C₇⋊D₇×D₃₁⋊C₅ #44477153

order	30380
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₇

Aliases

direct product	C₇⋊D₇×D₃₁⋊C₅
unambiguous semidirect product	D₃₁⋊(C₅×C₇⋊D₇)
unambiguous semidirect product	(C₇×D₃₁⋊C₅)⋊D₇
unambiguous semidirect product	(D₃₁×C₇⋊D₇)⋊C₅

Splittings

Fitting, socle, derived, Hall	C₇²×C₃₁ ⋊ C₂²×C₅
Hall	C₇²×ΓL(1,𝔽₃₂) ⋊ C₂²
Hall	D₃₁×C₇⋊D₇ ⋊ C₅
Hall	ΓL(1,𝔽₃₂) ⋊ C₂×C₇⋊D₇
Sylow	C₇² ⋊ C₂×D₃₁⋊C₅
Sylow	C₃₁ ⋊ C₂×C₅×C₇⋊D₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₃₁	C₂²×C₅
LCS	G ▷ C₇²×C₃₁
UCS	C₁

200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 1519

A minimal faithful permutation representation — 3 generators on 45 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30380, 62);

C7⋊D7×D31⋊C5 #44477153