C₂×#4697 #78462

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#4697

Splittings

Fitting	C₂×C₃×C₄×C₈ ⋊ C₂
Sylow	C₃ ⋊ C₂×#488

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle	C₂³×C₃	C₂×D₄
prime radical	C₂²×D₁₂	C₂²
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	C₂×#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

412 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 29
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 29 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10694738, #10694740, #10694745, …
maximal normal subgroup of	#10694738, #10694740, #10694745, …
maximal quotient of	C₂×#69763, C₂×#69767, C₂×#69768, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11038);