C₂×#7716428 #72324792

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

Aliases

direct product

C₂×#7716428

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

63985 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)(13,15)(14,16)(17,18)(19,20)(21,24)(22,25)(23,26);(3,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,23)(22,26)(24,25);(4,5)(8,9)(12,13)(14,18)(19,20)(21,23)(22,25)(24,26);(6,10)(8,9)(14,18)(16,17)(19,21)(20,24)(22,26)(23,25);(1,2);(3,5,7,4)(6,9,10,8)(11,13,15,12)(14,16,18,17)(19,20)(21,23)(22,25)(24,26);(3,6)(4,8)(5,9)(7,10)(11,14,15,18)(12,16,13,17)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232068
maximal normal subgroup of	C₂×#72232068
maximal quotient of	C₂×#72232144

GAP

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

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

C2×#7716428 #72324792