C₂×#4496 #70688

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#4496

Splittings

Sylow

C₃ ⋊ C₂×#362

Subgroups

	subgroup	G/H
center	C₂²	#809
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₂×C₃×#108	C₂
socle	C₂²×C₃	#102
prime radical	C₂³×C₃	SD₁₆
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂×#108	D₃
hypocenter	C₃	C₂×#362
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

216 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 37
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 37 points:

(3,4,5)(6,8,9,17,11,20,23,19)(7,12,13,28,15,30,21,29)(10,25,22,27,26,36,31,18)(14,32,24,35,33,34,37,16);(4,5)(6,7,11,15)(8,16,20,35)(9,21,23,13)(10,24,26,37)(12,27,30,18)(14,31,33,22)(17,34,19,32)(25,29,36,28);(1,2)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10710444, C₄×#4496, #10710463, …
maximal normal subgroup of	#10710444, C₄×#4496, #10710463, …
maximal quotient of	C₂×#67466, #10704965, #10705196, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 3264);