C₇×C₁₉⋊#1035 #27210854

order	13300
automorphism	?
homology	Abelianization (H₁) = C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×C₁₉⋊#1035
unambiguous semidirect product	C₁₉⋊(C₇×#1035)

Splittings

Fitting, Hall	C₇×C₁₉×C₂₅ ⋊ C₄
derived, Hall	C₁₉×C₂₅ ⋊ C₄×C₇
Hall	C₇×C₂₅ ⋊ Dic₁₉
Hall	C₇×C₁₉ ⋊ #1035
Sylow	C₂₅ ⋊ C₇×Dic₁₉
Sylow	C₁₉ ⋊ C₇×#1035

Subgroups

	subgroup	G/H
center, hypercenter	C₇	C₁₉⋊#1035
commutator, hypocenter	C₁₉×C₂₅	C₄×C₇
Frattini	C₅	C₇×C₁₉⋊AGL(1,𝔽₅)
Fitting	C₇×C₁₉×C₂₅	C₄
socle	C₅×C₇×C₁₉	AGL(1,𝔽₅)
prime radical	C₇×C₁₉×D₂₅	C₂
cosocle	C₁₉×D₂₅	C₂×C₇
LCS	G ▷ C₁₉×C₂₅
UCS	C₁ ◁ C₇

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 3325

A minimal faithful permutation representation — 2 generators on 51 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13300, 12);

C7×C19⋊#1035 #27210854