C₄×#829 #70908

order	384
isoclinism	isoclinic class of #826
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₄×#829

Splittings

Sylow

C₃ ⋊ C₄×Q₃₂

Subgroups

	subgroup	G/H
center	C₂×C₄	#213
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×C₄×Q₁₆	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₄×Q₁₆	D₃
hypocenter	C₃	C₄×Q₃₂
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

150 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 39
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3)(8,10,13,21)(9,14,17,31)(11,23,26,39)(12,28,24,20)(15,33,34,38)(16,35,27,30)(18,36,37,22)(19,32,29,25);(2,3)(8,9,13,17)(10,18,21,37)(11,22,26,36)(12,27,24,16)(14,29,31,19)(15,32,34,25)(20,38,28,33)(23,35,39,30);(4,5,6,7)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10706918, C₈×#829, #10706930, …
maximal normal subgroup of	#10706918, C₈×#829, #10706930, …
maximal quotient of	C₈×#829, #10707101, #10707103, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 3484);