D₇×#5429 #72246261

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

direct product

D₇×#5429

Splittings

Fitting	C₇×H(5,𝔽₂) ⋊ C₂×D₃
Sylow	C₇ ⋊ C₂×#5429

Subgroups

	subgroup	G/H
center	C₂	D₇×GL(2,ℤ/4)
commutator, cosocle	C₇×#41⋊C₃	C₂³
Frattini	C₄	C₂×D₇×S₄
Fitting	C₇×H(5,𝔽₂)	C₂×D₃
socle	C₂×C₇	C₂×GL(2,ℤ/4)
hypercenter	Q₈	D₇×S₄
hypocenter	C₇×SL(2,𝔽₃)	C₂×D₄
LCS	G ▷ C₇×#41⋊C₃ ▷ C₇×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

690 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 31
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 3 generators on 31 points:

(1,17,19,2,11,22,7,16,24,6,14,12)(3,9,23,5,15,21,10,13,18,8,4,20)(25,26,28,30,31,29,27);(26,27)(28,29)(30,31);(2,5)(3,10)(4,12)(6,8)(9,18)(11,20)(13,23)(14,21)(15,22)(16,19)(17,24)

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×SO⁺(4,𝔽₃), AGL(1,𝔽₇)×#5429
maximal normal subgroup of	AGL(1,𝔽₇)×#5429

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,17,19,2,11,22,7,16,24,6,14,12)(3,9,23,5,15,21,10,13,18,8,4,20)(25,26,28,30,31,29,27), (26,27)(28,29)(30,31), (2,5)(3,10)(4,12)(6,8)(9,18)(11,20)(13,23)(14,21)(15,22)(16,19)(17,24));