C₂×#553714 #72323947

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#553714

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2846	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

131905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297914, C₂×#72232341, C₂×#72323253
maximal normal subgroup of	C₂×#72297914, C₂×#72232341, C₂×#72323253

GAP

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

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

C2×#553714 #72323947