C₂×#72252082 #72295304

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂×#72252082

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 8132 conjugacy classes of subgroups; only the 12312 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72293994, C₂×#72278822
maximal normal subgroup of	C₂×#72293994, C₂×#72278822

GAP

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

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

C2×#72252082 #72295304