C₂×#483023 #72292773

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#483023

Subgroups

	subgroup	G/H
center, socle	C₂²	#17675
commutator	C₄²	C₂⁴×C₄
Frattini, cosocle	#97	C₂⁵
prime radical	C₂×#60704	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#2873 ◁ G
LECS	G ▷ #97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

4665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72292716, C₂×#72292717
maximal normal subgroup of	C₂×#72292716, C₂×#72292717

GAP

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

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

C2×#483023 #72292773