C₈₃×D₇₅ #27159481

order	12450
isoclinism	isoclinic class of D₇₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₈₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₈₃×D₇₅
unambiguous semidirect product	C₃⋊(D₂₅×C₈₃)
unambiguous semidirect product	C₂₅⋊(D₃×C₈₃)

Splittings

Fitting, Hall	C₃×C₂₅×C₈₃ ⋊ C₂
derived, Hall	C₃×C₂₅ ⋊ C₂×C₈₃
Hall	C₂₅×C₈₃ ⋊ D₃
Hall	C₃×C₈₃ ⋊ D₂₅
Sylow	C₂₅ ⋊ D₃×C₈₃
Sylow	C₃ ⋊ D₂₅×C₈₃

Subgroups

	subgroup	G/H
center, hypercenter	C₈₃	D₇₅
commutator, cosocle, hypocenter	C₃×C₂₅	C₂×C₈₃
Frattini	C₅	D₁₅×C₈₃
Fitting	C₃×C₂₅×C₈₃	C₂
socle	C₃×C₅×C₈₃	D₅
LCS	G ▷ C₃×C₂₅
UCS	C₁ ◁ C₈₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 6225

A minimal faithful permutation representation — 2 generators on 111 points:

(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,12)(14,20)(17,21)(18,22)(19,24)(23,25)(27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(1,21,12,24,9,7,14,5,23,4,16,6,22,18,13,8,2,25,10,20,3,15,19,11,17)(26,27,28)

Conjugacy classes & character table

3237 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇₅×D₈₃, D₃×D₂₅×C₈₃, D₇₅⋊D₈₃, …
maximal normal subgroup of	D₇₅×D₈₃, D₃×D₂₅×C₈₃, D₇₅⋊D₈₃, …
maximal quotient of	C₈₃×Dic₇₅, C₂×C₈₃×D₇₅, C₈₃×D₂₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12450, 7);

C83×D75 #27159481