C₂×D₄×#4207 #72276290

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

direct product

C₂×D₄×#4207

Splittings

Fitting	C₂×D₄×#737 ⋊ C₂
Sylow	#737 ⋊ C₂²×D₄

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#4207
commutator, cosocle	C₂×C₃×C₉	C₂⁴×C₃
Frattini	C₂×C₃	C₂³×C₃×C₃⋊D₃
Fitting	C₂×D₄×#737	C₂
socle	C₂²×C₃	C₂²×C₃×C₃⋊D₃
hypercenter	C₂×D₄	#4207
hypocenter	C₃×C₉	C₂²×C₃×D₄
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂×D₄

2013 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 4 generators on 33 points:

(8,19,21,11,12,13)(9,26,17,15,16,28)(10,18)(14,24)(20,32,29,25,23,31)(22,30)(27,33);(4,6)(7,9,15)(8,14,25)(10,17,26)(11,20,24)(12,22,31)(13,23,33)(16,28,18)(19,29,30)(21,27,32);(1,2)(7,21,19,10,8,11,18,12,13)(9,27,29,17,14,20,16,22,23)(15,32,30,26,25,24,28,31,33);(3,4)(5,6)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal quotient of

C₂×D₄×#92299

GAP

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

gap> G := Group((8,19,21,11,12,13)(9,26,17,15,16,28)(10,18)(14,24)(20,32,29,25,23,31)(22,30)(27,33), (4,6)(7,9,15)(8,14,25)(10,17,26)(11,20,24)(12,22,31)(13,23,33)(16,28,18)(19,29,30)(21,27,32), (1,2)(7,21,19,10,8,11,18,12,13)(9,27,29,17,14,20,16,22,23)(15,32,30,26,25,24,28,31,33), (3,4)(5,6));

C2×D4×#4207 #72276290