C₂²×#8522 #72306479

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#8522

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#379
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
prime radical	C₂³×#535	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

39493 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(13,14)(15,21)(16,20)(17,19)(18,23)(22,27)(24,26)(25,28);(5,6,7,10,8,9,11,12)(13,15)(14,18,19,21)(16,22)(17,23)(20,26,28,27)(24,25);(1,2);(3,4);(6,9)(7,11)(15,22)(16,25)(18,26)(20,28)(21,27)(23,24)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#246927, C₂²×#152018, C₂²×#152017
maximal normal subgroup of	C₂²×#246927, C₂²×#152018, C₂²×#152017

GAP

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

gap> G := Group((13,14)(15,21)(16,20)(17,19)(18,23)(22,27)(24,26)(25,28), (5,6,7,10,8,9,11,12)(13,15)(14,18,19,21)(16,22)(17,23)(20,26,28,27)(24,25), (1,2), (3,4), (6,9)(7,11)(15,22)(16,25)(18,26)(20,28)(21,27)(23,24));

C22×#8522 #72306479