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C₂²×#101 #2886

order	128
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	Aut(G) order 49152
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×#101

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²⋊C₄
commutator	C₂²	C₂³×C₄
Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂⁴	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	A₄×#101, #11711616, #11711617, …
maximal subgroup of	#10741, #10750, #10763, …
maximal normal subgroup of	#10741, #10750, #10763, …
maximal quotient of	C₂²×#324, C₂×#1461, C₂×#1463, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1618);