C₂×#72233704 #72334673

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

Aliases

direct product

C₂×#72233704

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 9406 conjugacy classes of subgroups; only the 39728 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72233717, C₂×#72335129, C₂×#72237469
maximal normal subgroup of	C₂×#72233717, C₂×#72335129, C₂×#72237469

GAP

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

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

C2×#72233704 #72334673