C₂×#4919 #82026

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#4919

Splittings

Fitting	C₂×C₃×Q₃₂ ⋊ C₂
Sylow	C₃ ⋊ C₂×#508

Subgroups

	subgroup	G/H
center	C₂²	D₃×D₈
commutator, cosocle	C₃×C₈	C₂⁴
Frattini	C₈	C₂³×D₃
Fitting	C₂×C₃×Q₃₂	C₂
socle	C₂²×C₃	C₂×D₈
hypercenter	C₂×Q₃₂	D₃
hypocenter	C₃	C₂×#508
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

368 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 117 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 — 4 generators on 37 points:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10715620, #10715625, #10715662, …
maximal normal subgroup of	#10715620, #10715625, #10715662, …
maximal quotient of	C₂×#69220, C₂×#69325, C₂×#69317, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14602);