C₂×#149198 #72322331

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

Aliases

direct product

C₂×#149198

Subgroups

	subgroup	G/H
center, socle	C₂³	#1853
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂⁵	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3213 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72224903, C₂×#72249887
maximal normal subgroup of	C₂×#72224903, C₂×#72249887

GAP

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

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

C2×#149198 #72322331