C₂²×#11404 #72287087

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₈ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂²×#11404

Subgroups

	subgroup	G/H
center	C₂²×C₄²	C₄²
commutator	C₄	C₂³×C₄×C₈
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂⁴	C₂×#97
prime radical	C₂⁵	#97
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4381 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

352 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#483961, C₂²×#483590, C₂²×#479977
maximal normal subgroup of	C₂²×#483961, C₂²×#483590, C₂²×#479977
maximal quotient of	C₂²×#128259

GAP

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

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

C22×#11404 #72287087