C₄×#7511 #72234368

order	1024
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₆ · Schur multiplier (H₂) = C₄²

Aliases

direct product

C₄×#7511

Subgroups

	subgroup	G/H
center	C₄×C₁₆	C₄²
commutator	C₄	C₄²×C₁₆
Frattini, cosocle	C₂²×C₃₂	C₂³
socle	C₂²	C₂×#1397
prime radical	C₂³	C₂×M₆₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

359 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2,3,4);(5,6,8,15,10,17,27,41,11,18,28,39,33,45,56,62,12,16,29,40,34,43,57,63,24,44,25,52,51,65,53,67,13,19,30,42,35,46,58,64,22,14,36,37,50,38,60,61,9,31,26,54,32,59,55,68,7,20,21,47,23,48,49,66);(5,7,9,22,13,24,12,11)(6,14,16,20,19,18,31,44)(8,25,26,28,30,21,29,36)(10,23,32,50,35,51,34,33)(15,39,40,52,42,37,54,47)(17,38,43,48,46,45,59,65)(27,53,55,56,58,49,57,60)(41,62,63,67,64,61,68,66)

Conjugacy classes & character table

352 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#72234385, #72234399
maximal normal subgroup of	#72234385, #72234399
maximal quotient of	#72234371

GAP

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

gap> G := Group((1,2,3,4), (5,6,8,15,10,17,27,41,11,18,28,39,33,45,56,62,12,16,29,40,34,43,57,63,24,44,25,52,51,65,53,67,13,19,30,42,35,46,58,64,22,14,36,37,50,38,60,61,9,31,26,54,32,59,55,68,7,20,21,47,23,48,49,66), (5,7,9,22,13,24,12,11)(6,14,16,20,19,18,31,44)(8,25,26,28,30,21,29,36)(10,23,32,50,35,51,34,33)(15,39,40,52,42,37,54,47)(17,38,43,48,46,45,59,65)(27,53,55,56,58,49,57,60)(41,62,63,67,64,61,68,66));

C4×#7511 #72234368