C₂×#72214006 #72258740

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₈

Aliases

direct product

C₂×#72214006

Subgroups

	subgroup	G/H
center, socle	C₂³	#8813
commutator	C₈²	C₂³×C₄
Frattini, cosocle	C₂×C₈²	C₂⁴
prime radical	C₂⁴	#1705
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

2092 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 34 points:

(3,4,5,10,8,12,7,17)(6,14,13,18,16,9,15,11)(20,24,25,30)(21,27)(22,29)(26,33,32,31)(28,34);(4,9,12,14)(5,7)(6,15)(10,18,17,11)(13,16)(20,24,25,30)(21,27)(22,29)(26,33,32,31)(28,34);(1,2);(19,20,21,24,22,26,27,31,23,25,28,30,29,32,34,33)

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72214055
maximal normal subgroup of	C₂×#72214055

GAP

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

gap> G := Group((3,4,5,10,8,12,7,17)(6,14,13,18,16,9,15,11)(20,24,25,30)(21,27)(22,29)(26,33,32,31)(28,34), (4,9,12,14)(5,7)(6,15)(10,18,17,11)(13,16)(20,24,25,30)(21,27)(22,29)(26,33,32,31)(28,34), (1,2), (19,20,21,24,22,26,27,31,23,25,28,30,29,32,34,33));

C2×#72214006 #72258740