C₂×#10682486 #72270176

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

Aliases

direct product

C₂×#10682486

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×#1781	C₂
socle	C₂²×C₃	#1304
prime radical	C₂⁴×C₃×D₄	C₂²
cosocle	C₂³×C₃×D₄	C₂³
hypercenter	C₂×#1781	D₃
hypocenter	C₃	C₂×#7316
LCS	G ▷ C₂⁴×C₃ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ C₂×#1781

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270180, #72270200, #72270272, …
maximal normal subgroup of	#72270180, #72270200, #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)(7,13)(8,15)(10,11)(12,19)(14,18)(16,17)(20,21), (4,5)(6,8,9,15)(7,12,18,19)(10,17,11,16)(13,21,14,20), (6,7)(8,12,16,20)(9,18,10,13)(11,14)(15,19)(17,21));

C2×#10682486 #72270176