C₂×#72278726 #72297078

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂×#72278726

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 16712 conjugacy classes of subgroups; only the 7584 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 30 points:

(11,12)(13,14)(16,19)(20,22)(24,25)(26,30);(8,10)(12,14)(16,19)(17,21)(24,25)(28,29);(4,6)(7,8)(9,10)(16,19)(20,22)(23,24,27,25)(26,29,30,28);(4,6)(8,10)(11,12,13,14)(15,16,18,19)(17,22,21,20)(23,25,27,24)(26,28,30,29);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17)(16,20)(18,21)(19,22)(23,26)(24,28)(25,29)(27,30);(1,2)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72278822
maximal normal subgroup of	C₂×#72278822

GAP

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

gap> G := Group((11,12)(13,14)(16,19)(20,22)(24,25)(26,30), (8,10)(12,14)(16,19)(17,21)(24,25)(28,29), (4,6)(7,8)(9,10)(16,19)(20,22)(23,24,27,25)(26,29,30,28), (4,6)(8,10)(11,12,13,14)(15,16,18,19)(17,22,21,20)(23,25,27,24)(26,28,30,29), (3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17)(16,20)(18,21)(19,22)(23,26)(24,28)(25,29)(27,30), (1,2));

C2×#72278726 #72297078