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## G = D5×CSU2(𝔽3)  order 480 = 25·3·5

### Direct product of D5 and CSU2(𝔽3)

Aliases: D5×CSU2(𝔽3), D10.5S4, SL2(𝔽3).4D10, C2.8(D5×S4), C10.5(C2×S4), (Q8×D5).1S3, Q8.5(S3×D5), (C5×Q8).5D6, Q8.D155C2, C51(C2×CSU2(𝔽3)), (C5×CSU2(𝔽3))⋊2C2, (D5×SL2(𝔽3)).1C2, (C5×SL2(𝔽3)).4C22, SmallGroup(480,971)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — D5×CSU2(𝔽3)
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — D5×SL2(𝔽3) — D5×CSU2(𝔽3)
 Lower central C5×SL2(𝔽3) — D5×CSU2(𝔽3)
 Upper central C1 — C2

Generators and relations for D5×CSU2(𝔽3)
G = < a,b,c,d,e,f | a5=b2=c4=e3=1, d2=f2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fdf-1=c-1, ece-1=cd, fcf-1=c2d, ede-1=c, fef-1=e-1 >

Subgroups: 538 in 78 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C8, C2×C4, Q8, Q8, D5, C10, Dic3, C2×C6, C15, C2×C8, Q16, C2×Q8, Dic5, C20, D10, SL2(𝔽3), C2×Dic3, C3×D5, C30, C2×Q16, C52C8, C40, Dic10, C4×D5, C5×Q8, C5×Q8, CSU2(𝔽3), CSU2(𝔽3), C2×SL2(𝔽3), C5×Dic3, Dic15, C6×D5, C8×D5, Dic20, C5⋊Q16, C5×Q16, Q8×D5, Q8×D5, C2×CSU2(𝔽3), D5×Dic3, C5×SL2(𝔽3), D5×Q16, C5×CSU2(𝔽3), Q8.D15, D5×SL2(𝔽3), D5×CSU2(𝔽3)
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, CSU2(𝔽3), C2×S4, S3×D5, C2×CSU2(𝔽3), D5×S4, D5×CSU2(𝔽3)

Smallest permutation representation of D5×CSU2(𝔽3)
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 11)(2 15)(3 14)(4 13)(5 12)(6 65)(7 64)(8 63)(9 62)(10 61)(16 59)(17 58)(18 57)(19 56)(20 60)(21 79)(22 78)(23 77)(24 76)(25 80)(26 48)(27 47)(28 46)(29 50)(30 49)(31 73)(32 72)(33 71)(34 75)(35 74)(36 66)(37 70)(38 69)(39 68)(40 67)(41 54)(42 53)(43 52)(44 51)(45 55)
(1 54 12 42)(2 55 13 43)(3 51 14 44)(4 52 15 45)(5 53 11 41)(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 69 50 40)(27 70 46 36)(28 66 47 37)(29 67 48 38)(30 68 49 39)
(1 66 12 37)(2 67 13 38)(3 68 14 39)(4 69 15 40)(5 70 11 36)(6 60 64 19)(7 56 65 20)(8 57 61 16)(9 58 62 17)(10 59 63 18)(21 75 79 34)(22 71 80 35)(23 72 76 31)(24 73 77 32)(25 74 78 33)(26 45 50 52)(27 41 46 53)(28 42 47 54)(29 43 48 55)(30 44 49 51)
(16 74 25)(17 75 21)(18 71 22)(19 72 23)(20 73 24)(26 52 69)(27 53 70)(28 54 66)(29 55 67)(30 51 68)(31 76 60)(32 77 56)(33 78 57)(34 79 58)(35 80 59)(36 46 41)(37 47 42)(38 48 43)(39 49 44)(40 50 45)
(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)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A 10B 15A 15B 20A 20B 20C 20D 30A 30B 40A 40B 40C 40D order 1 2 2 2 3 4 4 4 4 5 5 6 6 6 8 8 8 8 10 10 15 15 20 20 20 20 30 30 40 40 40 40 size 1 1 5 5 8 6 12 30 60 2 2 8 40 40 6 6 30 30 2 2 16 16 12 12 24 24 16 16 12 12 12 12

32 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 3 3 4 4 4 6 8 type + + + + + + + + - + + - + - + - image C1 C2 C2 C2 S3 D5 D6 D10 CSU2(𝔽3) S4 C2×S4 CSU2(𝔽3) S3×D5 D5×CSU2(𝔽3) D5×S4 D5×CSU2(𝔽3) kernel D5×CSU2(𝔽3) C5×CSU2(𝔽3) Q8.D15 D5×SL2(𝔽3) Q8×D5 CSU2(𝔽3) C5×Q8 SL2(𝔽3) D5 D10 C10 D5 Q8 C1 C2 C1 # reps 1 1 1 1 1 2 1 2 4 2 2 2 2 4 4 2

Matrix representation of D5×CSU2(𝔽3) in GL4(𝔽241) generated by

 189 1 0 0 240 0 0 0 0 0 1 0 0 0 0 1
,
 1 189 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 226 16 0 0 16 15
,
 1 0 0 0 0 1 0 0 0 0 0 240 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 225 0 0 0 1 15
,
 240 0 0 0 0 240 0 0 0 0 165 54 0 0 76 76
G:=sub<GL(4,GF(241))| [189,240,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,189,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,226,16,0,0,16,15],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,240,0],[1,0,0,0,0,1,0,0,0,0,225,1,0,0,0,15],[240,0,0,0,0,240,0,0,0,0,165,76,0,0,54,76] >;

D5×CSU2(𝔽3) in GAP, Magma, Sage, TeX

D_5\times {\rm CSU}_2({\mathbb F}_3)
% in TeX

G:=Group("D5xCSU(2,3)");
// GroupNames label

G:=SmallGroup(480,971);
// by ID

G=gap.SmallGroup(480,971);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1688,93,1347,2111,3168,172,1272,1909,285,124]);
// Polycyclic

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

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