C₂×#7626839 #72324751

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7626839

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#3484	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 6270 conjugacy classes of subgroups; only the 56037 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72232070
maximal normal subgroup of	C₂×#72232070
maximal quotient of	C₂×#72232145

GAP

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

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

C2×#7626839 #72324751