C₄×#41×C₇₁ #17985648

order	4544
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₄×#41×C₇₁

Subgroups

	subgroup	G/H
center	C₄²×C₇₁	C₂²
commutator	C₂	C₂³×C₄×C₇₁
Frattini, cosocle	C₂²	C₂⁴×C₇₁
socle	C₂²×C₇₁	C₂⁴
prime radical	C₂×#41×C₇₁	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₇₁ ◁ G

310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 2272

A minimal faithful permutation representation — 4 generators on 83 points:

(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(5,6)(7,10)(8,9)(11,12);(6,9)(10,12);(5,7,8,11)(6,10,9,12)

Conjugacy classes & character table

2840 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇₁×#1476, C₇₁×#1477, C₇₁×#1488, …
maximal normal subgroup of	C₇₁×#1476, C₇₁×#1477, C₇₁×#1488, …
maximal quotient of	C₄×#117×C₇₁, C₄²×D₄×C₇₁, C₄²×Q₈×C₇₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4544, 198);

C4×#41×C71 #17985648