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## G = D5×C4.A4order 480 = 25·3·5

### Direct product of D5 and C4.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×Q8 — D5×C4.A4
 Chief series C1 — C2 — C10 — C5×Q8 — C5×SL2(𝔽3) — D5×SL2(𝔽3) — D5×C4.A4
 Lower central C5×Q8 — D5×C4.A4
 Upper central C1 — C4

Generators and relations for D5×C4.A4
G = < a,b,c,d,e,f | a5=b2=c4=f3=1, d2=e2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=c2d, fdf-1=c2de, fef-1=d >

Subgroups: 638 in 98 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C2×C4, D4, Q8, Q8, C23, D5, D5, C10, C10, C12, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, SL2(𝔽3), C2×C12, C3×D5, C30, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C2×SL2(𝔽3), C4.A4, C4.A4, C3×Dic5, C60, C6×D5, C2×C4×D5, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, C2×C4.A4, C5×SL2(𝔽3), D5×C12, D5×C4○D4, Dic5.A4, D5×SL2(𝔽3), C5×C4.A4, D5×C4.A4
Quotients: C1, C2, C3, C22, C6, D5, A4, C2×C6, D10, C2×A4, C3×D5, C4.A4, C22×A4, C6×D5, C2×C4.A4, D5×A4, C2×D5×A4, D5×C4.A4

Smallest permutation representation of D5×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 5)(2 4)(6 9)(7 8)(11 14)(12 13)(16 17)(18 20)(21 24)(22 23)(26 27)(28 30)(31 34)(32 33)(36 37)(38 40)(41 44)(42 43)(46 47)(48 50)(51 54)(52 53)(56 57)(58 60)(61 64)(62 63)(66 67)(68 70)(71 74)(72 73)(76 77)(78 80)
(1 53 13 47)(2 54 14 48)(3 55 15 49)(4 51 11 50)(5 52 12 46)(6 41 80 40)(7 42 76 36)(8 43 77 37)(9 44 78 38)(10 45 79 39)(16 62 22 56)(17 63 23 57)(18 64 24 58)(19 65 25 59)(20 61 21 60)(26 72 32 66)(27 73 33 67)(28 74 34 68)(29 75 35 69)(30 71 31 70)
(1 17 13 23)(2 18 14 24)(3 19 15 25)(4 20 11 21)(5 16 12 22)(6 71 80 70)(7 72 76 66)(8 73 77 67)(9 74 78 68)(10 75 79 69)(26 42 32 36)(27 43 33 37)(28 44 34 38)(29 45 35 39)(30 41 31 40)(46 56 52 62)(47 57 53 63)(48 58 54 64)(49 59 55 65)(50 60 51 61)
(1 37 13 43)(2 38 14 44)(3 39 15 45)(4 40 11 41)(5 36 12 42)(6 50 80 51)(7 46 76 52)(8 47 77 53)(9 48 78 54)(10 49 79 55)(16 32 22 26)(17 33 23 27)(18 34 24 28)(19 35 25 29)(20 31 21 30)(56 72 62 66)(57 73 63 67)(58 74 64 68)(59 75 65 69)(60 71 61 70)
(6 71 61)(7 72 62)(8 73 63)(9 74 64)(10 75 65)(16 36 26)(17 37 27)(18 38 28)(19 39 29)(20 40 30)(21 41 31)(22 42 32)(23 43 33)(24 44 34)(25 45 35)(56 76 66)(57 77 67)(58 78 68)(59 79 69)(60 80 70)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 6D 6E 6F 10A 10B 10C 10D 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 30A 30B 30C 30D 60A ··· 60H order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 5 5 6 6 6 6 6 6 10 10 10 10 12 12 12 12 12 12 12 12 15 15 15 15 20 20 20 20 20 20 30 30 30 30 60 ··· 60 size 1 1 5 5 6 30 4 4 1 1 5 5 6 30 2 2 4 4 20 20 20 20 2 2 12 12 4 4 4 4 20 20 20 20 8 8 8 8 2 2 2 2 12 12 8 8 8 8 8 ··· 8

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 4 6 6 type + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D5 D10 C3×D5 C4.A4 C6×D5 A4 C2×A4 C2×A4 C2×A4 D5×C4.A4 D5×A4 C2×D5×A4 kernel D5×C4.A4 Dic5.A4 D5×SL2(𝔽3) C5×C4.A4 D5×C4○D4 Q8×D5 Q8⋊2D5 C5×C4○D4 C4.A4 SL2(𝔽3) C4○D4 D5 Q8 C4×D5 Dic5 C20 D10 C1 C4 C2 # reps 1 1 1 1 2 2 2 2 2 2 4 12 4 1 1 1 1 12 2 2

Matrix representation of D5×C4.A4 in GL4(𝔽61) generated by

 17 1 0 0 60 0 0 0 0 0 1 0 0 0 0 1
,
 1 17 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 60 0 0 0 0 60 0 0 0 0 50 0 0 0 0 50
,
 1 0 0 0 0 1 0 0 0 0 0 60 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 48 14 0 0 14 13
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 14 13
G:=sub<GL(4,GF(61))| [17,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,17,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,50,0,0,0,0,50],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,0],[1,0,0,0,0,1,0,0,0,0,48,14,0,0,14,13],[1,0,0,0,0,1,0,0,0,0,1,14,0,0,0,13] >;

D5×C4.A4 in GAP, Magma, Sage, TeX

D_5\times C_4.A_4
% in TeX

G:=Group("D5xC4.A4");
// GroupNames label

G:=SmallGroup(480,1042);
// by ID

G=gap.SmallGroup(480,1042);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-5,-2,1688,269,584,123,795,382,8069]);
// Polycyclic

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

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