C₂×#4723 #78588

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

Aliases

direct product

C₂×#4723

Splittings

Sylow

C₃ ⋊ C₂×#485

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	C₂⁴×C₃	D₄
hypercenter	C₂×C₂²⋊C₈	D₃
hypocenter	C₃	C₂×#485
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

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

normal subgroups only

This group has 416 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)(8,12)(14,16,17,22,18,23,24,28)(15,19,20,25,21,26,27,29);(1,2)(6,8)(7,10)(9,12)(11,13)(14,15,18,21)(16,19,23,26)(17,20,24,27)(22,25,28,29);(4,5)(6,7,9,11)(8,10,12,13)(16,22)(17,24)(19,25)(20,27)(23,28)(26,29);(1,2)

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10691500, #10691502, #10802350, …
maximal normal subgroup of	#10691500, #10691502, #10802350, …
maximal quotient of	C₂×#69926, C₂×#69929, C₂×#69931, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11164);