C₂×#72233692 #72334656

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#72233692

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 6464 conjugacy classes of subgroups; only the 42392 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

(3,4,5,6)(8,10)(11,12,15,16)(13,17,18,14)(19,20,22,24)(21,25,26,23);(3,4,5,6)(7,8,9,10)(11,13,15,18)(12,14,16,17)(19,21,22,26)(20,23,24,25);(12,16)(14,17)(19,20)(21,25)(22,24)(23,26);(3,4)(5,6)(11,14)(12,13)(15,17)(16,18)(19,20)(21,25)(22,24)(23,26);(7,8)(9,10);(1,2);(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,18)(16,17)(19,20)(21,25)(22,24)(23,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72233717, C₂×#72334382
maximal normal subgroup of	C₂×#72233717, C₂×#72334382

GAP

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

gap> G := Group((3,4,5,6)(8,10)(11,12,15,16)(13,17,18,14)(19,20,22,24)(21,25,26,23), (3,4,5,6)(7,8,9,10)(11,13,15,18)(12,14,16,17)(19,21,22,26)(20,23,24,25), (12,16)(14,17)(19,20)(21,25)(22,24)(23,26), (3,4)(5,6)(11,14)(12,13)(15,17)(16,18)(19,20)(21,25)(22,24)(23,26), (7,8)(9,10), (1,2), (3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,18)(16,17)(19,20)(21,25)(22,24)(23,26));

C2×#72233692 #72334656