C₂×#72214029 #72225148

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₈

Aliases

direct product

C₂×#72214029

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₄³
commutator	C₄²	C₂×C₄³
Frattini, cosocle	C₂×C₈²	C₂⁴
socle	C₂³	#7846
prime radical	C₂⁴	#1455
LCS	G ▷ C₄² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

2428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

320 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#72226962, C₂×#72214055
maximal normal subgroup of	#72226962, C₂×#72214055

GAP

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

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

C2×#72214029 #72225148