C₂²×#9492 #72306476

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

Aliases

direct product

C₂²×#9492

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#386
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
prime radical	C₂²×#392	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

12945 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

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

GAP

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

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

C22×#9492 #72306476