C₂×#5403 #85445

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

Aliases

direct product

C₂×#5403

Subgroups

	subgroup	G/H
center, socle	C₂²	C₄⋊S₄
commutator, cosocle	#41⋊C₃	C₂³
Frattini	C₄	C₂²×S₄
Fitting	C₂×#131	D₃
prime radical	C₂×#41⋊C₃	C₂²
hypercenter	C₂×C₈	S₄
hypocenter	SL(2,𝔽₃)	C₂×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

243 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 66 points:

(3,18,52,28,26,48,6,43,25,5,41,66,9,47,51,17,7,46,16,53,27,20,22,65)(4,29,60,40,36,58,11,38,35,10,49,64,14,57,59,15,12,56,21,63,37,31,32,62)(8,19,42,54,24,45,23,44)(13,30,50,61,34,55,33,39);(3,4,9,14)(5,15,20,40)(6,21,16,11)(7,13,26,34)(8,12,24,36)(10,28,31,17)(18,39,47,61)(19,38,45,63)(22,50,41,33)(23,49,42,32)(25,37,27,35)(29,54,57,44)(30,53,55,43)(46,64,48,62)(51,59,52,60)(56,65,58,66);(1,2)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11711571, #11711576, #11711605, …
maximal normal subgroup of	#11711571, #11711576, #11711605, …
maximal quotient of	C₂×#73015, C₂×#73013, C₂×#73011, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18021);