C₂×#4733 #78609

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#4733

Splittings

Fitting	C₂×C₃×C₂²⋊C₈ ⋊ C₂
Sylow	C₃ ⋊ C₂×#452

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₂²⋊C₈	C₂
socle	C₂³×C₃	C₂×D₄
prime radical	C₂²×#235	C₂
hypercenter	C₂×C₂²⋊C₈	D₃
hypocenter	C₃	C₂×#452
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

760 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 165 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) = 25
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 25 points:

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10691493, #10691495, #10800814, …
maximal normal subgroup of	#10691493, #10691495, #10800814, …
maximal quotient of	C₂×#68132, C₂×#68136, C₂×#68109, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11185);