C₂²×#73028 #72342872

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂²×#73028

Splittings

Fitting

C₂²×B(4,𝔽₂) ⋊ D₃

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₂³	#1020⋊C₂
commutator, cosocle, hypocenter	#997	C₂⁴
Frattini	C₂³	C₂³×S₄
Fitting	C₂²×B(4,𝔽₂)	D₃
LCS	G ▷ #997
UCS	C₁ ◁ C₂³

3760 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 3760 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#11714174, C₂²×#11714182, C₂²×#11714171
maximal normal subgroup of	C₂²×#11714174, C₂²×#11714182, C₂²×#11714171
maximal quotient of	C₂×#72245595

GAP

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

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