C₂×#11861057 #72227382

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#11861057

Splittings

Fitting	C₂×C₃²×#97 ⋊ C₂²
Sylow	C₃² ⋊ C₂²×#444

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#3802
commutator	C₃²×C₄	C₂⁴×C₄
Frattini	C₂×C₄	C₂³×D₃²
Fitting	C₂×C₃²×#97	C₂²
socle	C₂²×C₃²	C₂×C₄×D₄
prime radical	C₂×#10594684	C₂
cosocle	C₂×C₃²×C₄	C₂⁵
hypercenter	C₂×#97	D₃²
hypocenter	C₃²	C₂²×#444
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#97

5760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

216 conjugacy classes — view the full character table.

GAP

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

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

C2×#11861057 #72227382