C₂×#129596 #72307272

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#129596

Subgroups

	subgroup	G/H
center	C₂×C₄	#1790
commutator	C₂×Q₈	C₂³×C₈
Frattini, cosocle	C₂×#131	C₂⁴
socle	C₂²	#8316
prime radical	C₂×#3483	C₄
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

4150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307300, #72307386, #72307431, …
maximal normal subgroup of	#72307300, #72307386, #72307431, …

GAP

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

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

C2×#129596 #72307272