Copied to
clipboard

## G = C2×D5×SL2(𝔽3)  order 480 = 25·3·5

### Direct product of C2, D5 and SL2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×Q8 — C2×D5×SL2(𝔽3)
 Chief series C1 — C2 — C10 — C5×Q8 — C5×SL2(𝔽3) — D5×SL2(𝔽3) — C2×D5×SL2(𝔽3)
 Lower central C5×Q8 — C2×D5×SL2(𝔽3)
 Upper central C1 — C22

Generators and relations for C2×D5×SL2(𝔽3)
G = < a,b,c,d,e,f | a2=b5=c2=d4=f3=1, e2=d2, 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=d-1, fdf-1=e, fef-1=de >

Subgroups: 718 in 122 conjugacy classes, 35 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C2×C4, Q8, Q8, C23, D5, C10, C10, C2×C6, C15, C22×C4, C2×Q8, C2×Q8, Dic5, C20, D10, C2×C10, SL2(𝔽3), C22×C6, C3×D5, C30, C22×Q8, Dic10, C4×D5, C2×Dic5, C2×C20, C5×Q8, C5×Q8, C22×D5, C2×SL2(𝔽3), C2×SL2(𝔽3), C6×D5, C2×C30, C2×Dic10, C2×C4×D5, Q8×D5, Q8×D5, Q8×C10, C22×SL2(𝔽3), C5×SL2(𝔽3), D5×C2×C6, C2×Q8×D5, D5×SL2(𝔽3), C10×SL2(𝔽3), C2×D5×SL2(𝔽3)
Quotients: C1, C2, C3, C22, C6, D5, A4, C2×C6, D10, SL2(𝔽3), C2×A4, C3×D5, C2×SL2(𝔽3), C22×A4, C6×D5, C22×SL2(𝔽3), D5×A4, D5×SL2(𝔽3), C2×D5×A4, C2×D5×SL2(𝔽3)

Smallest permutation representation of C2×D5×SL2(𝔽3)
On 80 points
Generators in S80
(1 53)(2 54)(3 55)(4 51)(5 52)(6 40)(7 36)(8 37)(9 38)(10 39)(11 50)(12 46)(13 47)(14 48)(15 49)(16 62)(17 63)(18 64)(19 65)(20 61)(21 60)(22 56)(23 57)(24 58)(25 59)(26 72)(27 73)(28 74)(29 75)(30 71)(31 70)(32 66)(33 67)(34 68)(35 69)(41 80)(42 76)(43 77)(44 78)(45 79)
(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 52)(2 51)(3 55)(4 54)(5 53)(6 38)(7 37)(8 36)(9 40)(10 39)(11 48)(12 47)(13 46)(14 50)(15 49)(16 63)(17 62)(18 61)(19 65)(20 64)(21 58)(22 57)(23 56)(24 60)(25 59)(26 73)(27 72)(28 71)(29 75)(30 74)(31 68)(32 67)(33 66)(34 70)(35 69)(41 78)(42 77)(43 76)(44 80)(45 79)
(1 23 13 17)(2 24 14 18)(3 25 15 19)(4 21 11 20)(5 22 12 16)(6 70 80 71)(7 66 76 72)(8 67 77 73)(9 68 78 74)(10 69 79 75)(26 36 32 42)(27 37 33 43)(28 38 34 44)(29 39 35 45)(30 40 31 41)(46 62 52 56)(47 63 53 57)(48 64 54 58)(49 65 55 59)(50 61 51 60)
(1 43 13 37)(2 44 14 38)(3 45 15 39)(4 41 11 40)(5 42 12 36)(6 51 80 50)(7 52 76 46)(8 53 77 47)(9 54 78 48)(10 55 79 49)(16 26 22 32)(17 27 23 33)(18 28 24 34)(19 29 25 35)(20 30 21 31)(56 66 62 72)(57 67 63 73)(58 68 64 74)(59 69 65 75)(60 70 61 71)
(6 61 71)(7 62 72)(8 63 73)(9 64 74)(10 65 75)(16 26 36)(17 27 37)(18 28 38)(19 29 39)(20 30 40)(21 31 41)(22 32 42)(23 33 43)(24 34 44)(25 35 45)(56 66 76)(57 67 77)(58 68 78)(59 69 79)(60 70 80)

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 4 4 6 6 type + + + + + - + + + - + + image C1 C2 C2 C3 C6 C6 D5 D10 SL2(𝔽3) SL2(𝔽3) C3×D5 C6×D5 A4 C2×A4 C2×A4 D5×SL2(𝔽3) D5×SL2(𝔽3) D5×A4 C2×D5×A4 kernel C2×D5×SL2(𝔽3) D5×SL2(𝔽3) C10×SL2(𝔽3) C2×Q8×D5 Q8×D5 Q8×C10 C2×SL2(𝔽3) SL2(𝔽3) D10 D10 C2×Q8 Q8 C22×D5 D10 C2×C10 C2 C2 C22 C2 # reps 1 2 1 2 4 2 2 2 4 8 4 4 1 2 1 4 8 2 2

Matrix representation of C2×D5×SL2(𝔽3) in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 43 1 0 0 60 0 0 0 0 0 1 0 0 0 0 1
,
 60 18 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 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 13 47 0 0 47 48
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 13 47
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[43,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,18,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,13,47,0,0,47,48],[1,0,0,0,0,1,0,0,0,0,1,13,0,0,0,47] >;

C2×D5×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_2\times D_5\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C2xD5xSL(2,3)");
// GroupNames label

G:=SmallGroup(480,1039);
// by ID

G=gap.SmallGroup(480,1039);
# by ID

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

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

׿
×
𝔽