C₅×D₂₉×C₇⋊D₇ #43249334

order	28420
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₇

Aliases

direct product	C₅×D₂₉×C₇⋊D₇
unambiguous semidirect product	(C₇×D₂₉)⋊(C₅×D₇)

Splittings

Fitting, socle, Hall	C₅×C₇²×C₂₉ ⋊ C₂²
derived, Hall	C₇²×C₂₉ ⋊ C₂²×C₅
Hall	C₅×C₇² ⋊ C₂×D₂₉
Hall	C₅×C₂₉ ⋊ C₂×C₇⋊D₇
Sylow	C₇² ⋊ C₂×C₅×D₂₉
Sylow	C₂₉ ⋊ C₂×C₅×C₇⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₂₉×C₇⋊D₇
commutator, cosocle, hypocenter	C₇²×C₂₉	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₂₉	C₂²
LCS	G ▷ C₇²×C₂₉
UCS	C₁ ◁ C₅

200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 7105

A minimal faithful permutation representation — 3 generators on 48 points:

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

Conjugacy classes & character table

2080 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28420, 109);

C5×D29×C7⋊D7 #43249334