C₂×#181415 #72239973

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

direct product

C₂×#181415

Subgroups

	subgroup	G/H
center	C₂×C₄²	#95
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂³	C₂²×#95
prime radical	C₂⁶	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

10945 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#72210004
maximal subgroup of	#72239976, C₂×#72210004, #72240027
maximal normal subgroup of	#72239976, C₂×#72210004, #72240027

GAP

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

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

C2×#181415 #72239973