C₂×#7672774 #72330516

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#7672774

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

130649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72267577, #72330942, C₂×#72327577
maximal normal subgroup of	C₂×#72267577, #72330942, C₂×#72327577
maximal quotient of	C₂×#72240157, C₂×#72323284, C₂×#72263534, …

GAP

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

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

C2×#7672774 #72330516