C₂×#636924 #72334464

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

Aliases

direct product

C₂×#636924

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

71781 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)(13,16)(14,15)(17,18)(19,20,23,24)(21,25,26,22);(7,8)(9,10)(12,15)(16,17)(20,24)(22,25);(4,5)(8,9)(11,12)(13,17)(14,15)(16,18)(19,21,23,26)(20,25,24,22);(1,2);(3,5,6,4)(7,9,10,8)(11,12)(13,17)(14,15)(16,18)(19,22)(20,21)(23,25)(24,26);(11,13)(12,16)(14,18)(15,17)(19,21)(20,25)(22,24)(23,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236761, C₂×#72237112
maximal normal subgroup of	C₂×#72236761, C₂×#72237112
maximal quotient of	C₂×#72334313

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

C2×#636924 #72334464