C₂×#10682487 #72271601

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#10682487

Splittings

Sylow

C₃ ⋊ C₂×#7316

Subgroups

	subgroup	G/H
center	C₂²	C₃⋊#1304
commutator	C₂⁴×C₃	C₂×C₄²
Frattini	C₂³×D₄	C₂²×D₃
Fitting	C₂×C₃×#1783	C₂
socle	C₂²×C₃	#1304
prime radical	C₂⁴×C₃×D₄	C₂²
cosocle	C₂³×C₃×D₄	C₂³
hypercenter	C₂×#1783	D₃
hypocenter	C₃	C₂×#7316
LCS	G ▷ C₂⁴×C₃ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ C₂×#1783

3992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271603, C₂×D₃×#7316, #72271624, …
maximal normal subgroup of	#72271603, C₂×D₃×#7316, #72271624, …

GAP

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

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

C2×#10682487 #72271601