C₂×#4896 #79255

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

Aliases

direct product

C₂×#4896

Splittings

Fitting	C₂×C₃×#108 ⋊ C₂
Sylow	C₃ ⋊ C₂×#473

Subgroups

	subgroup	G/H
center	C₂²	#895
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×#108	C₂
socle	C₂²×C₃	#121
prime radical	C₂×#235	C₂²
hypercenter	C₂×#108	D₃
hypocenter	C₃	C₂×#473
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

448 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 141 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:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10712158, #10712160, #10712207, …
maximal normal subgroup of	#10712158, #10712160, #10712207, …
maximal quotient of	C₂×#68577, C₂×#68578, C₂×#68539, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11831);