D₄×C₂²⋊C₄×C₃₇ #17991156

order	4736
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

D₄×C₂²⋊C₄×C₃₇

Subgroups

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

1252 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);(5,6,8,10)(7,9,11,12);(6,9)(10,12)

Conjugacy classes & character table

1850 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#9255, C₃₇×#9259, D₄×#99×C₃₇, …
maximal normal subgroup of	C₃₇×#9255, C₃₇×#9259, D₄×#99×C₃₇, …
maximal quotient of	C₂²⋊C₄²×C₃₇, C₂²⋊C₄×C₄⋊C₄×C₃₇, D₄×#95×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1070);

D4×C22⋊C4×C37 #17991156