C₂×C₃²×#2177 #72246002

order	2304
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃²×C₄ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

direct product

C₂×C₃²×#2177

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃²	C₂×D₈
commutator	C₈	C₂³×C₃²×C₄
Frattini, cosocle	C₂×C₈	C₂⁴×C₃²
prime radical	C₂²×C₃²×D₁₆	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ C₂×C₃²×C₄² ◁ C₂×C₃²×C₄×C₈ ◁ G

2394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 1152

A minimal faithful permutation representation — 4 generators on 28 points:

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

Conjugacy classes & character table

576 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	C₂×C₃⋊D₃×#2177, C₂×#72222464
maximal subgroup of	C₂×C₃⋊D₃×#2177, C₂×C₃²×#12067, C₂×#72222464
maximal normal subgroup of	C₂×C₃⋊D₃×#2177, C₂×C₃²×#12067, C₂×#72222464

GAP

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

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

C2×C32×#2177 #72246002