C₂×#5428 #85524

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

solvable

Aliases

direct product

C₂×#5428

Subgroups

	subgroup	G/H
center, socle	C₂²	GL(2,ℤ/4)
commutator, cosocle	#41⋊C₃	C₂³
Frattini	C₄	C₂²×S₄
Fitting	C₂×H(5,𝔽₂)	D₃
prime radical	C₂×#994	C₂
hypercenter	C₂×Q₈	S₄
hypocenter	SL(2,𝔽₃)	C₂×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

299 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 50
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 50 points:

(3,36,38,15,25,40,10,34,46,5,20,43)(4,28,48,9,33,45,16,26,50,11,31,41)(6,17,49,14,7,44,22,37,42,19,27,23)(8,13,32,18,29,39,21,35,24,12,30,47);(3,4,10,16)(5,8,15,21)(6,18,22,12)(7,23,27,44)(9,14,11,19)(13,32,35,24)(17,38,37,46)(20,40,25,43)(26,39,28,47)(29,48,30,50)(31,41,33,45)(34,49,36,42);(1,2)

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#5428, #11712833, #11713086, …
maximal normal subgroup of	C₄×#5428, #11712833, #11713086, …
maximal quotient of	C₂×#73089, C₂×#73091, C₂×#73097, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18100);