C₂×#72240888 #72340923

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#72240888

Splittings

Fitting

C₂×#52301 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	#85559
commutator, cosocle, hypocenter	#2679⋊C₃	C₂³
Frattini	C₂³	C₂²×#1020
Fitting	C₂×#52301	D₃
socle	C₂⁴	C₂×#1020
prime radical	C₂×#11714279	C₂
LCS	G ▷ #2679⋊C₃
UCS	C₁ ◁ C₂×C₄

5883 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 5883 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72240895, C₂×#72240898
maximal normal subgroup of	C₂×#72240895, C₂×#72240898

GAP

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

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