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## G = C5×SL2(𝔽3)  order 120 = 23·3·5

### Direct product of C5 and SL2(𝔽3)

Aliases: C5×SL2(𝔽3), Q8⋊C15, C10.A4, (C5×Q8)⋊C3, C2.(C5×A4), SmallGroup(120,15)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×SL2(𝔽3)
G = < a,b,c,d | a5=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

Smallest permutation representation of C5×SL2(𝔽3)
On 40 points
Generators in S40
(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)
(1 28 12 32)(2 29 13 33)(3 30 14 34)(4 26 15 35)(5 27 11 31)(6 19 21 40)(7 20 22 36)(8 16 23 37)(9 17 24 38)(10 18 25 39)
(1 16 12 37)(2 17 13 38)(3 18 14 39)(4 19 15 40)(5 20 11 36)(6 35 21 26)(7 31 22 27)(8 32 23 28)(9 33 24 29)(10 34 25 30)
(6 19 26)(7 20 27)(8 16 28)(9 17 29)(10 18 30)(21 40 35)(22 36 31)(23 37 32)(24 38 33)(25 39 34)

G:=sub<Sym(40)| (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), (1,28,12,32)(2,29,13,33)(3,30,14,34)(4,26,15,35)(5,27,11,31)(6,19,21,40)(7,20,22,36)(8,16,23,37)(9,17,24,38)(10,18,25,39), (1,16,12,37)(2,17,13,38)(3,18,14,39)(4,19,15,40)(5,20,11,36)(6,35,21,26)(7,31,22,27)(8,32,23,28)(9,33,24,29)(10,34,25,30), (6,19,26)(7,20,27)(8,16,28)(9,17,29)(10,18,30)(21,40,35)(22,36,31)(23,37,32)(24,38,33)(25,39,34)>;

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), (1,28,12,32)(2,29,13,33)(3,30,14,34)(4,26,15,35)(5,27,11,31)(6,19,21,40)(7,20,22,36)(8,16,23,37)(9,17,24,38)(10,18,25,39), (1,16,12,37)(2,17,13,38)(3,18,14,39)(4,19,15,40)(5,20,11,36)(6,35,21,26)(7,31,22,27)(8,32,23,28)(9,33,24,29)(10,34,25,30), (6,19,26)(7,20,27)(8,16,28)(9,17,29)(10,18,30)(21,40,35)(22,36,31)(23,37,32)(24,38,33)(25,39,34) );

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)], [(1,28,12,32),(2,29,13,33),(3,30,14,34),(4,26,15,35),(5,27,11,31),(6,19,21,40),(7,20,22,36),(8,16,23,37),(9,17,24,38),(10,18,25,39)], [(1,16,12,37),(2,17,13,38),(3,18,14,39),(4,19,15,40),(5,20,11,36),(6,35,21,26),(7,31,22,27),(8,32,23,28),(9,33,24,29),(10,34,25,30)], [(6,19,26),(7,20,27),(8,16,28),(9,17,29),(10,18,30),(21,40,35),(22,36,31),(23,37,32),(24,38,33),(25,39,34)])

C5×SL2(𝔽3) is a maximal subgroup of   Q8.D15  Q8⋊D15  Dic5.A4

35 conjugacy classes

 class 1 2 3A 3B 4 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 15A ··· 15H 20A 20B 20C 20D 30A ··· 30H order 1 2 3 3 4 5 5 5 5 6 6 10 10 10 10 15 ··· 15 20 20 20 20 30 ··· 30 size 1 1 4 4 6 1 1 1 1 4 4 1 1 1 1 4 ··· 4 6 6 6 6 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 2 2 2 3 3 type + - + image C1 C3 C5 C15 SL2(𝔽3) SL2(𝔽3) C5×SL2(𝔽3) A4 C5×A4 kernel C5×SL2(𝔽3) C5×Q8 SL2(𝔽3) Q8 C5 C5 C1 C10 C2 # reps 1 2 4 8 1 2 12 1 4

Matrix representation of C5×SL2(𝔽3) in GL2(𝔽11) generated by

 3 0 0 3
,
 1 4 5 10
,
 7 10 6 4
,
 0 6 9 10
G:=sub<GL(2,GF(11))| [3,0,0,3],[1,5,4,10],[7,6,10,4],[0,9,6,10] >;

C5×SL2(𝔽3) in GAP, Magma, Sage, TeX

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

G:=Group("C5xSL(2,3)");
// GroupNames label

G:=SmallGroup(120,15);
// by ID

G=gap.SmallGroup(120,15);
# by ID

G:=PCGroup([5,-3,-5,-2,2,-2,452,72,903,133,58]);
// Polycyclic

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

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