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## G = SL2(𝔽3).11D10order 480 = 25·3·5

### 1st non-split extension by SL2(𝔽3) of D10 acting through Inn(SL2(𝔽3))

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×Q8 — SL2(𝔽3).11D10
 Chief series C1 — C2 — C10 — C5×Q8 — C5×SL2(𝔽3) — D5×SL2(𝔽3) — SL2(𝔽3).11D10
 Lower central C5×Q8 — SL2(𝔽3).11D10
 Upper central C1 — C2 — C22

Generators and relations for SL2(𝔽3).11D10
G = < a,b,c,d,e | a4=c3=d10=1, b2=e2=a2, bab-1=a-1, cac-1=b, ad=da, ae=ea, cbc-1=ab, bd=db, be=eb, cd=dc, ce=ec, ede-1=a2d-1 >

Subgroups: 574 in 92 conjugacy classes, 23 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C2×C4, D4, Q8, Q8, D5, C10, C10, C12, C2×C6, C15, C2×Q8, C2×Q8, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, SL2(𝔽3), C3×D4, C3×D5, C30, 2- 1+4, Dic10, C4×D5, D20, C5⋊D4, C5⋊D4, C2×C20, C5×Q8, C5×Q8, C2×SL2(𝔽3), C2×SL2(𝔽3), C4.A4, C3×Dic5, C6×D5, C2×C30, C4○D20, Q8×D5, Q8×D5, Q82D5, Q82D5, Q8×C10, D4.A4, C5×SL2(𝔽3), C3×C5⋊D4, Q8.10D10, Dic5.A4, D5×SL2(𝔽3), C10×SL2(𝔽3), SL2(𝔽3).11D10
Quotients: C1, C2, C3, C22, C6, D5, A4, C2×C6, D10, C2×A4, C3×D5, C22×A4, C6×D5, D4.A4, D5×A4, C2×D5×A4, SL2(𝔽3).11D10

Smallest permutation representation of SL2(𝔽3).11D10
On 80 points
Generators in S80
(1 22 12 32)(2 23 13 33)(3 24 14 34)(4 25 15 35)(5 21 11 31)(6 38 18 28)(7 39 19 29)(8 40 20 30)(9 36 16 26)(10 37 17 27)(41 62 46 67)(42 63 47 68)(43 64 48 69)(44 65 49 70)(45 66 50 61)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(1 7 12 19)(2 8 13 20)(3 9 14 16)(4 10 15 17)(5 6 11 18)(21 28 31 38)(22 29 32 39)(23 30 33 40)(24 26 34 36)(25 27 35 37)(41 52 46 57)(42 53 47 58)(43 54 48 59)(44 55 49 60)(45 56 50 51)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(6 21 28)(7 22 29)(8 23 30)(9 24 26)(10 25 27)(16 34 36)(17 35 37)(18 31 38)(19 32 39)(20 33 40)(51 61 71)(52 62 72)(53 63 73)(54 64 74)(55 65 75)(56 66 76)(57 67 77)(58 68 78)(59 69 79)(60 70 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 50 12 45)(2 44 13 49)(3 48 14 43)(4 42 15 47)(5 46 11 41)(6 57 18 52)(7 51 19 56)(8 55 20 60)(9 59 16 54)(10 53 17 58)(21 67 31 62)(22 61 32 66)(23 65 33 70)(24 69 34 64)(25 63 35 68)(26 79 36 74)(27 73 37 78)(28 77 38 72)(29 71 39 76)(30 75 40 80)

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 10A ··· 10F 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L order 1 2 2 2 2 3 3 4 4 4 4 5 5 6 6 6 6 6 6 10 ··· 10 12 12 15 15 15 15 20 20 20 20 30 ··· 30 size 1 1 2 10 30 4 4 6 6 10 30 2 2 4 4 8 8 40 40 2 ··· 2 40 40 8 8 8 8 12 12 12 12 8 ··· 8

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 6 6 type + + + + + + + + + + - + + image C1 C2 C2 C2 C3 C6 C6 C6 D5 D10 C3×D5 C6×D5 A4 C2×A4 C2×A4 C2×A4 D4.A4 D4.A4 SL2(𝔽3).11D10 D5×A4 C2×D5×A4 kernel SL2(𝔽3).11D10 Dic5.A4 D5×SL2(𝔽3) C10×SL2(𝔽3) Q8.10D10 Q8×D5 Q8⋊2D5 Q8×C10 C2×SL2(𝔽3) SL2(𝔽3) C2×Q8 Q8 C5⋊D4 Dic5 D10 C2×C10 C5 C5 C1 C22 C2 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 1 1 1 2 12 2 2

Matrix representation of SL2(𝔽3).11D10 in GL4(𝔽61) generated by

 14 13 0 0 13 47 0 0 0 0 14 13 0 0 13 47
,
 0 60 0 0 1 0 0 0 0 0 0 60 0 0 1 0
,
 48 14 0 0 1 0 0 0 0 0 48 14 0 0 1 0
,
 9 0 13 0 0 9 0 13 0 0 27 0 0 0 0 27
,
 13 0 38 0 0 13 0 38 18 0 48 0 0 18 0 48
G:=sub<GL(4,GF(61))| [14,13,0,0,13,47,0,0,0,0,14,13,0,0,13,47],[0,1,0,0,60,0,0,0,0,0,0,1,0,0,60,0],[48,1,0,0,14,0,0,0,0,0,48,1,0,0,14,0],[9,0,0,0,0,9,0,0,13,0,27,0,0,13,0,27],[13,0,18,0,0,13,0,18,38,0,48,0,0,38,0,48] >;

SL2(𝔽3).11D10 in GAP, Magma, Sage, TeX

{\rm SL}_2({\mathbb F}_3)._{11}D_{10}
% in TeX

G:=Group("SL(2,3).11D10");
// GroupNames label

G:=SmallGroup(480,1040);
// by ID

G=gap.SmallGroup(480,1040);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-5,-2,3389,269,584,123,795,382,8069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^3=d^10=1,b^2=e^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,a*d=d*a,a*e=e*a,c*b*c^-1=a*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^2*d^-1>;
// generators/relations

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