C₂×#10682473 #72270215

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#10682473

Splittings

Sylow

C₃ ⋊ C₂×#7310

Subgroups

	subgroup	G/H
center	C₂²	C₃⋊#1304
commutator	C₂⁴×C₃	C₂×C₄²
Frattini	C₂³×D₄	C₂²×D₃
Fitting	C₂×C₃×#1781	C₂
socle	C₂²×C₃	#1304
prime radical	C₂⁴×C₃×D₄	C₂²
cosocle	C₂³×C₃×D₄	C₂³
hypercenter	C₂×#1781	D₃
hypocenter	C₃	C₂×#7310
LCS	G ▷ C₂⁴×C₃ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ C₂×#1781

4148 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)(6,9)(7,12)(8,16)(10,19)(11,20)(13,14)(15,21)(17,18);(4,5)(6,8,10,18)(7,11)(9,16)(12,20,13,15)(14,21)(17,19);(6,7,9,13)(8,15,17,21)(10,14,19,12)(11,18,20,16)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270221, #72270226, #72270272, …
maximal normal subgroup of	#72270221, #72270226, #72270272, …

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

C2×#10682473 #72270215