C₂×#120×C₂²⋊Q₈ #72341877

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×#120×C₂²⋊Q₈

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁴×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

719908 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#120×#525
maximal normal subgroup of	C₂×#120×#525

GAP

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

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

C2×#120×C22⋊Q8 #72341877