C₄×#904 #78460

order	384
isoclinism	isoclinic class of #204
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₄×#904

Splittings

Fitting	C₂×C₃×C₄×C₈ ⋊ C₂
Sylow	C₃ ⋊ C₄×#135

Subgroups

	subgroup	G/H
center	C₄²	D₁₂
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×#904	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	C₄×#135
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

500 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 197 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) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10704925, #10704928, #10704930, …
maximal normal subgroup of	#10704925, #10704928, #10704930, …
maximal quotient of	C₈×#904, #10764720, #10764938, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11036);