C₂×C₄×Q₈×C₃₇ #14557181

order	2368
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₄×Q₈×C₃₇

Subgroups

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

298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 4 generators on 51 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);(3,4,5,6);(7,8,9,10)(11,14,13,12);(7,11,9,13)(8,12,10,14)

Conjugacy classes & character table

1480 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#1461, C₃₇×#1467, C₂×C₃₇×Q₈⋊C₈, …
maximal normal subgroup of	C₃₇×#1461, C₃₇×#1467, C₂×C₃₇×Q₈⋊C₈, …
maximal quotient of	C₂×C₄×C₄⋊C₄×C₃₇, C₄²×Q₈×C₃₇, C₂×C₃₇×#387, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 197);

C2×C4×Q8×C37 #14557181