C₂×#6342519 #72238848

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹²×C₈

Aliases

direct product

C₂×#6342519

Subgroups

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

16033 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 34 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×#72209989
maximal subgroup of	#72238850, C₂×#72209989
maximal normal subgroup of	#72238850, C₂×#72209989

GAP

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

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

C2×#6342519 #72238848