++-
imageC1C2C2C2D4D5D5SD16D10D20C5⋊D4C40⋊C2D4.D5D52C5⋊D20C52⋊3SD16
kernelC52⋊3SD16C5×C5⋊2C8C5×D20C52⋊4Q8C5×C10C5⋊2C8D20C52C20C10C10C5C5C4C2C1
# reps1111122244482448

Matrix representation of C52⋊3SD16 ►in GL6(𝔽41)

100000
010000
001000
000100
0000040
0000134
,
100000
010000
00353800
00124000
000010
000001
,
0110000
15110000
00391400
0026200
00002827
00001813
,
1960000
22220000
0013800
0004000
0000171
00004024

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,34],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,12,0,0,0,0,38,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,15,0,0,0,0,11,11,0,0,0,0,0,0,39,26,0,0,0,0,14,2,0,0,0,0,0,0,28,18,0,0,0,0,27,13],[19,22,0,0,0,0,6,22,0,0,0,0,0,0,1,0,0,0,0,0,38,40,0,0,0,0,0,0,17,40,0,0,0,0,1,24] >;
 

C52⋊3SD16 in GAP, Magma, Sage, TeX 

C_5^2\rtimes_3{\rm SD}_{16}
 
% in TeX
 
G:=Group("C5^2:3SD16");
 
// GroupNames label
 
G:=SmallGroup(400,67);
 
// by ID
 
G=gap.SmallGroup(400,67);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,73,31,218,50,970,11525]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^3>;
 
// generators/relations
 

Export

Subgroup lattice of C52⋊3SD16 in TeX

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