C₂×#7659684 #72324733

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂×#7659684

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 6780 conjugacy classes of subgroups; only the 32077 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232070
maximal normal subgroup of	C₂×#72232070
maximal quotient of	C₂×#72232168

GAP

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

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

C2×#7659684 #72324733