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## G = C5×C4.A4order 240 = 24·3·5

### Direct product of C5 and C4.A4

Aliases: C5×C4.A4, C20.A4, Q8.C30, SL2(𝔽3)⋊2C10, C4○D4⋊C15, C4.(C5×A4), C2.3(C10×A4), C10.6(C2×A4), (C5×Q8).2C6, (C5×SL2(𝔽3))⋊5C2, (C5×C4○D4)⋊C3, SmallGroup(240,154)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×C4.A4
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — C5×C4.A4
 Lower central Q8 — C5×C4.A4
 Upper central C1 — C20

Generators and relations for C5×C4.A4
G = < a,b,c,d,e | a5=b4=e3=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b2c, ece-1=b2cd, ede-1=c >

Smallest permutation representation of C5×C4.A4
On 80 points
Generators in S80
(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 49 12 20)(2 50 13 16)(3 46 14 17)(4 47 15 18)(5 48 11 19)(6 69 80 40)(7 70 76 36)(8 66 77 37)(9 67 78 38)(10 68 79 39)(21 44 62 28)(22 45 63 29)(23 41 64 30)(24 42 65 26)(25 43 61 27)(31 59 72 54)(32 60 73 55)(33 56 74 51)(34 57 75 52)(35 58 71 53)
(1 60 12 55)(2 56 13 51)(3 57 14 52)(4 58 15 53)(5 59 11 54)(6 62 80 21)(7 63 76 22)(8 64 77 23)(9 65 78 24)(10 61 79 25)(16 33 50 74)(17 34 46 75)(18 35 47 71)(19 31 48 72)(20 32 49 73)(26 38 42 67)(27 39 43 68)(28 40 44 69)(29 36 45 70)(30 37 41 66)
(1 37 12 66)(2 38 13 67)(3 39 14 68)(4 40 15 69)(5 36 11 70)(6 18 80 47)(7 19 76 48)(8 20 77 49)(9 16 78 50)(10 17 79 46)(21 35 62 71)(22 31 63 72)(23 32 64 73)(24 33 65 74)(25 34 61 75)(26 51 42 56)(27 52 43 57)(28 53 44 58)(29 54 45 59)(30 55 41 60)
(6 62 71)(7 63 72)(8 64 73)(9 65 74)(10 61 75)(21 35 80)(22 31 76)(23 32 77)(24 33 78)(25 34 79)(26 51 67)(27 52 68)(28 53 69)(29 54 70)(30 55 66)(36 45 59)(37 41 60)(38 42 56)(39 43 57)(40 44 58)

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

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

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

C5×C4.A4 is a maximal subgroup of   C52U2(𝔽3)  SL2(𝔽3).Dic5  C20.2S4  C20.6S4  C20.3S4  Dic10.A4  D20.A4
C5×C4.A4 is a maximal quotient of   C20×SL2(𝔽3)

70 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 12A 12B 12C 12D 15A ··· 15H 20A ··· 20H 20I 20J 20K 20L 30A ··· 30H 60A ··· 60P order 1 2 2 3 3 4 4 4 5 5 5 5 6 6 10 10 10 10 10 10 10 10 12 12 12 12 15 ··· 15 20 ··· 20 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 6 4 4 1 1 6 1 1 1 1 4 4 1 1 1 1 6 6 6 6 4 4 4 4 4 ··· 4 1 ··· 1 6 6 6 6 4 ··· 4 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 3 3 3 3 type + + + + image C1 C2 C3 C5 C6 C10 C15 C30 C4.A4 C5×C4.A4 A4 C2×A4 C5×A4 C10×A4 kernel C5×C4.A4 C5×SL2(𝔽3) C5×C4○D4 C4.A4 C5×Q8 SL2(𝔽3) C4○D4 Q8 C5 C1 C20 C10 C4 C2 # reps 1 1 2 4 2 4 8 8 6 24 1 1 4 4

Matrix representation of C5×C4.A4 in GL2(𝔽41) generated by

 16 0 0 16
,
 32 0 0 32
,
 1 25 36 40
,
 32 0 4 9
,
 40 39 21 0
G:=sub<GL(2,GF(41))| [16,0,0,16],[32,0,0,32],[1,36,25,40],[32,4,0,9],[40,21,39,0] >;

C5×C4.A4 in GAP, Magma, Sage, TeX

C_5\times C_4.A_4
% in TeX

G:=Group("C5xC4.A4");
// GroupNames label

G:=SmallGroup(240,154);
// by ID

G=gap.SmallGroup(240,154);
# by ID

G:=PCGroup([6,-2,-3,-5,-2,2,-2,720,729,117,1360,202,88]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=e^3=1,c^2=d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e^-1=b^2*c*d,e*d*e^-1=c>;
// generators/relations

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