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## G = C5×D4.A4order 480 = 25·3·5

### Direct product of C5 and D4.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×D4.A4
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — C10×SL2(𝔽3) — C5×D4.A4
 Lower central Q8 — C5×D4.A4
 Upper central C1 — C10 — C5×D4

Generators and relations for C5×D4.A4
G = < a,b,c,d,e,f | a5=b4=c2=f3=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=b2d, fdf-1=b2de, fef-1=d >

Subgroups: 246 in 92 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C2×C4, D4, D4, Q8, Q8, C10, C10, C12, C2×C6, C15, C2×Q8, C2×Q8, C4○D4, C4○D4, C20, C20, C2×C10, C2×C10, SL2(𝔽3), C3×D4, C30, 2- 1+4, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C2×SL2(𝔽3), C4.A4, C60, C2×C30, Q8×C10, Q8×C10, C5×C4○D4, C5×C4○D4, D4.A4, C5×SL2(𝔽3), D4×C15, C5×2- 1+4, C10×SL2(𝔽3), C5×C4.A4, C5×D4.A4
Quotients: C1, C2, C3, C22, C5, C6, C10, A4, C2×C6, C15, C2×C10, C2×A4, C30, C22×A4, C5×A4, C2×C30, D4.A4, C10×A4, A4×C2×C10, C5×D4.A4

Smallest permutation representation of C5×D4.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 65 12 7)(2 61 13 8)(3 62 14 9)(4 63 15 10)(5 64 11 6)(16 55 57 43)(17 51 58 44)(18 52 59 45)(19 53 60 41)(20 54 56 42)(21 68 79 39)(22 69 80 40)(23 70 76 36)(24 66 77 37)(25 67 78 38)(26 35 50 71)(27 31 46 72)(28 32 47 73)(29 33 48 74)(30 34 49 75)
(1 7)(2 8)(3 9)(4 10)(5 6)(11 64)(12 65)(13 61)(14 62)(15 63)(16 55)(17 51)(18 52)(19 53)(20 54)(21 68)(22 69)(23 70)(24 66)(25 67)(26 71)(27 72)(28 73)(29 74)(30 75)(31 46)(32 47)(33 48)(34 49)(35 50)(36 76)(37 77)(38 78)(39 79)(40 80)(41 60)(42 56)(43 57)(44 58)(45 59)
(1 47 12 28)(2 48 13 29)(3 49 14 30)(4 50 15 26)(5 46 11 27)(6 31 64 72)(7 32 65 73)(8 33 61 74)(9 34 62 75)(10 35 63 71)(16 25 57 78)(17 21 58 79)(18 22 59 80)(19 23 60 76)(20 24 56 77)(36 53 70 41)(37 54 66 42)(38 55 67 43)(39 51 68 44)(40 52 69 45)
(1 37 12 66)(2 38 13 67)(3 39 14 68)(4 40 15 69)(5 36 11 70)(6 76 64 23)(7 77 65 24)(8 78 61 25)(9 79 62 21)(10 80 63 22)(16 33 57 74)(17 34 58 75)(18 35 59 71)(19 31 60 72)(20 32 56 73)(26 52 50 45)(27 53 46 41)(28 54 47 42)(29 55 48 43)(30 51 49 44)
(16 33 78)(17 34 79)(18 35 80)(19 31 76)(20 32 77)(21 58 75)(22 59 71)(23 60 72)(24 56 73)(25 57 74)(26 69 45)(27 70 41)(28 66 42)(29 67 43)(30 68 44)(36 53 46)(37 54 47)(38 55 48)(39 51 49)(40 52 50)

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

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

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

95 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 10A 10B 10C 10D 10E ··· 10L 10M 10N 10O 10P 12A 12B 15A ··· 15H 20A 20B 20C 20D 20E ··· 20P 30A ··· 30H 30I ··· 30X 60A ··· 60H order 1 2 2 2 2 3 3 4 4 4 4 5 5 5 5 6 6 6 6 6 6 10 10 10 10 10 ··· 10 10 10 10 10 12 12 15 ··· 15 20 20 20 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 2 2 6 4 4 2 6 6 6 1 1 1 1 4 4 8 8 8 8 1 1 1 1 2 ··· 2 6 6 6 6 8 8 4 ··· 4 2 2 2 2 6 ··· 6 4 ··· 4 8 ··· 8 8 ··· 8

95 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 4 4 4 type + + + + + + - image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 A4 C2×A4 C2×A4 C5×A4 C10×A4 C10×A4 D4.A4 D4.A4 C5×D4.A4 kernel C5×D4.A4 C10×SL2(𝔽3) C5×C4.A4 C5×2- 1+4 D4.A4 Q8×C10 C5×C4○D4 C2×SL2(𝔽3) C4.A4 2- 1+4 C2×Q8 C4○D4 C5×D4 C20 C2×C10 D4 C4 C22 C5 C5 C1 # reps 1 2 1 2 4 4 2 8 4 8 16 8 1 1 2 4 4 8 1 2 12

Matrix representation of C5×D4.A4 in GL4(𝔽61) generated by

 34 0 0 0 0 34 0 0 0 0 34 0 0 0 0 34
,
 34 42 1 3 0 53 0 59 2 0 27 19 0 2 0 8
,
 27 19 59 58 0 8 0 1 59 0 34 42 0 59 0 53
,
 48 14 50 47 14 13 11 11 0 0 48 14 0 0 14 13
,
 0 1 40 21 60 0 21 21 0 0 0 1 0 0 60 0
,
 1 14 0 58 0 13 0 0 0 0 1 14 0 0 0 13
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[34,0,2,0,42,53,0,2,1,0,27,0,3,59,19,8],[27,0,59,0,19,8,0,59,59,0,34,0,58,1,42,53],[48,14,0,0,14,13,0,0,50,11,48,14,47,11,14,13],[0,60,0,0,1,0,0,0,40,21,0,60,21,21,1,0],[1,0,0,0,14,13,0,0,0,0,1,0,58,0,14,13] >;

C5×D4.A4 in GAP, Magma, Sage, TeX

C_5\times D_4.A_4
% in TeX

G:=Group("C5xD4.A4");
// GroupNames label

G:=SmallGroup(480,1132);
// by ID

G=gap.SmallGroup(480,1132);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,3389,1068,172,1909,285,124]);
// Polycyclic

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

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