C₂×#72214568 #72321158

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#72214568

Subgroups

	subgroup	G/H
center, socle	C₂²	#483875
commutator	C₂×C₁₆	C₂⁴×C₄
Frattini, cosocle	#346	C₂⁵
prime radical	C₂×#513022	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂×#11404 ◁ G
LECS	G ▷ #346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72225118
maximal normal subgroup of	C₂×#72225118

GAP

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

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

C2×#72214568 #72321158