C₂×#5047 #82734

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#5047

Splittings

Fitting	C₂×C₃×C₂²⋊Q₈ ⋊ C₂
Sylow	C₃ ⋊ C₂×#452

Subgroups

	subgroup	G/H
center	C₂³	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₂²⋊Q₈	C₂
socle	C₂³×C₃	C₂×D₄
prime radical	C₂²×#235	C₂
hypercenter	C₂×C₂²⋊Q₈	D₃
hypocenter	C₃	C₂×#452
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

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

normal subgroups only

This group has 760 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)(7,9)(10,13,15,12)(11,17,19,25)(14,20,24,22)(16,23,21,18);(1,2)(6,7)(8,9)(10,12,15,13)(11,16,19,21)(14,20,24,22)(17,23,25,18);(4,5)(10,11)(12,16)(13,21)(14,23)(15,19)(17,22)(18,24)(20,25);(1,2)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10691491, #10691497, C₄×#5047, …
maximal normal subgroup of	#10691491, #10691497, C₄×#5047, …
maximal quotient of	C₂×#69705, C₂×#69717, C₂×#69700, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 15310);