C₂×#72214059 #72229241

order	5408
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×#72214059

Splittings

derived, Sylow

C₁₃² ⋊ C₂×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	D₁₃⋊#261
commutator, hypocenter	C₁₃²	C₂×C₄²
Frattini	C₂	C₂×D₁₃⋊#261
Fitting, socle	C₂²×C₁₃²	C₂×C₄
prime radical	C₂²×C₁₃⋊D₁₃	C₂²
cosocle	C₂×C₁₃⋊D₁₃	C₂³
LCS	G ▷ C₁₃²
UCS	C₁ ◁ C₂²

378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72214067, #72229293, C₂×#72214198, …
maximal normal subgroup of	C₂×#72214067, #72229293, C₂×#72214198, …
maximal quotient of	#72259204

GAP

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

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

C2×#72214059 #72229241