C₂×#72234244 #72342817

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹×C₄

Aliases

direct product

C₂×#72234244

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

2790692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72256567
maximal normal subgroup of	C₂×#72256567

GAP

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

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

C2×#72234244 #72342817