C₂²×#85558 #72266501

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#85558

Splittings

Fitting	C₂²×H(3,𝔽₄) ⋊ D₃
derived	#5464 ⋊ C₂³

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#85558
commutator, cosocle, hypocenter	#5464	C₂³
Frattini	C₂²	C₂²×#1020
Fitting	C₂²×H(3,𝔽₄)	D₃
socle	C₂⁴	#1020
LCS	G ▷ #5464
UCS	C₁ ◁ C₂²

2894 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 2894 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 26
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#11715758
maximal normal subgroup of	C₂²×#11715758
maximal quotient of	C₂²×#11714280, #72340836

GAP

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

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