C₅×D₇×D₂₁₇ #44477164

order	30380
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×D₇×D₂₁₇
unambiguous semidirect product	(C₇×D₇)⋊(C₅×D₃₁)

Splittings

Fitting, socle, Hall	C₅×C₇²×C₃₁ ⋊ C₂²
derived, Hall	C₇²×C₃₁ ⋊ C₂²×C₅
Hall	C₅×C₇² ⋊ C₂×D₃₁
Hall	C₅×C₃₁ ⋊ D₇²
Sylow	C₇² ⋊ C₂×C₅×D₃₁
Sylow	C₃₁ ⋊ C₅×D₇²

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₇×D₂₁₇
commutator, cosocle, hypocenter	C₇²×C₃₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₃₁	C₂²
LCS	G ▷ C₇²×C₃₁
UCS	C₁ ◁ C₅

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 2170

A minimal faithful permutation representation — 2 generators on 50 points:

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

Conjugacy classes & character table

2750 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30380, 73);

C5×D7×D217 #44477164