C5^2:3Q16 - GroupNames

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

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

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

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

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

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

49 conjugacy classes

class	1	2	4A	4B	4C	5A	5B	5C	5D	5E	5F	5G	5H	8A	8B	10A	10B	10C	10D	10E	10F	10G	10H	20A	20B	20C	20D	20E	···	20N	20O	20P	20Q	20R	40A	···	40H
order	1	2	4	4	4	5	5	5	5	5	5	5	5	8	8	10	10	10	10	10	10	10	10	20	20	20	20	20	···	20	20	20	20	20	40	···	40
size	1	1	2	20	100	2	2	2	2	4	4	4	4	10	10	2	2	2	2	4	4	4	4	2	2	2	2	4	···	4	20	20	20	20	10	···	10

49 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	-	+	+		-	-	+	+	-
image	C₁	C₂	C₂	C₂	D₄	D₅	D₅	Q₁₆	D₁₀	D₂₀	C₅⋊D₄	Dic₂₀	C₅⋊Q₁₆	D₅²	C₅⋊D₂₀	C₅²⋊₃Q₁₆
kernel	C₅²⋊₃Q₁₆	C₅×C₅⋊₂C₈	C₅×Dic₁₀	C₅²⋊₄Q₈	C₅×C₁₀	C₅⋊₂C₈	Dic₁₀	C₅²	C₂₀	C₁₀	C₁₀	C₅	C₅	C₄	C₂	C₁
# reps	1	1	1	1	1	2	2	2	4	4	4	8	2	4	4	8

Matrix representation of C₅²⋊₃Q₁₆ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	40
0	0	0	0	1	34

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	35	0	0
0	0	6	35	0	0
0	0	0	0	1	0
0	0	0	0	0	1

13	34	0	0	0	0
22	4	0	0	0	0
0	0	2	28	0	0
0	0	13	39	0	0
0	0	0	0	13	18
0	0	0	0	27	28

25	2	0	0	0	0
15	16	0	0	0	0
0	0	25	25	0	0
0	0	39	16	0	0
0	0	0	0	24	40
0	0	0	0	1	17

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,34],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,6,0,0,0,0,35,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,22,0,0,0,0,34,4,0,0,0,0,0,0,2,13,0,0,0,0,28,39,0,0,0,0,0,0,13,27,0,0,0,0,18,28],[25,15,0,0,0,0,2,16,0,0,0,0,0,0,25,39,0,0,0,0,25,16,0,0,0,0,0,0,24,1,0,0,0,0,40,17] >;

C₅²⋊₃Q₁₆ in GAP, Magma, Sage, TeX

C_5^2\rtimes_3Q_{16}

% in TeX

G:=Group("C5^2:3Q16");

// GroupNames label

G:=SmallGroup(400,70);

// by ID

G=gap.SmallGroup(400,70);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,73,79,218,50,970,11525]);

// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^8=1,d^2=c^4,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;

// generators/relations

Export

Subgroup lattice of C₅²⋊₃Q₁₆ in TeX

G = C52⋊3Q16 order 400 = 24·52

2nd semidirect product of C52 and Q16 acting via Q16/C4=C22

G = C₅²⋊₃Q₁₆ order 400 = 2⁴·5²

2^nd semidirect product of C₅² and Q₁₆ acting via Q₁₆/C₄=C₂²