C₂×#72293088 #72293100

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×#72293088

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁴×D₄²	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

922272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72293201, C₂×#72334357, #72337321
maximal normal subgroup of	#72293201, C₂×#72334357, #72337321
maximal quotient of	C₂×#72334421, #72337321

GAP

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

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

C2×#72293088 #72293100