C₂×#11711936 #72245855

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

solvable

Aliases

direct product

C₂×#11711936

Splittings

Fitting

C₂²×#392 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×#857⋊C₂
commutator, cosocle, hypocenter	#95⋊C₃	C₂⁴
Frattini	C₂³	C₂³×S₄
Fitting	C₂²×#392	D₃
socle	C₂⁴	C₂²×S₄
hypercenter	C₂³	#857⋊C₂
LCS	G ▷ #95⋊C₃
UCS	C₁ ◁ C₂² ◁ C₂³

4100 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 4100 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 4 generators on 26 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal quotient of

C₂×GL(2,ℤ/8)

GAP

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

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