D₄×D₈×C₃₇ #17992097

order	4736
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₄×D₈×C₃₇

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₄×D₄×C₃₇ ◁ G

620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 4 generators on 49 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);(2,4);(5,6,7,8,9,10,11,12);(6,12)(7,11)(8,10)

Conjugacy classes & character table

1295 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#12316, C₃₇×#12320, C₃₇×#12368, …
maximal normal subgroup of	C₃₇×#12316, C₃₇×#12320, C₃₇×#12368, …
maximal quotient of	C₃₇×#19755, C₃₇×#19756, C₃₇×#19757, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2011);

D4×D8×C37 #17992097