C₂²×#12778 #72337479

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂²×#12778

Subgroups

	subgroup	G/H
center, socle	C₂³	#1790
commutator	C₂×Q₈	C₂⁴×C₄
Frattini, cosocle	C₂×#41	C₂⁵
prime radical	C₂²×#3549	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

24845 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72337481, C₂×#72236900, #72337529, …
maximal normal subgroup of	#72337481, C₂×#72236900, #72337529, …
maximal quotient of	#72338792, #72340182

GAP

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

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

C22×#12778 #72337479