C₂×#72237100 #72289194

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

Aliases

direct product

C₂×#72237100

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 6376 conjugacy classes of subgroups; only the 41416 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72268588, C₂×#72268592, C₂×#72237114
maximal normal subgroup of	C₂×#72268588, C₂×#72268592, C₂×#72237114
maximal quotient of	C₂×#72287912

GAP

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

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

C2×#72237100 #72289194