C4xSL(2,5) - GroupNames

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

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

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

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

36 conjugacy classes

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

36 irreducible representations

dim	1	1	1	2	2	3	3	3	4	4	4	4	4	5	5	5	6	6
type	+	+		-		+	+		+	-	+			+	+		-
image	C₁	C₂	C₄	SL₂(𝔽₅)	C₄.A₅	A₅	C₂×A₅	C₄×A₅	A₅	SL₂(𝔽₅)	C₂×A₅	C₄×A₅	C₄.A₅	A₅	C₂×A₅	C₄×A₅	SL₂(𝔽₅)	C₄.A₅
kernel	C₄×SL₂(𝔽₅)	C₂×SL₂(𝔽₅)	SL₂(𝔽₅)	C₄	C₂	C₂×C₄	C₂²	C₂	C₂×C₄	C₄	C₂²	C₂	C₂	C₂×C₄	C₂²	C₂	C₄	C₂
# reps	1	1	2	4	4	2	2	4	1	2	1	2	2	1	1	2	2	2

Matrix representation of C₄×SL₂(𝔽₅) ►in GL₃(𝔽₆₁) generated by

11	0	0
0	38	36
0	10	40

11	0	0
0	36	21
0	40	42

G:=sub<GL(3,GF(61))| [11,0,0,0,38,10,0,36,40],[11,0,0,0,36,40,0,21,42] >;

C₄×SL₂(𝔽₅) in GAP, Magma, Sage, TeX

C_4\times {\rm SL}_2({\mathbb F}_5)

% in TeX

G:=Group("C4xSL(2,5)");

// GroupNames label

G:=SmallGroup(480,222);

// by ID

G=gap.SmallGroup(480,222);

# by ID

Export

Subgroup lattice of C₄×SL₂(𝔽₅) in TeX

G = C4×SL2(𝔽5) order 480 = 25·3·5

Direct product of C4 and SL2(𝔽5)

G = C₄×SL₂(𝔽₅) order 480 = 2⁵·3·5

Direct product of C₄ and SL₂(𝔽₅)