C₈×C₄⋊C₄×C₃₇ #17990587

order	4736
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈×C₃₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₈×C₄⋊C₄×C₃₇

Subgroups

	subgroup	G/H
center	C₂²×C₈×C₃₇	C₂²
commutator	C₂	C₂×C₄×C₈×C₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
socle, prime radical	C₂³×C₃₇	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₃₇ ◁ G

296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 4736

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5,6,7,8)(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);(10,12)(13,14,15,16);(9,10,11,12)

Conjugacy classes & character table

2960 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#7167, C₃₇×#7169, C₈×#102×C₃₇, …
maximal normal subgroup of	C₃₇×#7167, C₃₇×#7169, C₈×#102×C₃₇, …
maximal quotient of	C₈×C₄⋊C₈×C₃₇, C₃₇×#8261, C₃₇×#8264, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 501);

C8×C4⋊C4×C37 #17990587