C₂×#6342473 #72239953

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#6342473

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×#72210002
maximal subgroup of	#72239957, C₂×#72210002
maximal normal subgroup of	#72239957, C₂×#72210002

GAP

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

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

C2×#6342473 #72239953