C₂×#72213768 #72233669

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂×#72213768

Subgroups

	subgroup	G/H
center, socle	C₂³	#14694
commutator	C₂³×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁵
prime radical	C₂×#619101	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

At least 8121 conjugacy classes of subgroups; only the 2520 normal subgroups are shown.

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:

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

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72215640
maximal normal subgroup of	C₂×#72215640

GAP

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

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

C2×#72213768 #72233669