C₂²×C₂²⋊Q₈×C₃₇ #17992251

order	4736
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃₇ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×C₂²⋊Q₈×C₃₇

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₃₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₇
prime radical	C₂⁵×C₃₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₇ ◁ G

1840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 5 generators on 53 points:

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

Conjugacy classes & character table

2072 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#8308, C₃₇×#8329, C₃₇×#8550, …
maximal normal subgroup of	C₃₇×#8308, C₃₇×#8329, C₃₇×#8550, …
maximal quotient of	C₂²×C₃₇×#380, C₂²×C₃₇×#385, C₂²×C₃₇×#387, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2165);

C22×C22⋊Q8×C37 #17992251