C₂×#72240153 #72334441

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²³

Aliases

direct product

C₂×#72240153

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁵×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4681832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72334301, #72337394
maximal normal subgroup of	C₂×#72334301, #72337394
maximal quotient of	C₂×#72334421, #72337381, #72337394, …

GAP

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

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

C2×#72240153 #72334441