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C₂×C₂²⋊#95 #13751

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂¹⁰

Aliases

C₂×C₂²⋊#95

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂³
commutator	C₂²	C₂²×C₄²
Frattini, cosocle	C₂⁴	C₂⁴
prime radical	C₂⁶	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3815 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 18 points:

(1,2);(4,6)(7,8,9,10)(12,14);(11,12)(13,14)(15,16,17,18);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72238069, #72267761, #72267779
Frattini subgroup of	#72238066, #72238067, #72238068, …
maximal subgroup of	C₂×#7032, #95112, #95118, …
maximal normal subgroup of	C₂×#7032, #95112, #95118, …
maximal quotient of	C₂×#7554, C₄×C₂²⋊#95, C₂×#7555, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6739);