C₂×#261² #72229232

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

Aliases

direct product

C₂×#261²

Splittings

Fitting, socle	C₂×C₁₃² ⋊ C₄²
derived, Sylow	C₁₃² ⋊ C₂×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#261²
commutator, hypocenter	C₁₃²	C₂×C₄²
Frattini	C₁	G
Fitting, socle	C₂×C₁₃²	C₄²
prime radical	C₂×D₁₃²	C₂²
cosocle	D₁₃²	C₂³
LCS	G ▷ C₁₃²
UCS	C₁ ◁ C₂

240 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 3 generators on 28 points:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72214122, #72229297, D₃×#261²
maximal normal subgroup of	C₂×#72214122, #72229297
maximal quotient of	#72229293, #72229297, D₃×#261²

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