C₂×#152324 #72250671

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#152324

Subgroups

	subgroup	G/H
center, socle	C₂²	#12852
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂×#124	C₂⁴
prime radical	C₂²×#535	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂²×#535 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216948, C₂×#72216942, C₂×#72215737
maximal normal subgroup of	C₂×#72216948, C₂×#72216942, C₂×#72215737

GAP

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

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

C2×#152324 #72250671