C₂×#5028 #82440

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

Aliases

direct product

C₂×#5028

Splittings

Fitting	C₂×C₃×C₄×Q₈ ⋊ C₂
Sylow	C₃ ⋊ C₂×#464

Subgroups

	subgroup	G/H
center	C₂³	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄×Q₈	C₂
socle	C₂³×C₃	C₂×D₄
prime radical	C₂²×D₁₂	C₂²
hypercenter	C₂×C₄×Q₈	D₃
hypocenter	C₃	C₂×#464
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

500 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 165 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) = 25
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 25 points:

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10691527, #10691529, #10801415, …
maximal normal subgroup of	#10691527, #10691529, #10801415, …
maximal quotient of	C₂×#70940, C₂×#70946, C₂×#70941, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 15016);