C₇×#66774 #14558669

order	2380
isoclinism	isoclinic class of #66774
automorphism	?
homology	Abelianization (H₁) = C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₇×#66774

Splittings

Fitting, socle, Hall	C₅×C₇×C₁₇ ⋊ C₄
derived, Hall	C₅×C₁₇ ⋊ C₄×C₇
Hall	C₇×C₁₇ ⋊ AGL(1,𝔽₅)
Hall	C₅×C₇ ⋊ Dic₁₇
Sylow	C₁₇ ⋊ C₇×AGL(1,𝔽₅)
Sylow	C₅ ⋊ C₇×Dic₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	#66774
commutator, hypocenter	C₅×C₁₇	C₄×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₁₇	C₄
prime radical	C₇×D₅×C₁₇	C₂
cosocle	D₅×C₁₇	C₂×C₇
LCS	G ▷ C₅×C₁₇
UCS	C₁ ◁ C₇

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 595

A minimal faithful permutation representation — 2 generators on 29 points:

(1,2,3,4,5,6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(26,27,28,29);(8,9,11,13,15,17,19,21,23,24,22,20,18,16,14,12,10)(25,26,29,27,28)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×#13133322, C₅×C₇×#66774, C₇×#13133345, …
maximal normal subgroup of	C₅×C₇×#66774, C₄₉×#66774, C₇²×#66774, …
maximal quotient of	C₇×#13133322, C₅×C₇×#66774, C₇×#13133345, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2380, 16);

C7×#66774 #14558669