C₂×C₄×Q₈×C₇₁ #17985647

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

Aliases

direct product

C₂×C₄×Q₈×C₇₁

Subgroups

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

298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 4544

A minimal faithful permutation representation — 4 generators on 85 points:

(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(3,4,5,6);(7,8,9,10)(11,14,13,12);(7,11,9,13)(8,12,10,14)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇₁×#1461, C₇₁×#1467, C₂×Q₈⋊C₈×C₇₁, …
maximal normal subgroup of	C₇₁×#1461, C₇₁×#1467, C₂×Q₈⋊C₈×C₇₁, …
maximal quotient of	C₂×C₄×C₄⋊C₄×C₇₁, C₄²×Q₈×C₇₁, C₂×#387×C₇₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4544, 197);

C2×C4×Q8×C71 #17985647