C₇×#4044 #11765142

order	1120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#4044

Splittings

Fitting	C₅×C₇×C₄⋊C₄ ⋊ C₂
Hall	C₅×C₇ ⋊ #126
Sylow	C₅ ⋊ C₇×#126

Subgroups

	subgroup	G/H
center	C₂²×C₇	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂³×C₇
Frattini	C₂²	C₂²×C₇×D₅
Fitting	C₅×C₇×C₄⋊C₄	C₂
socle	C₂²×C₅×C₇	C₂³
prime radical	C₂²×C₇×D₅	C₂²
hypercenter	C₇×C₄⋊C₄	D₅
hypocenter	C₅	C₇×#126
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₇ ◁ C₇×C₄⋊C₄

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

normal subgroups only

This group has 120 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)(14,18)(17,20)(21,23,24,27)(22,26,25,28);(13,15,16,19)(14,17,18,20)(22,25)(23,27);(9,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,25)(27,28)

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, 748);