C₂×#72232108 #72325697

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂×#72232108

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 21172 conjugacy classes of subgroups; only the 36168 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:

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

Conjugacy classes & character table

320 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((8,10)(11,12)(13,17)(14,18)(15,16)(19,20)(21,24)(22,23)(25,26), (4,6)(8,10)(11,12)(13,17)(14,18)(15,16)(20,23)(24,25), (12,16)(14,17), (7,8)(9,10)(19,21)(20,24)(22,26)(23,25), (1,2), (4,6)(11,13)(12,14)(15,18)(16,17)(19,21)(20,24)(22,26)(23,25), (3,4)(5,6)(11,14)(12,13)(15,17)(16,18)(19,20)(21,25)(22,23)(24,26));

C2×#72232108 #72325697