C₂×#149234 #72322836

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#149234

Subgroups

	subgroup	G/H
center, socle	C₂³	#1864
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂⁵	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3149 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72249887, C₂×#72249861
maximal normal subgroup of	C₂×#72249887, C₂×#72249861

GAP

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

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

C2×#149234 #72322836