C₂×#7752091 #72324746

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7752091

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

At least 10824 conjugacy classes of subgroups; only the 31925 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232070, C₂×#72232068
maximal normal subgroup of	C₂×#72232070, C₂×#72232068
maximal quotient of	C₂×#72232143, C₂×#72232118, C₂×#72232148

GAP

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

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

C2×#7752091 #72324746