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G = C3×C25⋊C4order 300 = 22·3·52

Direct product of C3 and C25⋊C4

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3×C25⋊C4, C25⋊C12, C752C4, D25.C6, C15.2F5, C5.(C3×F5), (C3×D25).2C2, SmallGroup(300,5)

Series: Derived Chief Lower central Upper central

C1C25 — C3×C25⋊C4
C1C5C25D25C3×D25 — C3×C25⋊C4
C25 — C3×C25⋊C4
C1C3

Generators and relations for C3×C25⋊C4
 G = < a,b,c | a3=b25=c4=1, ab=ba, ac=ca, cbc-1=b18 >

25C2
25C4
25C6
5D5
25C12
5F5
5C3×D5
5C3×F5

Character table of C3×C25⋊C4

 class 123A3B4A4B56A6B12A12B12C12D15A15B25A25B25C25D25E75A75B75C75D75E75F75G75H75I75J
 size 125112525425252525252544444444444444444
ρ1111111111111111111111111111111    trivial
ρ21111-1-1111-1-1-1-111111111111111111    linear of order 2
ρ311ζ3ζ32-1-11ζ3ζ32ζ6ζ65ζ65ζ6ζ3ζ3211111ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 6
ρ411ζ3ζ32111ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ3211111ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 3
ρ511ζ32ζ3111ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ311111ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 3
ρ611ζ32ζ3-1-11ζ32ζ3ζ65ζ6ζ6ζ65ζ32ζ311111ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 6
ρ71-111i-i1-1-1i-ii-i11111111111111111    linear of order 4
ρ81-111-ii1-1-1-ii-ii11111111111111111    linear of order 4
ρ91-1ζ3ζ32-ii1ζ65ζ6ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ3ζ3211111ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 12
ρ101-1ζ32ζ3i-i1ζ6ζ65ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ32ζ311111ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 12
ρ111-1ζ32ζ3-ii1ζ6ζ65ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ32ζ311111ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 12
ρ121-1ζ3ζ32i-i1ζ65ζ6ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ3ζ3211111ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 12
ρ13404400400000044-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ14404400-1000000-1-1ζ25192517258256ζ252325142511252ζ251625132512259ζ25222521254253ζ2524251825725ζ251625132512259ζ252325142511252ζ251625132512259ζ25222521254253ζ25222521254253ζ2524251825725ζ2524251825725ζ25192517258256ζ252325142511252ζ25192517258256    orthogonal lifted from C25⋊C4
ρ15404400-1000000-1-1ζ252325142511252ζ251625132512259ζ25222521254253ζ2524251825725ζ25192517258256ζ25222521254253ζ251625132512259ζ25222521254253ζ2524251825725ζ2524251825725ζ25192517258256ζ25192517258256ζ252325142511252ζ251625132512259ζ252325142511252    orthogonal lifted from C25⋊C4
ρ16404400-1000000-1-1ζ251625132512259ζ25222521254253ζ2524251825725ζ25192517258256ζ252325142511252ζ2524251825725ζ25222521254253ζ2524251825725ζ25192517258256ζ25192517258256ζ252325142511252ζ252325142511252ζ251625132512259ζ25222521254253ζ251625132512259    orthogonal lifted from C25⋊C4
ρ17404400-1000000-1-1ζ25222521254253ζ2524251825725ζ25192517258256ζ252325142511252ζ251625132512259ζ25192517258256ζ2524251825725ζ25192517258256ζ252325142511252ζ252325142511252ζ251625132512259ζ251625132512259ζ25222521254253ζ2524251825725ζ25222521254253    orthogonal lifted from C25⋊C4
ρ18404400-1000000-1-1ζ2524251825725ζ25192517258256ζ252325142511252ζ251625132512259ζ25222521254253ζ252325142511252ζ25192517258256ζ252325142511252ζ251625132512259ζ251625132512259ζ25222521254253ζ25222521254253ζ2524251825725ζ25192517258256ζ2524251825725    orthogonal lifted from C25⋊C4
ρ1940-2-2-3-2+2-3004000000-2-2-3-2+2-3-1-1-1-1-1ζ6ζ65ζ65ζ6ζ65ζ65ζ6ζ6ζ6ζ65    complex lifted from C3×F5
ρ2040-2+2-3-2-2-3004000000-2+2-3-2-2-3-1-1-1-1-1ζ65ζ6ζ6ζ65ζ6ζ6ζ65ζ65ζ65ζ6    complex lifted from C3×F5
ρ2140-2+2-3-2-2-300-1000000ζ65ζ6ζ25222521254253ζ2524251825725ζ25192517258256ζ252325142511252ζ251625132512259ζ3ζ25193ζ25173ζ2583ζ256ζ32ζ252432ζ251832ζ25732ζ25ζ32ζ251932ζ251732ζ25832ζ256ζ3ζ25233ζ25143ζ25113ζ252ζ32ζ252332ζ251432ζ251132ζ252ζ32ζ251632ζ251332ζ251232ζ259ζ3ζ25163ζ25133ζ25123ζ259ζ3ζ25223ζ25213ζ2543ζ253ζ3ζ25243ζ25183ζ2573ζ25ζ32ζ252232ζ252132ζ25432ζ253    complex faithful
ρ2240-2-2-3-2+2-300-1000000ζ6ζ65ζ251625132512259ζ25222521254253ζ2524251825725ζ25192517258256ζ252325142511252ζ32ζ252432ζ251832ζ25732ζ25ζ3ζ25223ζ25213ζ2543ζ253ζ3ζ25243ζ25183ζ2573ζ25ζ32ζ251932ζ251732ζ25832ζ256ζ3ζ25193ζ25173ζ2583ζ256ζ3ζ25233ζ25143ζ25113ζ252ζ32ζ252332ζ251432ζ251132ζ252ζ32ζ251632ζ251332ζ251232ζ259ζ32ζ252232ζ252132ζ25432ζ253ζ3ζ25163ζ25133ζ25123ζ259    complex faithful
ρ2340-2-2-3-2+2-300-1000000ζ6ζ65ζ25222521254253ζ2524251825725ζ25192517258256ζ252325142511252ζ251625132512259ζ32ζ251932ζ251732ζ25832ζ256ζ3ζ25243ζ25183ζ2573ζ25ζ3ζ25193ζ25173ζ2583ζ256ζ32ζ252332ζ251432ζ251132ζ252ζ3ζ25233ζ25143ζ25113ζ252ζ3ζ25163ζ25133ζ25123ζ259ζ32ζ251632ζ251332ζ251232ζ259ζ32ζ252232ζ252132ζ25432ζ253ζ32ζ252432ζ251832ζ25732ζ25ζ3ζ25223ζ25213ζ2543ζ253    complex faithful
ρ2440-2+2-3-2-2-300-1000000ζ65ζ6ζ251625132512259ζ25222521254253ζ2524251825725ζ25192517258256ζ252325142511252ζ3ζ25243ζ25183ζ2573ζ25ζ32ζ252232ζ252132ζ25432ζ253ζ32ζ252432ζ251832ζ25732ζ25ζ3ζ25193ζ25173ζ2583ζ256ζ32ζ251932ζ251732ζ25832ζ256ζ32ζ252332ζ251432ζ251132ζ252ζ3ζ25233ζ25143ζ25113ζ252ζ3ζ25163ζ25133ζ25123ζ259ζ3ζ25223ζ25213ζ2543ζ253ζ32ζ251632ζ251332ζ251232ζ259    complex faithful
ρ2540-2+2-3-2-2-300-1000000ζ65ζ6ζ2524251825725ζ25192517258256ζ252325142511252ζ251625132512259ζ25222521254253ζ3ζ25233ζ25143ζ25113ζ252ζ32ζ251932ζ251732ζ25832ζ256ζ32ζ252332ζ251432ζ251132ζ252ζ3ζ25163ζ25133ζ25123ζ259ζ32ζ251632ζ251332ζ251232ζ259ζ32ζ252232ζ252132ζ25432ζ253ζ3ζ25223ζ25213ζ2543ζ253ζ3ζ25243ζ25183ζ2573ζ25ζ3ζ25193ζ25173ζ2583ζ256ζ32ζ252432ζ251832ζ25732ζ25    complex faithful
ρ2640-2+2-3-2-2-300-1000000ζ65ζ6ζ25192517258256ζ252325142511252ζ251625132512259ζ25222521254253ζ2524251825725ζ3ζ25163ζ25133ζ25123ζ259ζ32ζ252332ζ251432ζ251132ζ252ζ32ζ251632ζ251332ζ251232ζ259ζ3ζ25223ζ25213ζ2543ζ253ζ32ζ252232ζ252132ζ25432ζ253ζ32ζ252432ζ251832ζ25732ζ25ζ3ζ25243ζ25183ζ2573ζ25ζ3ζ25193ζ25173ζ2583ζ256ζ3ζ25233ζ25143ζ25113ζ252ζ32ζ251932ζ251732ζ25832ζ256    complex faithful
ρ2740-2-2-3-2+2-300-1000000ζ6ζ65ζ252325142511252ζ251625132512259ζ25222521254253ζ2524251825725ζ25192517258256ζ32ζ252232ζ252132ζ25432ζ253ζ3ζ25163ζ25133ζ25123ζ259ζ3ζ25223ζ25213ζ2543ζ253ζ32ζ252432ζ251832ζ25732ζ25ζ3ζ25243ζ25183ζ2573ζ25ζ3ζ25193ζ25173ζ2583ζ256ζ32ζ251932ζ251732ζ25832ζ256ζ32ζ252332ζ251432ζ251132ζ252ζ32ζ251632ζ251332ζ251232ζ259ζ3ζ25233ζ25143ζ25113ζ252    complex faithful
ρ2840-2-2-3-2+2-300-1000000ζ6ζ65ζ2524251825725ζ25192517258256ζ252325142511252ζ251625132512259ζ25222521254253ζ32ζ252332ζ251432ζ251132ζ252ζ3ζ25193ζ25173ζ2583ζ256ζ3ζ25233ζ25143ζ25113ζ252ζ32ζ251632ζ251332ζ251232ζ259ζ3ζ25163ζ25133ζ25123ζ259ζ3ζ25223ζ25213ζ2543ζ253ζ32ζ252232ζ252132ζ25432ζ253ζ32ζ252432ζ251832ζ25732ζ25ζ32ζ251932ζ251732ζ25832ζ256ζ3ζ25243ζ25183ζ2573ζ25    complex faithful
ρ2940-2+2-3-2-2-300-1000000ζ65ζ6ζ252325142511252ζ251625132512259ζ25222521254253ζ2524251825725ζ25192517258256ζ3ζ25223ζ25213ζ2543ζ253ζ32ζ251632ζ251332ζ251232ζ259ζ32ζ252232ζ252132ζ25432ζ253ζ3ζ25243ζ25183ζ2573ζ25ζ32ζ252432ζ251832ζ25732ζ25ζ32ζ251932ζ251732ζ25832ζ256ζ3ζ25193ζ25173ζ2583ζ256ζ3ζ25233ζ25143ζ25113ζ252ζ3ζ25163ζ25133ζ25123ζ259ζ32ζ252332ζ251432ζ251132ζ252    complex faithful
ρ3040-2-2-3-2+2-300-1000000ζ6ζ65ζ25192517258256ζ252325142511252ζ251625132512259ζ25222521254253ζ2524251825725ζ32ζ251632ζ251332ζ251232ζ259ζ3ζ25233ζ25143ζ25113ζ252ζ3ζ25163ζ25133ζ25123ζ259ζ32ζ252232ζ252132ζ25432ζ253ζ3ζ25223ζ25213ζ2543ζ253ζ3ζ25243ζ25183ζ2573ζ25ζ32ζ252432ζ251832ζ25732ζ25ζ32ζ251932ζ251732ζ25832ζ256ζ32ζ252332ζ251432ζ251132ζ252ζ3ζ25193ζ25173ζ2583ζ256    complex faithful

Smallest permutation representation of C3×C25⋊C4
On 75 points
Generators in S75
(1 60 44)(2 61 45)(3 62 46)(4 63 47)(5 64 48)(6 65 49)(7 66 50)(8 67 26)(9 68 27)(10 69 28)(11 70 29)(12 71 30)(13 72 31)(14 73 32)(15 74 33)(16 75 34)(17 51 35)(18 52 36)(19 53 37)(20 54 38)(21 55 39)(22 56 40)(23 57 41)(24 58 42)(25 59 43)
(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)
(2 8 25 19)(3 15 24 12)(4 22 23 5)(6 11 21 16)(7 18 20 9)(10 14 17 13)(26 43 37 45)(27 50 36 38)(28 32 35 31)(29 39 34 49)(30 46 33 42)(40 41 48 47)(51 72 69 73)(52 54 68 66)(53 61 67 59)(55 75 65 70)(56 57 64 63)(58 71 62 74)

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

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

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

Matrix representation of C3×C25⋊C4 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
6616
5536
1060
4216
,
1500
1600
4215
2306
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,5,1,4,6,5,0,2,1,3,6,1,6,6,0,6],[1,1,4,2,5,6,2,3,0,0,1,0,0,0,5,6] >;

C3×C25⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_{25}\rtimes C_4
% in TeX

G:=Group("C3xC25:C4");
// GroupNames label

G:=SmallGroup(300,5);
// by ID

G=gap.SmallGroup(300,5);
# by ID

G:=PCGroup([5,-2,-3,-2,-5,-5,30,1683,973,118,3004,1014]);
// Polycyclic

G:=Group<a,b,c|a^3=b^25=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^18>;
// generators/relations

Export

Subgroup lattice of C3×C25⋊C4 in TeX
Character table of C3×C25⋊C4 in TeX

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