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C₂×C₅×#4296 #13133756

order	1710
isoclinism	isoclinic class of #4296
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₉ · Schur multiplier (H₂) = C₁

Aliases

C₂×C₅×#4296

Splittings

Fitting, socle, Hall	C₂×C₅×C₁₉ ⋊ C₉
derived, Sylow	C₁₉ ⋊ C₂×C₅×C₉
Hall	C₅×C₁₉ ⋊ C₂×C₉
Hall	C₂×C₁₉ ⋊ C₅×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	#4296
commutator, hypocenter	C₁₉	C₂×C₅×C₉
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₁₉	C₉
prime radical	C₂×C₅×C₁₉⋊C₃	C₃
cosocle	C₁₉⋊C₃	C₂×C₃×C₅
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂×C₅

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 190

A minimal faithful permutation representation — 2 generators on 26 points:

(1,2)(3,4,5,6,7)(9,10,13,11,14,20,16,21,15)(12,17,24,18,25,26,19,23,22);(8,9,12,19,10,15,21,11,17,14,22,16,23,24,18,26,13,20,25)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#4296⋊C₄, Dic₅×#4296, #4296⋊Dic₅, …
maximal normal subgroup of	C₅×#4296⋊C₄, Dic₅×#4296, #4296⋊Dic₅, …
maximal quotient of	C₄×C₅×#4296, C₂²×C₅×#4296, C₅×#14558803, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1710, 16);