C₂²×#87516 #72275312

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂²×#87516

Splittings

Fitting

C₂³×H(5,𝔽₂) ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#1020
commutator, cosocle, hypocenter	#997	C₂⁴
Frattini	C₂	C₂³×#1020
Fitting	C₂³×H(5,𝔽₂)	D₃
hypercenter	C₂⁴	#1020
LCS	G ▷ #997
UCS	C₁ ◁ C₂³ ◁ C₂⁴

13277 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 13277 conjugacy classes of subgroups — too many to draw. Only its 158 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)(6,10)(7,17,14)(8,20,15)(11,12,18,16,13,19);(3,4)(5,7,9,15)(6,11,10,16)(8,20,17,14)(12,19,13,18);(1,2)(3,4)(6,10)(11,16)(12,13)(18,19);(5,6,9,10)(7,13,15,12)(8,16,17,11)(14,19,20,18)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#72209772, C₂²×#11714170, C₂²×#11715828, …
maximal normal subgroup of	C₂²×#11714170, C₂²×#11715828, C₂²×#11714171, …
maximal quotient of	C₂×#72245388

GAP

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

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