C₂×#5731 #86208

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#5731

Splittings

Fitting	C₂×C₃×C₄⋊Q₈ ⋊ C₂
Sylow	C₃ ⋊ C₂×#541

Subgroups

	subgroup	G/H
center	C₂³	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂×C₃×C₄⋊Q₈	C₂
socle	C₂³×C₃	C₂⁴
prime radical	C₂³×D₃	C₂³
hypercenter	C₂×C₄⋊Q₈	D₃
hypocenter	C₃	C₂×#541
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

1004 conjugacy classes of subgroups; only the 485 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 485 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 — 5 generators on 29 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10687728, #10800341, #10800466, …
maximal normal subgroup of	#10687728, #10800341, #10800466, …
maximal quotient of	C₂×#74358, C₂×#76483, C₂×#76611, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18784);