C₂×#72213800 #72233725

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂×#72213800

Subgroups

	subgroup	G/H
center, socle	C₂³	#14715
commutator, Frattini, cosocle	C₂²×C₄²	C₂⁵
LCS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

At least 8716 conjugacy classes of subgroups; only the 1680 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);(4,9)(6,16)(10,20)(11,21)(12,22)(14,26)(17,29)(19,24)(23,32)(25,33)(28,34)(30,31);(5,13)(10,23)(14,26)(15,27)(19,30)(20,32)(24,31)(28,34);(4,10,12,24)(6,17)(9,19,22,20)(11,23,25,31)(14,28)(16,29)(21,30,33,32)(26,34);(3,4,6,9,8,11,16,21)(5,10,14,20,13,23,26,32)(7,12,17,22,18,25,29,33)(15,24,28,19,27,31,34,30)

Conjugacy classes & character table

176 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), (4,9)(6,16)(10,20)(11,21)(12,22)(14,26)(17,29)(19,24)(23,32)(25,33)(28,34)(30,31), (5,13)(10,23)(14,26)(15,27)(19,30)(20,32)(24,31)(28,34), (4,10,12,24)(6,17)(9,19,22,20)(11,23,25,31)(14,28)(16,29)(21,30,33,32)(26,34), (3,4,6,9,8,11,16,21)(5,10,14,20,13,23,26,32)(7,12,17,22,18,25,29,33)(15,24,28,19,27,31,34,30));

C2×#72213800 #72233725