C₂×#148152 #72322600

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#148152

Subgroups

	subgroup	G/H
center, socle	C₂³	#1864
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂⁴×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4977 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 38 points:

(4,6)(7,8,10,16,28,35,13,17,14,20,29,36,12,18,11,19)(9,21,22,15,24,31,23,32,26,38,30,34,25,37,27,33);(1,2);(3,4,5,6)(7,9)(8,15,20,34)(10,27)(11,22)(12,24)(13,30)(14,26)(16,21,36,38)(17,37,19,31)(18,32,35,33)(23,29)(25,28);(4,6)(8,16)(10,11)(12,28)(13,29)(15,21)(17,18)(19,35)(20,36)(22,27)(23,30)(24,25)(31,33)(32,37)(34,38)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72249881, C₂×#72249861
maximal normal subgroup of	C₂×#72249881, C₂×#72249861

GAP

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

gap> G := Group((4,6)(7,8,10,16,28,35,13,17,14,20,29,36,12,18,11,19)(9,21,22,15,24,31,23,32,26,38,30,34,25,37,27,33), (1,2), (3,4,5,6)(7,9)(8,15,20,34)(10,27)(11,22)(12,24)(13,30)(14,26)(16,21,36,38)(17,37,19,31)(18,32,35,33)(23,29)(25,28), (4,6)(8,16)(10,11)(12,28)(13,29)(15,21)(17,18)(19,35)(20,36)(22,27)(23,30)(24,25)(31,33)(32,37)(34,38));

C2×#148152 #72322600