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## G = C10×GL2(𝔽3)  order 480 = 25·3·5

### Direct product of C10 and GL2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C10×GL2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C5×SL2(𝔽3) — C5×GL2(𝔽3) — C10×GL2(𝔽3)
 Lower central SL2(𝔽3) — C10×GL2(𝔽3)
 Upper central C1 — C2×C10

Generators and relations for C10×GL2(𝔽3)
G = < a,b,c,d,e | a10=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece=b-1, dbd-1=bc, ebe=b2c, dcd-1=b, ede=d-1 >

Subgroups: 386 in 102 conjugacy classes, 24 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C8, C2×C4, D4, Q8, Q8, C23, C10, C10, C10, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, C20, C2×C10, C2×C10, SL2(𝔽3), C22×S3, C5×S3, C30, C2×SD16, C40, C2×C20, C5×D4, C5×Q8, C5×Q8, C22×C10, GL2(𝔽3), C2×SL2(𝔽3), S3×C10, C2×C30, C2×C40, C5×SD16, D4×C10, Q8×C10, C2×GL2(𝔽3), C5×SL2(𝔽3), S3×C2×C10, C10×SD16, C5×GL2(𝔽3), C10×SL2(𝔽3), C10×GL2(𝔽3)
Quotients: C1, C2, C22, C5, S3, C10, D6, C2×C10, S4, C5×S3, GL2(𝔽3), C2×S4, S3×C10, C2×GL2(𝔽3), C5×S4, C5×GL2(𝔽3), C10×S4, C10×GL2(𝔽3)

Smallest permutation representation of C10×GL2(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 18 80 48)(2 19 71 49)(3 20 72 50)(4 11 73 41)(5 12 74 42)(6 13 75 43)(7 14 76 44)(8 15 77 45)(9 16 78 46)(10 17 79 47)(21 59 31 65)(22 60 32 66)(23 51 33 67)(24 52 34 68)(25 53 35 69)(26 54 36 70)(27 55 37 61)(28 56 38 62)(29 57 39 63)(30 58 40 64)
(1 22 80 32)(2 23 71 33)(3 24 72 34)(4 25 73 35)(5 26 74 36)(6 27 75 37)(7 28 76 38)(8 29 77 39)(9 30 78 40)(10 21 79 31)(11 69 41 53)(12 70 42 54)(13 61 43 55)(14 62 44 56)(15 63 45 57)(16 64 46 58)(17 65 47 59)(18 66 48 60)(19 67 49 51)(20 68 50 52)
(11 25 69)(12 26 70)(13 27 61)(14 28 62)(15 29 63)(16 30 64)(17 21 65)(18 22 66)(19 23 67)(20 24 68)(31 59 47)(32 60 48)(33 51 49)(34 52 50)(35 53 41)(36 54 42)(37 55 43)(38 56 44)(39 57 45)(40 58 46)
(1 80)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 21)(18 22)(19 23)(20 24)(31 47)(32 48)(33 49)(34 50)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)

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

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

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

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4 type + + + + + + + + image C1 C2 C2 C5 C10 C10 S3 D6 C5×S3 GL2(𝔽3) S3×C10 C5×GL2(𝔽3) S4 C2×S4 C5×S4 C10×S4 GL2(𝔽3) C5×GL2(𝔽3) kernel C10×GL2(𝔽3) C5×GL2(𝔽3) C10×SL2(𝔽3) C2×GL2(𝔽3) GL2(𝔽3) C2×SL2(𝔽3) Q8×C10 C5×Q8 C2×Q8 C10 Q8 C2 C2×C10 C10 C22 C2 C10 C2 # reps 1 2 1 4 8 4 1 1 4 4 4 16 2 2 8 8 2 8

Matrix representation of C10×GL2(𝔽3) in GL4(𝔽241) generated by

 154 0 0 0 0 154 0 0 0 0 98 0 0 0 0 98
,
 1 0 0 0 0 1 0 0 0 0 135 174 0 0 67 106
,
 1 0 0 0 0 1 0 0 0 0 173 136 0 0 67 68
,
 0 240 0 0 1 240 0 0 0 0 240 1 0 0 240 0
,
 0 1 0 0 1 0 0 0 0 0 1 240 0 0 0 240
G:=sub<GL(4,GF(241))| [154,0,0,0,0,154,0,0,0,0,98,0,0,0,0,98],[1,0,0,0,0,1,0,0,0,0,135,67,0,0,174,106],[1,0,0,0,0,1,0,0,0,0,173,67,0,0,136,68],[0,1,0,0,240,240,0,0,0,0,240,240,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,240,240] >;

C10×GL2(𝔽3) in GAP, Magma, Sage, TeX

C_{10}\times {\rm GL}_2({\mathbb F}_3)
% in TeX

G:=Group("C10xGL(2,3)");
// GroupNames label

G:=SmallGroup(480,1017);
// by ID

G=gap.SmallGroup(480,1017);
# by ID

G:=PCGroup([7,-2,-2,-5,-3,-2,2,-2,1123,4204,655,172,2525,404,285,124]);
// Polycyclic

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

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