C₇₃×D₈₅ #27137201

order	12410
isoclinism	isoclinic class of D₈₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₇₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇₃×D₈₅
unambiguous semidirect product	C₅⋊(D₁₇×C₇₃)
unambiguous semidirect product	C₁₇⋊(D₅×C₇₃)

Splittings

Fitting, socle, Hall	C₅×C₁₇×C₇₃ ⋊ C₂
derived, Hall	C₅×C₁₇ ⋊ C₂×C₇₃
Hall	C₁₇×C₇₃ ⋊ D₅
Hall	C₅×C₇₃ ⋊ D₁₇
Sylow	C₁₇ ⋊ D₅×C₇₃
Sylow	C₅ ⋊ D₁₇×C₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₇₃	D₈₅
commutator, cosocle, hypocenter	C₅×C₁₇	C₂×C₇₃
Frattini	C₁	G
Fitting, socle	C₅×C₁₇×C₇₃	C₂
LCS	G ▷ C₅×C₁₇
UCS	C₁ ◁ C₇₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 6205

A minimal faithful permutation representation — 2 generators on 95 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(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);(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,19,21,22,20)

Conjugacy classes & character table

3212 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇₃×#66775, #37613865, C₇₃×#66776, …
maximal normal subgroup of	C₇₃×#66775, #37613865, C₇₃×#66776, …
maximal quotient of	C₇₃×Dic₈₅, C₂×C₇₃×D₈₅, C₃×C₇₃×D₈₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12410, 7);

C73×D85 #27137201