C₂²×D₈×C₃₇ #14557234

order	2368
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×D₈×C₃₇

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₇	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

338 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)(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);(3,4);(5,6,7,8,9,10,11,12);(6,12)(7,11)(8,10)

Conjugacy classes & character table

1036 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#1879, C₃₇×#1970, C₃₇×#1972, …
maximal normal subgroup of	C₃₇×#1879, C₃₇×#1970, C₃₇×#1972, …
maximal quotient of	C₂²×#102×C₃₇, C₂²×#107×C₃₇, C₂×C₄×D₈×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 250);

C22×D8×C37 #14557234