C₂×#72210824 #72267331

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₄

solvable

Aliases

direct product

C₂×#72210824

Splittings

Fitting, Sylow	C₂×#147496 ⋊ C₃
derived	#13826 ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center	C₂²	#10935⋊C₃
commutator, cosocle, hypocenter	#13826	C₂²×C₃
Frattini	C₂³×Q₈	C₂²×A₄
Fitting	C₂×#147496	C₃
socle	C₂⁴	B(4,𝔽₂)⋊C₃
hypercenter	C₂³	#1437⋊C₃
LCS	G ▷ #13826
UCS	C₁ ◁ C₂² ◁ C₂³

4306 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 4306 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 28
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 28 points:

(1,2,10,4,5,6)(3,8,11,9,7,12)(15,16,25)(18,19,22)(20,28,27)(23,24,26);(1,3,4,9)(2,5)(6,11,10,12)(7,8)(13,20,26,19,17,22,16,27)(14,25,18,24,21,23,28,15)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72267346, #72267523, #72267749, …

GAP

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

gap> G := Group((1,2,10,4,5,6)(3,8,11,9,7,12)(15,16,25)(18,19,22)(20,28,27)(23,24,26), (1,3,4,9)(2,5)(6,11,10,12)(7,8)(13,20,26,19,17,22,16,27)(14,25,18,24,21,23,28,15));