AGL(1,𝔽₅)×C₆₇ #13050698

order	1340
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₄×C₆₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

AGL(1,𝔽₅)×C₆₇

Splittings

Fitting, socle, Hall	C₅×C₆₇ ⋊ C₄
derived, Sylow	C₅ ⋊ C₄×C₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₆₇	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄×C₆₇
Frattini	C₁	G
Fitting, socle	C₅×C₆₇	C₄
prime radical	D₅×C₆₇	C₂
cosocle	D₅	C₂×C₆₇
LCS	G ▷ C₅
UCS	C₁ ◁ C₆₇

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 335

A minimal faithful permutation representation — 2 generators on 72 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	AGL(1,𝔽₅)×C₆₇⋊C₃, C₃×AGL(1,𝔽₅)×C₆₇, ΓL(1,𝔽₁₆)×C₆₇, …
maximal normal subgroup of	AGL(1,𝔽₅)×C₆₇⋊C₃, C₃×AGL(1,𝔽₅)×C₆₇, AGL(1,𝔽₅)×C₆₇⋊C₁₁, …
maximal quotient of	C₃×AGL(1,𝔽₅)×C₆₇, ΓL(1,𝔽₁₆)×C₆₇, C₆₇×#4380, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1340, 5);

AGL(1,𝔽5)×C67 #13050698