ΓL(1,𝔽₁₆)×C₆₇ #16860852

order	4020
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₄×C₆₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	ΓL(1,𝔽₁₆)×C₆₇
unambiguous semidirect product	C₃⋊(AGL(1,𝔽₅)×C₆₇)

Splittings

Fitting, socle, Hall	C₃×C₅×C₆₇ ⋊ C₄
derived, Hall	C₃×C₅ ⋊ C₄×C₆₇
Hall	C₅×C₆₇ ⋊ Dic₃
Hall	C₃×C₆₇ ⋊ AGL(1,𝔽₅)
Sylow	C₅ ⋊ Dic₃×C₆₇
Sylow	C₃ ⋊ AGL(1,𝔽₅)×C₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₆₇	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₄×C₆₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₆₇	C₄
prime radical	C₃×D₅×C₆₇	C₂
cosocle	C₃×D₅	C₂×C₆₇
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₆₇

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 1005

A minimal faithful permutation representation — 2 generators on 75 points:

(2,3,4,5)(7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(1,2,5,3,4)(6,7,8)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₆₇×C₉⋊AGL(1,𝔽₅), ΓL(1,𝔽₁₆)×C₆₇⋊C₃, C₃×ΓL(1,𝔽₁₆)×C₆₇, …
maximal normal subgroup of	ΓL(1,𝔽₁₆)×C₆₇⋊C₃, C₃×ΓL(1,𝔽₁₆)×C₆₇, C₅×ΓL(1,𝔽₁₆)×C₆₇, …
maximal quotient of	C₆₇×C₉⋊AGL(1,𝔽₅), C₃×ΓL(1,𝔽₁₆)×C₆₇, C₆₇×C₃⋊ΓL(1,𝔽₁₆), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4020, 24);

ΓL(1,𝔽16)×C67 #16860852