C₂×#11711254 #72228211

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

solvable

Aliases

direct product

C₂×#11711254

Splittings

Fitting

C₂×#2657 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂²	#73084
commutator, cosocle	#376⋊C₃	C₂³
Frattini	C₂²×C₄	C₂²×S₄
Fitting	C₂×#2657	D₃
socle	C₂⁴	GL(2,ℤ/4)
hypercenter	C₂×C₄	#857⋊C₂
hypocenter	#95⋊C₃	C₂×D₄
LCS	G ▷ #376⋊C₃ ▷ #95⋊C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄

1530 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 1530 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:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258501
maximal normal subgroup of	#72258501
maximal quotient of	#72228212

GAP

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

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