C₂×#6342378 #72339346

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#6342378

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

79627 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

(4,6)(8,10)(13,17)(15,18)(19,20,22,24)(21,25,26,23);(3,4,5,6)(8,10)(13,17)(15,18)(19,21,22,26)(20,23,24,25);(7,8)(9,10)(11,12,14,16)(13,15,17,18);(1,2);(3,4)(5,6)(7,8)(9,10);(11,13)(12,15)(14,17)(16,18)(19,20)(21,25)(22,24)(23,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72237466, C₂×#72237451
maximal normal subgroup of	C₂×#72237466, C₂×#72237451
maximal quotient of	C₂×#72339322

GAP

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

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

C2×#6342378 #72339346