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G = C5×CSU2(𝔽3)  order 240 = 24·3·5

Direct product of C5 and CSU2(𝔽3)

direct product, non-abelian, soluble

Aliases: C5×CSU2(𝔽3), C10.4S4, SL2(𝔽3).C10, Q8.(C5×S3), C2.2(C5×S4), (C5×Q8).2S3, (C5×SL2(𝔽3)).2C2, SmallGroup(240,102)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — C5×CSU2(𝔽3)
C1C2Q8SL2(𝔽3)C5×SL2(𝔽3) — C5×CSU2(𝔽3)
SL2(𝔽3) — C5×CSU2(𝔽3)
C1C10

Generators and relations for C5×CSU2(𝔽3)
 G = < a,b,c,d,e | a5=b4=d3=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=bc, ebe-1=b2c, dcd-1=b, ede-1=d-1 >

4C3
3C4
6C4
4C6
4C15
3Q8
3C8
4Dic3
3C20
6C20
4C30
3Q16
3C5×Q8
3C40
4C5×Dic3
3C5×Q16

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

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

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

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

C5×CSU2(𝔽3) is a maximal subgroup of   CSU2(𝔽3)⋊D5  Dic5.7S4  D10.2S4

40 conjugacy classes

class 1  2  3 4A4B5A5B5C5D 6 8A8B10A10B10C10D15A15B15C15D20A20B20C20D20E20F20G20H30A30B30C30D40A···40H
order123445555688101010101515151520202020202020203030303040···40
size11861211118661111888866661212121288886···6

40 irreducible representations

dim111122223344
type+++-+-
imageC1C2C5C10S3C5×S3CSU2(𝔽3)C5×CSU2(𝔽3)S4C5×S4CSU2(𝔽3)C5×CSU2(𝔽3)
kernelC5×CSU2(𝔽3)C5×SL2(𝔽3)CSU2(𝔽3)SL2(𝔽3)C5×Q8Q8C5C1C10C2C5C1
# reps114414282814

Matrix representation of C5×CSU2(𝔽3) in GL2(𝔽31) generated by

160
016
,
199
812
,
2815
203
,
811
3022
,
267
145
G:=sub<GL(2,GF(31))| [16,0,0,16],[19,8,9,12],[28,20,15,3],[8,30,11,22],[26,14,7,5] >;

C5×CSU2(𝔽3) in GAP, Magma, Sage, TeX

C_5\times {\rm CSU}_2({\mathbb F}_3)
% in TeX

G:=Group("C5xCSU(2,3)");
// GroupNames label

G:=SmallGroup(240,102);
// by ID

G=gap.SmallGroup(240,102);
# by ID

G:=PCGroup([6,-2,-5,-3,-2,2,-2,720,362,1443,447,117,904,286,202,88]);
// Polycyclic

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

Export

Subgroup lattice of C5×CSU2(𝔽3) in TeX

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