C₂×#72224676 #72230366

order	31752
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃²

Aliases

direct product

C₂×#72224676

Splittings

Fitting	C₂×C₃²×C₇² ⋊ C₂²×C₃²
derived	C₃²×C₇² ⋊ C₂³×C₃²
Hall	#72224636 ⋊ C₂³
Sylow	C₇² ⋊ C₂²×C₃×(C₃⋊D₃)⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72224676
commutator, cosocle, hypocenter	C₃²×C₇²	C₂³×C₃²
Frattini	C₃	C₂×D₃×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
Fitting	C₂×C₃²×C₇²	C₂²×C₃²
socle	C₂×C₃×C₇²	C₂×C₃²×D₃
LCS	G ▷ C₃²×C₇²
UCS	C₁ ◁ C₂

2104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 126

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

holomorph of

C₂×C₃×ΓL(1,𝔽₈)

GAP

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

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

C2×#72224676 #72230366