C₂×#72232282 #72338056

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

Aliases

direct product

C₂×#72232282

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 13900 conjugacy classes of subgroups; only the 36904 normal subgroups are shown.

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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72232371
maximal normal subgroup of	C₂×#72232371

GAP

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

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

C2×#72232282 #72338056