C₃×C₇×Dic₂₅ #14523527

order	2100
isoclinism	isoclinic class of D₂₅
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×C₇×Dic₂₅

Splittings

derived, Sylow	C₂₅ ⋊ C₃×C₄×C₇
Hall	C₃×C₇×C₂₅ ⋊ C₄
Hall	C₇×C₂₅ ⋊ C₃×C₄
Hall	C₃×C₂₅ ⋊ C₄×C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₇	D₂₅
commutator, hypocenter	C₂₅	C₃×C₄×C₇
Frattini	C₂×C₅	C₃×C₇×D₅
Fitting	C₂×C₃×C₇×C₂₅	C₂
socle, prime radical	C₂×C₃×C₅×C₇	D₅
cosocle	C₂×C₂₅	C₂×C₃×C₇
LCS	G ▷ C₂₅
UCS	C₁ ◁ C₂×C₃×C₇

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 2100

A minimal faithful permutation representation — 2 generators on 39 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×C₄₉×Dic₂₅, C₇×ΓL(1,𝔽₈)⋊Dic₂₅, C₇×ΓL(1,𝔽₈)×Dic₂₅, …
maximal normal subgroup of	C₃×C₄₉×Dic₂₅, C₃×C₇²×Dic₂₅, C₃×C₇×C₁₁×Dic₂₅, …
maximal quotient of	C₃×C₄₉×Dic₂₅, C₇×ΓL(1,𝔽₈)⋊Dic₂₅, C₇×ΓL(1,𝔽₈)×Dic₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 7);

C3×C7×Dic25 #14523527