C₂×#8422210 #72331693

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#8422210

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 6448 conjugacy classes of subgroups; only the 29597 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:

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

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72241227
maximal normal subgroup of	C₂×#72241227
maximal quotient of	C₂×#72327618

GAP

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

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

C2×#8422210 #72331693