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## G = C5×C23.A4order 480 = 25·3·5

### Direct product of C5 and C23.A4

Aliases: C5×C23.A4, C422C30, C41D4⋊C15, (C4×C20)⋊5C6, C42⋊C33C10, C23.2(C5×A4), C22.4(C10×A4), (C22×C10).2A4, (C5×C41D4)⋊C3, (C5×C42⋊C3)⋊7C2, (C2×C10).8(C2×A4), SmallGroup(480,658)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C5×C23.A4
 Chief series C1 — C22 — C42 — C4×C20 — C5×C42⋊C3 — C5×C23.A4
 Lower central C42 — C5×C23.A4
 Upper central C1 — C5

Generators and relations for C5×C23.A4
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=g3=1, e2=dc=gcg-1=cd, f2=gdg-1=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf-1=bc=cb, bd=db, ebe-1=bcd, bg=gb, ce=ec, cf=fc, gfg-1=de=ed, df=fd, ef=fe, geg-1=cef >

Subgroups: 264 in 56 conjugacy classes, 14 normal (all characteristic)
C1, C2, C3, C4, C22, C22, C5, C6, C2×C4, D4, C23, C23, C10, A4, C15, C42, C2×D4, C20, C2×C10, C2×C10, C2×A4, C30, C41D4, C2×C20, C5×D4, C22×C10, C22×C10, C42⋊C3, C5×A4, C4×C20, D4×C10, C23.A4, C10×A4, C5×C41D4, C5×C42⋊C3, C5×C23.A4
Quotients: C1, C2, C3, C5, C6, C10, A4, C15, C2×A4, C30, C5×A4, C23.A4, C10×A4, C5×C23.A4

Smallest permutation representation of C5×C23.A4
On 60 points
Generators in S60
(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)
(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)(46 51)(47 52)(48 53)(49 54)(50 55)
(1 21)(2 22)(3 23)(4 24)(5 25)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)
(6 56)(7 57)(8 58)(9 59)(10 60)(11 16)(12 17)(13 18)(14 19)(15 20)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)
(1 26 21 31)(2 27 22 32)(3 28 23 33)(4 29 24 34)(5 30 25 35)(6 11 56 16)(7 12 57 17)(8 13 58 18)(9 14 59 19)(10 15 60 20)
(1 26 21 31)(2 27 22 32)(3 28 23 33)(4 29 24 34)(5 30 25 35)(6 56)(7 57)(8 58)(9 59)(10 60)(11 16)(12 17)(13 18)(14 19)(15 20)(36 46 41 51)(37 47 42 52)(38 48 43 53)(39 49 44 54)(40 50 45 55)
(1 36 56)(2 37 57)(3 38 58)(4 39 59)(5 40 60)(6 21 41)(7 22 42)(8 23 43)(9 24 44)(10 25 45)(11 26 46)(12 27 47)(13 28 48)(14 29 49)(15 30 50)(16 31 51)(17 32 52)(18 33 53)(19 34 54)(20 35 55)

G:=sub<Sym(60)| (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), (11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35)(46,51)(47,52)(48,53)(49,54)(50,55), (1,21)(2,22)(3,23)(4,24)(5,25)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55), (6,56)(7,57)(8,58)(9,59)(10,60)(11,16)(12,17)(13,18)(14,19)(15,20)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55), (1,26,21,31)(2,27,22,32)(3,28,23,33)(4,29,24,34)(5,30,25,35)(6,11,56,16)(7,12,57,17)(8,13,58,18)(9,14,59,19)(10,15,60,20), (1,26,21,31)(2,27,22,32)(3,28,23,33)(4,29,24,34)(5,30,25,35)(6,56)(7,57)(8,58)(9,59)(10,60)(11,16)(12,17)(13,18)(14,19)(15,20)(36,46,41,51)(37,47,42,52)(38,48,43,53)(39,49,44,54)(40,50,45,55), (1,36,56)(2,37,57)(3,38,58)(4,39,59)(5,40,60)(6,21,41)(7,22,42)(8,23,43)(9,24,44)(10,25,45)(11,26,46)(12,27,47)(13,28,48)(14,29,49)(15,30,50)(16,31,51)(17,32,52)(18,33,53)(19,34,54)(20,35,55)>;

G:=Group( (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), (11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35)(46,51)(47,52)(48,53)(49,54)(50,55), (1,21)(2,22)(3,23)(4,24)(5,25)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55), (6,56)(7,57)(8,58)(9,59)(10,60)(11,16)(12,17)(13,18)(14,19)(15,20)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55), (1,26,21,31)(2,27,22,32)(3,28,23,33)(4,29,24,34)(5,30,25,35)(6,11,56,16)(7,12,57,17)(8,13,58,18)(9,14,59,19)(10,15,60,20), (1,26,21,31)(2,27,22,32)(3,28,23,33)(4,29,24,34)(5,30,25,35)(6,56)(7,57)(8,58)(9,59)(10,60)(11,16)(12,17)(13,18)(14,19)(15,20)(36,46,41,51)(37,47,42,52)(38,48,43,53)(39,49,44,54)(40,50,45,55), (1,36,56)(2,37,57)(3,38,58)(4,39,59)(5,40,60)(6,21,41)(7,22,42)(8,23,43)(9,24,44)(10,25,45)(11,26,46)(12,27,47)(13,28,48)(14,29,49)(15,30,50)(16,31,51)(17,32,52)(18,33,53)(19,34,54)(20,35,55) );

G=PermutationGroup([[(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)], [(11,16),(12,17),(13,18),(14,19),(15,20),(26,31),(27,32),(28,33),(29,34),(30,35),(46,51),(47,52),(48,53),(49,54),(50,55)], [(1,21),(2,22),(3,23),(4,24),(5,25),(26,31),(27,32),(28,33),(29,34),(30,35),(36,41),(37,42),(38,43),(39,44),(40,45),(46,51),(47,52),(48,53),(49,54),(50,55)], [(6,56),(7,57),(8,58),(9,59),(10,60),(11,16),(12,17),(13,18),(14,19),(15,20),(36,41),(37,42),(38,43),(39,44),(40,45),(46,51),(47,52),(48,53),(49,54),(50,55)], [(1,26,21,31),(2,27,22,32),(3,28,23,33),(4,29,24,34),(5,30,25,35),(6,11,56,16),(7,12,57,17),(8,13,58,18),(9,14,59,19),(10,15,60,20)], [(1,26,21,31),(2,27,22,32),(3,28,23,33),(4,29,24,34),(5,30,25,35),(6,56),(7,57),(8,58),(9,59),(10,60),(11,16),(12,17),(13,18),(14,19),(15,20),(36,46,41,51),(37,47,42,52),(38,48,43,53),(39,49,44,54),(40,50,45,55)], [(1,36,56),(2,37,57),(3,38,58),(4,39,59),(5,40,60),(6,21,41),(7,22,42),(8,23,43),(9,24,44),(10,25,45),(11,26,46),(12,27,47),(13,28,48),(14,29,49),(15,30,50),(16,31,51),(17,32,52),(18,33,53),(19,34,54),(20,35,55)]])

50 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 15A ··· 15H 20A ··· 20H 30A ··· 30H order 1 2 2 2 3 3 4 4 5 5 5 5 6 6 10 10 10 10 10 10 10 10 10 10 10 10 15 ··· 15 20 ··· 20 30 ··· 30 size 1 3 4 12 16 16 6 6 1 1 1 1 16 16 3 3 3 3 4 4 4 4 12 12 12 12 16 ··· 16 6 ··· 6 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 C23.A4 C5×C23.A4 kernel C5×C23.A4 C5×C42⋊C3 C5×C4⋊1D4 C23.A4 C4×C20 C42⋊C3 C4⋊1D4 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×C23.A4 in GL9(𝔽61)

 9 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 60 0 60 0 60 60
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 1
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 60 0 0 0 0 60 60 0 0 0 60
,
 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 60 60 60 60 60 59 0 0 0 0 1 0 0 1 1
,
 60 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 60 0 0 0 0 60 0 60 0 0 60
,
 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 60 60 60 60 60 59 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1

G:=sub<GL(9,GF(61))| [9,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,0,0,1,0,0,0,0,60,0,0,0,1,0,0,0,0,0,60,0,0,1,0,0,0,0,0,0,60,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,60,0,0,0,0,1,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,1,0,0,60,0,0,0,0,60,0,0,0,60,1,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,59,1],[60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,60,0,0,0,60,0,0,0,0,0,0,0,0,0,0,0,1,0,60,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,60,0,1,0,0,0,0,0,0,60,0,0,1,0,0,0,0,0,60,0,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,59,0,0,0,1] >;

C5×C23.A4 in GAP, Magma, Sage, TeX

C_5\times C_2^3.A_4
% in TeX

G:=Group("C5xC2^3.A4");
// GroupNames label

G:=SmallGroup(480,658);
// by ID

G=gap.SmallGroup(480,658);
# by ID

G:=PCGroup([7,-2,-3,-5,-2,2,-2,2,10923,850,360,6304,5786,102,5052,8833]);
// Polycyclic

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

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