C₂×#4510 #71417

order	384
isoclinism	isoclinic class of D₄₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#4510

Splittings

Fitting	C₂²×C₃×C₁₆ ⋊ C₂
Sylow	C₃ ⋊ C₂×#359

Subgroups

	subgroup	G/H
center	C₂×C₄	D₂₄
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₂²×C₃×C₁₆	C₂
socle	C₂²×C₃	#102
prime radical	C₂×#904	C₂
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂²×C₁₆	D₃
hypocenter	C₃	C₂×#359
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

224 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 37
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10710410, C₄×#4510, #10710415, …
maximal normal subgroup of	#10710410, C₄×#4510, #10710415, …
maximal quotient of	C₂×#67504, C₂×#67505, #10704928, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 3993);