C₂²×#9428 #72306469

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂²×#9428

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#386
commutator	C₄²	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
prime radical	C₂²×#3151	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

13625 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#152191, C₂²×#246927
maximal normal subgroup of	C₂²×#152191, C₂²×#246927

GAP

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

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

C22×#9428 #72306469