C₃₇×D₄⋊C₈ #14556990

order	2368
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₃₇×D₄⋊C₈
unambiguous semidirect product	D₄⋊(C₈×C₃₇)

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	D₄
commutator	C₄	C₂×C₈×C₃₇
Frattini, cosocle	C₄²	C₂²×C₃₇
socle	C₂²×C₃₇	C₂²⋊C₄
prime radical	C₂×D₄×C₃₇	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₄²×C₃₇ ◁ G

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 2 generators on 53 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)(38,39)(40,42)(41,44)(43,45)(46,47,48,50,49,51,52,53);(39,42)(40,45)(43,44)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₃₇×D₄⋊C₈, C₃₇×#1476, C₃₇×#1477, …
maximal normal subgroup of	C₂×C₃₇×D₄⋊C₈, C₃₇×#1476, C₃₇×#1477, …
maximal quotient of	C₃₇×#1270, C₃₇×#1271, C₃₇×#1279, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 6);

C37×D4⋊C8 #14556990