C₇×#16810003 #37556868

order	23660
automorphism	?
homology	Abelianization (H₁) = C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₇×#16810003

Splittings

Fitting, socle, Hall	C₅×C₇×C₁₃² ⋊ C₄
derived, Hall	C₅×C₁₃² ⋊ C₄×C₇
Hall	C₇×C₁₃² ⋊ AGL(1,𝔽₅)
Sylow	C₁₃² ⋊ C₇×AGL(1,𝔽₅)
Hall	C₅×C₇ ⋊ #10621728
Sylow	C₅ ⋊ C₇×#10621728

Subgroups

	subgroup	G/H
center, hypercenter	C₇	#16810003
commutator, hypocenter	C₅×C₁₃²	C₄×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₁₃²	C₄
prime radical	C₇×C₅⋊(C₁₃⋊D₁₃)	C₂
cosocle	C₅⋊(C₁₃⋊D₁₃)	C₂×C₇
LCS	G ▷ C₅×C₁₃²
UCS	C₁ ◁ C₇

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 5915

A minimal faithful permutation representation — 3 generators on 38 points:

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

Conjugacy classes & character table

1505 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23660, 98);

C7×#16810003 #37556868