C3xD25 - GroupNames

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

(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)

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

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

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

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

C₃xD₂₅ is a maximal subgroup of C₇₅:C₄

42 conjugacy classes

class	1	2	3A	3B	5A	5B	6A	6B	15A	15B	15C	15D	25A	···	25J	75A	···	75T
order	1	2	3	3	5	5	6	6	15	15	15	15	25	···	25	75	···	75
size	1	25	1	1	2	2	25	25	2	2	2	2	2	···	2	2	···	2

42 irreducible representations

dim	1	1	1	1	2	2	2	2
type	+	+			+		+
image	C₁	C₂	C₃	C₆	D₅	C₃xD₅	D₂₅	C₃xD₂₅
kernel	C₃xD₂₅	C₇₅	D₂₅	C₂₅	C₁₅	C₅	C₃	C₁
# reps	1	1	2	2	2	4	10	20

Matrix representation of C₃xD₂₅ ►in GL₂(F₁₅₁) generated by

32	0
0	32

71	36
115	120

71	36
11	80

G:=sub<GL(2,GF(151))| [32,0,0,32],[71,115,36,120],[71,11,36,80] >;

C₃xD₂₅ in GAP, Magma, Sage, TeX

C_3\times D_{25}

% in TeX

G:=Group("C3xD25");

// GroupNames label

G:=SmallGroup(150,2);

// by ID

G=gap.SmallGroup(150,2);

# by ID

G:=PCGroup([4,-2,-3,-5,-5,650,250,1923]);

// Polycyclic

G:=Group<a,b,c|a^3=b^25=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;

// generators/relations

Export

Subgroup lattice of C₃xD₂₅ in TeX

G = C3xD25 order 150 = 2·3·52

Direct product of C3 and D25

G = C₃xD₂₅ order 150 = 2·3·5²

Direct product of C₃ and D₂₅