C₂×#153759 #72334587

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#153759

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁷	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

71077 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 22 points:

(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16,18,17)(19,20)(21,22);(4,6)(7,8,9,10)(12,14)(19,20,21,22);(4,6)(8,10)(11,12,13,14)(15,17,18,16)(19,20)(21,22);(1,2);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72332558, C₂×#72279059, C₂×#72317648, …
maximal normal subgroup of	C₂×#72332558, C₂×#72279059, C₂×#72317648, …

GAP

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

gap> G := Group((3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16,18,17)(19,20)(21,22), (4,6)(7,8,9,10)(12,14)(19,20,21,22), (4,6)(8,10)(11,12,13,14)(15,17,18,16)(19,20)(21,22), (1,2), (3,4)(5,6)(7,8)(9,10)(11,12)(13,14));

C2×#153759 #72334587