C₂×#72232152 #72325713

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

Aliases

direct product

C₂×#72232152

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 22470 conjugacy classes of subgroups; only the 35320 normal subgroups are shown.

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72232191
maximal normal subgroup of	C₂×#72232191

GAP

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

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

C2×#72232152 #72325713