C₂×C₅×C₃⋊(C₂₃⋊D₂₃) #44540774

order	31740
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₂₃

Aliases

direct product	C₂×C₅×C₃⋊(C₂₃⋊D₂₃)
unambiguous semidirect product	(C₂×C₃)⋊(C₅×C₂₃⋊D₂₃)
unambiguous semidirect product	(C₂×C₂₃)⋊(C₅×D₆₉)

Splittings

Fitting, socle	C₂×C₃×C₅×C₂₃² ⋊ C₂
derived, Hall	C₃×C₂₃² ⋊ C₂²×C₅
Hall	C₃×C₅×C₂₃² ⋊ C₂²
Hall	C₅×C₂₃² ⋊ C₂×D₃
Sylow	C₂₃² ⋊ C₂×C₅×D₃
Hall	C₃×C₅ ⋊ C₂×C₂₃⋊D₂₃
Sylow	C₃ ⋊ C₂×C₅×C₂₃⋊D₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	C₃⋊(C₂₃⋊D₂₃)
commutator, cosocle, hypocenter	C₃×C₂₃²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₂₃²	C₂
LCS	G ▷ C₃×C₂₃²
UCS	C₁ ◁ C₂×C₅

520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 15870

A minimal faithful permutation representation — 3 generators on 56 points:

(4,5)(6,7,8,9,10)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56);(1,2)(3,4,5)(34,35,37,39,41,43,45,47,49,51,53,55,56,54,52,50,48,46,44,42,40,38,36);(11,12,14,16,18,20,22,24,26,28,30,32,33,31,29,27,25,23,21,19,17,15,13)

Conjugacy classes & character table

7950 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31740, 112);

C2×C5×C3⋊(C23⋊D23) #44540774