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## G = C52⋊3SD16order 400 = 24·52

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

Aliases: D20.1D5, C523SD16, C20.12D10, C10.13D20, C4.2D52, C52C82D5, (C5×C10).9D4, C53(C40⋊C2), C51(D4.D5), (C5×D20).2C2, C524Q82C2, C10.2(C5⋊D4), (C5×C20).4C22, C2.5(C5⋊D20), (C5×C52C8)⋊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 C523SD16
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 >

Smallest permutation representation of C523SD16
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 2A 2B 4A 4B 5A 5B 5C 5D 5E 5F 5G 5H 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 20A 20B 20C 20D 20E ··· 20N 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 10 10 10 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 20 2 100 2 2 2 2 4 4 4 4 10 10 2 2 2 2 4 4 4 4 20 20 20 20 2 2 2 2 4 ··· 4 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 D4.D5 D52 C5⋊D20 C52⋊3SD16 kernel C52⋊3SD16 C5×C5⋊2C8 C5×D20 C52⋊4Q8 C5×C10 C5⋊2C8 D20 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 C523SD16 in GL6(𝔽41)

 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 40 0 0 0 0 1 34
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 35 38 0 0 0 0 12 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 11 0 0 0 0 15 11 0 0 0 0 0 0 39 14 0 0 0 0 26 2 0 0 0 0 0 0 28 27 0 0 0 0 18 13
,
 19 6 0 0 0 0 22 22 0 0 0 0 0 0 1 38 0 0 0 0 0 40 0 0 0 0 0 0 17 1 0 0 0 0 40 24

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] >;

C523SD16 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

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