C₂×#612720 #72334724

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#612720

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

66945 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72334367, C₂×#72334373, C₂×#72334380
maximal normal subgroup of	C₂×#72334367, C₂×#72334373, C₂×#72334380
maximal quotient of	C₂×#72334407

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

C2×#612720 #72334724