C₅×#6478 #11765073

order	1120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#6478

Splittings

Fitting	C₅×C₇×SD₁₆ ⋊ C₂
Hall	C₅×C₇ ⋊ #137
Sylow	C₇ ⋊ C₅×#137

Subgroups

	subgroup	G/H
center	C₂×C₅	D₄×D₇
commutator, cosocle	C₄×C₇	C₂³×C₅
Frattini	C₄	C₂²×C₅×D₇
Fitting	C₅×C₇×SD₁₆	C₂
socle	C₂×C₅×C₇	C₂×D₄
prime radical	C₅×#1159	C₂
hypercenter	C₅×SD₁₆	D₇
hypocenter	C₇	C₅×#137
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×SD₁₆

120 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 54 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)(6,7,9,11,12,10,8)(14,19)(15,21)(16,24)(18,25)(22,23)(26,28);(13,15,17,23)(14,18,19,26)(16,21,24,22)(20,28,27,25);(7,8)(9,10)(11,12)(13,14)(15,18)(16,20)(17,19)(21,28)(22,25)(23,26)(24,27)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 679);