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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 [×2], C2 [×2], C3, C4 [×2], C22, C22 [×4], C5, S3 [×4], C6 [×3], C8 [×2], C2×C4, D4 [×3], Q8, Q8, C23, C10, C10 [×2], C10 [×2], D6 [×6], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, C20 [×2], C2×C10, C2×C10 [×4], SL2(𝔽3), C22×S3, C5×S3 [×4], C30 [×3], C2×SD16, C40 [×2], C2×C20, C5×D4 [×3], C5×Q8, C5×Q8, C22×C10, GL2(𝔽3) [×2], C2×SL2(𝔽3), S3×C10 [×6], C2×C30, C2×C40, C5×SD16 [×4], D4×C10, Q8×C10, C2×GL2(𝔽3), C5×SL2(𝔽3), S3×C2×C10, C10×SD16, C5×GL2(𝔽3) [×2], C10×SL2(𝔽3), C10×GL2(𝔽3)
Quotients: C1, C2 [×3], C22, C5, S3, C10 [×3], D6, C2×C10, S4, C5×S3, GL2(𝔽3) [×2], C2×S4, S3×C10, C2×GL2(𝔽3), C5×S4, C5×GL2(𝔽3) [×2], 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 66 40 78)(2 67 31 79)(3 68 32 80)(4 69 33 71)(5 70 34 72)(6 61 35 73)(7 62 36 74)(8 63 37 75)(9 64 38 76)(10 65 39 77)(11 57 21 41)(12 58 22 42)(13 59 23 43)(14 60 24 44)(15 51 25 45)(16 52 26 46)(17 53 27 47)(18 54 28 48)(19 55 29 49)(20 56 30 50)
(1 14 40 24)(2 15 31 25)(3 16 32 26)(4 17 33 27)(5 18 34 28)(6 19 35 29)(7 20 36 30)(8 11 37 21)(9 12 38 22)(10 13 39 23)(41 75 57 63)(42 76 58 64)(43 77 59 65)(44 78 60 66)(45 79 51 67)(46 80 52 68)(47 71 53 69)(48 72 54 70)(49 73 55 61)(50 74 56 62)
(11 41 63)(12 42 64)(13 43 65)(14 44 66)(15 45 67)(16 46 68)(17 47 69)(18 48 70)(19 49 61)(20 50 62)(21 57 75)(22 58 76)(23 59 77)(24 60 78)(25 51 79)(26 52 80)(27 53 71)(28 54 72)(29 55 73)(30 56 74)
(1 40)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 61)(20 62)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 71)(28 72)(29 73)(30 74)

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

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

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

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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