C₇×#4096 #11765194

order	1120
isoclinism	isoclinic class of #686
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#4096

Splittings

Fitting	C₅×C₇×#41 ⋊ C₂
Hall	C₅×C₇ ⋊ #135
Sylow	C₅ ⋊ C₇×#135

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂³×C₇
Frattini	C₄	C₂²×C₇×D₅
Fitting	C₅×C₇×#41	C₂
socle	C₂×C₅×C₇	C₂×D₄
hypercenter	C₇×#41	D₅
hypocenter	C₅	C₇×#135
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#41

124 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 560

A minimal faithful permutation representation — 3 generators on 28 points:

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

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 800);