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## G = C22×SL2(𝔽5)  order 480 = 25·3·5

### Direct product of C22 and SL2(𝔽5)

Aliases: C22×SL2(𝔽5), C23.A5, C22.5(C2×A5), C2.7(C22×A5), SmallGroup(480,960)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C2 — C22 — C23 — C22×SL2(𝔽5)
 Derived series SL2(𝔽5) — C22×SL2(𝔽5)
 Lower central SL2(𝔽5) — C22×SL2(𝔽5)
 Upper central C1 — C23

Subgroups: 788 in 104 conjugacy classes, 21 normal (5 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C2×C4, Q8, C23, C10, Dic3, C2×C6, C22×C4, C2×Q8, Dic5, C2×C10, SL2(𝔽3), C2×Dic3, C22×C6, C22×Q8, C2×Dic5, C22×C10, C2×SL2(𝔽3), C22×Dic3, C22×Dic5, C22×SL2(𝔽3), SL2(𝔽5), C2×SL2(𝔽5), C22×SL2(𝔽5)
Quotients: C1, C2, C22, A5, SL2(𝔽5), C2×A5, C2×SL2(𝔽5), C22×A5, C22×SL2(𝔽5)

Smallest permutation representation of C22×SL2(𝔽5)
On 96 points
Generators in S96
(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 81 82 83 84 85 86)(87 88 89 90 91 92 93 94 95 96)
(1 53 60 24 83 5 87 28 46 72)(2 48 65 19 78 6 92 33 41 67)(3 23 94 75 61 10 45 50 86 29)(4 18 89 70 66 9 40 55 81 34)(7 69 37 27 90 12 80 25 59 56)(8 74 42 32 95 11 85 20 64 51)(13 36 77 49 38 15 58 76 93 26)(14 31 82 54 43 16 63 71 88 21)(17 39)(22 44)(30 62)(35 57)(47 91)(52 96)(68 79)(73 84)

G:=sub<Sym(96)| (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)(77,78,79,80,81,82,83,84,85,86)(87,88,89,90,91,92,93,94,95,96), (1,53,60,24,83,5,87,28,46,72)(2,48,65,19,78,6,92,33,41,67)(3,23,94,75,61,10,45,50,86,29)(4,18,89,70,66,9,40,55,81,34)(7,69,37,27,90,12,80,25,59,56)(8,74,42,32,95,11,85,20,64,51)(13,36,77,49,38,15,58,76,93,26)(14,31,82,54,43,16,63,71,88,21)(17,39)(22,44)(30,62)(35,57)(47,91)(52,96)(68,79)(73,84)>;

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)(77,78,79,80,81,82,83,84,85,86)(87,88,89,90,91,92,93,94,95,96), (1,53,60,24,83,5,87,28,46,72)(2,48,65,19,78,6,92,33,41,67)(3,23,94,75,61,10,45,50,86,29)(4,18,89,70,66,9,40,55,81,34)(7,69,37,27,90,12,80,25,59,56)(8,74,42,32,95,11,85,20,64,51)(13,36,77,49,38,15,58,76,93,26)(14,31,82,54,43,16,63,71,88,21)(17,39)(22,44)(30,62)(35,57)(47,91)(52,96)(68,79)(73,84) );

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),(77,78,79,80,81,82,83,84,85,86),(87,88,89,90,91,92,93,94,95,96)], [(1,53,60,24,83,5,87,28,46,72),(2,48,65,19,78,6,92,33,41,67),(3,23,94,75,61,10,45,50,86,29),(4,18,89,70,66,9,40,55,81,34),(7,69,37,27,90,12,80,25,59,56),(8,74,42,32,95,11,85,20,64,51),(13,36,77,49,38,15,58,76,93,26),(14,31,82,54,43,16,63,71,88,21),(17,39),(22,44),(30,62),(35,57),(47,91),(52,96),(68,79),(73,84)]])

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 2 3 3 4 4 4 5 5 6 type + + - + + + - + + + - image C1 C2 SL2(𝔽5) A5 C2×A5 A5 SL2(𝔽5) C2×A5 A5 C2×A5 SL2(𝔽5) kernel C22×SL2(𝔽5) C2×SL2(𝔽5) C22 C23 C22 C23 C22 C22 C23 C22 C22 # reps 1 3 8 2 6 1 4 3 1 3 4

Matrix representation of C22×SL2(𝔽5) in GL4(𝔽61) generated by

 1 0 0 0 0 60 0 0 0 0 28 49 0 0 59 51
,
 60 0 0 0 0 1 0 0 0 0 10 34 0 0 40 8
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,28,59,0,0,49,51],[60,0,0,0,0,1,0,0,0,0,10,40,0,0,34,8] >;

C22×SL2(𝔽5) in GAP, Magma, Sage, TeX

C_2^2\times {\rm SL}_2({\mathbb F}_5)
% in TeX

G:=Group("C2^2xSL(2,5)");
// GroupNames label

G:=SmallGroup(480,960);
// by ID

G=gap.SmallGroup(480,960);
# by ID

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