C₅×#6505 #11765102

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

Aliases

direct product

C₅×#6505

Splittings

Fitting	C₂×C₅×C₇×Q₈ ⋊ C₂
Hall	C₅×C₇ ⋊ #137
Sylow	C₇ ⋊ C₅×#137

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×C₂²⋊D₇
commutator, cosocle	C₄×C₇	C₂³×C₅
Frattini	C₄	C₂²×C₅×D₇
Fitting	C₂×C₅×C₇×Q₈	C₂
socle	C₂×C₅×C₇	C₂×D₄
prime radical	C₅×#1157	C₂
hypercenter	C₂×C₅×Q₈	D₇
hypocenter	C₇	C₅×#137
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×Q₈

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

Conjugacy classes & character table

205 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 708);