C₂²×C₄⋊C₄×C₇₁ #17985644

order	4544
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇₁ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×C₄⋊C₄×C₇₁

Subgroups

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

418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 4544

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

(1,2)(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);(3,4);(6,8)(9,10,11,12);(5,6,7,8)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇₁×C₄⋊#95, C₇₁×#1436, C₇₁×#1439, …
maximal normal subgroup of	C₇₁×C₄⋊#95, C₇₁×#1436, C₇₁×#1439, …
maximal quotient of	C₂²×#95×C₇₁, C₂×C₄×C₄⋊C₄×C₇₁, C₂×#380×C₇₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4544, 194);

C22×C4⋊C4×C71 #17985644