C₂×#72268551 #72296239

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

Aliases

direct product

C₂×#72268551

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 7442 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)(8,10)(12,15)(16,18)(20,24)(21,25);(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,26);(3,4,5,6)(7,8,9,10)(12,15)(13,17)(21,25)(22,26);(7,8)(9,10)(19,21)(20,22)(23,25)(24,26);(11,12)(13,16)(14,15)(17,18);(3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,17)(15,18)(19,22)(20,21)(23,26)(24,25);(1,2)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72268592
maximal normal subgroup of	C₂×#72268592
maximal quotient of	C₂×#72296157

GAP

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

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

C2×#72268551 #72296239