C₂²×#7919 #72308570

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

direct product

C₂²×#7919

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂⁴	C₂×#95
prime radical	C₂⁶	C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

13901 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#196123
maximal normal subgroup of	C₂²×#196123
maximal quotient of	#72309366

GAP

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

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

C22×#7919 #72308570