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G = C5×C42⋊C6order 480 = 25·3·5

Direct product of C5 and C42⋊C6

Aliases: C5×C42⋊C6, C421C30, (C4×C20)⋊2C6, C422C2⋊C15, C42⋊C31C10, C23.1(C5×A4), (C22×C10).1A4, C22.3(C10×A4), (C5×C42⋊C3)⋊2C2, (C5×C422C2)⋊C3, (C2×C10).7(C2×A4), SmallGroup(480,657)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C5×C42⋊C6
 Chief series C1 — C22 — C42 — C4×C20 — C5×C42⋊C3 — C5×C42⋊C6
 Lower central C42 — C5×C42⋊C6
 Upper central C1 — C5

Generators and relations for C5×C42⋊C6
G = < a,b,c,d | a5=b4=c4=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=c-1, dcd-1=b-1c >

Smallest permutation representation of C5×C42⋊C6
On 80 points
Generators in S80
(1 2 4 5 3)(6 13 17 9 18)(7 14 15 10 19)(8 12 16 11 20)(21 61 28 66 75)(22 62 29 67 76)(23 57 30 68 77)(24 58 31 63 78)(25 59 32 64 79)(26 60 27 65 80)(33 49 71 54 40)(34 50 72 55 41)(35 45 73 56 42)(36 46 74 51 43)(37 47 69 52 44)(38 48 70 53 39)
(1 50 7 57)(2 72 14 30)(3 34 19 23)(4 55 15 68)(5 41 10 77)(6 47 8 60)(9 44 11 80)(12 27 13 69)(16 65 17 52)(18 37 20 26)(21 22 24 36)(25 35 33 38)(28 29 31 74)(32 73 71 70)(39 79 42 40)(43 75 76 78)(45 49 48 59)(46 61 62 58)(51 66 67 63)(53 64 56 54)
(1 62 6 49)(2 29 13 71)(3 22 18 33)(4 67 17 54)(5 76 9 40)(7 46 8 59)(10 43 11 79)(12 32 14 74)(15 51 16 64)(19 36 20 25)(21 26 35 23)(24 37 38 34)(27 73 30 28)(31 69 70 72)(39 41 78 44)(42 77 75 80)(45 57 61 60)(47 48 50 58)(52 53 55 63)(56 68 66 65)
(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)

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

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

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

50 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 10E 10F 10G 10H 15A ··· 15H 20A ··· 20H 20I 20J 20K 20L 30A ··· 30H order 1 2 2 3 3 4 4 4 5 5 5 5 6 6 10 10 10 10 10 10 10 10 15 ··· 15 20 ··· 20 20 20 20 20 30 ··· 30 size 1 3 4 16 16 6 6 12 1 1 1 1 16 16 3 3 3 3 4 4 4 4 16 ··· 16 6 ··· 6 12 12 12 12 16 ··· 16

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 6 6 type + + + + image C1 C2 C3 C5 C6 C10 C15 C30 A4 C2×A4 C5×A4 C10×A4 C42⋊C6 C5×C42⋊C6 kernel C5×C42⋊C6 C5×C42⋊C3 C5×C42⋊2C2 C42⋊C6 C4×C20 C42⋊C3 C42⋊2C2 C42 C22×C10 C2×C10 C23 C22 C5 C1 # reps 1 1 2 4 2 4 8 8 1 1 4 4 2 8

Matrix representation of C5×C42⋊C6 in GL6(𝔽61)

 58 0 0 0 0 0 0 58 0 0 0 0 0 0 58 0 0 0 0 0 0 58 0 0 0 0 0 0 58 0 0 0 0 0 0 58
,
 0 50 0 0 1 6 0 50 0 0 0 0 0 0 0 1 0 56 1 60 0 0 0 56 0 0 1 0 0 55 0 0 0 0 0 50
,
 55 55 6 10 1 0 50 0 11 60 1 0 55 5 6 11 0 0 5 55 6 11 50 1 6 56 5 50 11 0 11 11 50 1 60 0
,
 1 0 60 50 10 55 0 0 60 0 60 0 0 1 60 50 11 5 0 0 0 0 11 55 0 0 0 0 50 56 0 0 0 60 1 11

G:=sub<GL(6,GF(61))| [58,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,58],[0,0,0,1,0,0,50,50,0,60,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,6,0,56,56,55,50],[55,50,55,5,6,11,55,0,5,55,56,11,6,11,6,6,5,50,10,60,11,11,50,1,1,1,0,50,11,60,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,60,60,60,0,0,0,50,0,50,0,0,60,10,60,11,11,50,1,55,0,5,55,56,11] >;

C5×C42⋊C6 in GAP, Magma, Sage, TeX

C_5\times C_4^2\rtimes C_6
% in TeX

G:=Group("C5xC4^2:C6");
// GroupNames label

G:=SmallGroup(480,657);
// by ID

G=gap.SmallGroup(480,657);
# by ID

G:=PCGroup([7,-2,-3,-5,-2,2,-2,2,4203,850,360,10504,5786,102,5052,8833]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=c^4=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=c^-1,d*c*d^-1=b^-1*c>;
// generators/relations

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