C₂²×#10738 #72318770

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×#10738

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂⁴	C₂×#95
prime radical	C₂⁷	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁶×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

46581 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72279500, C₂×#72279624
maximal normal subgroup of	C₂×#72279500, C₂×#72279624

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

C22×#10738 #72318770