D₇×C₂₅₇ #16813340

order	3598
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₂₅₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₇×C₂₅₇
unambiguous semidirect product	C₇⋊(C₂×C₂₅₇)

Splittings

Fitting, socle, Hall	C₇×C₂₅₇ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂×C₂₅₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂₅₇	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₂₅₇
Frattini	C₁	G
Fitting, socle	C₇×C₂₅₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂₅₇

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 264
transitive degree	μ_t(G) = 1799

A minimal faithful permutation representation — 2 generators on 264 points:

(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264);(2,7)(3,6)(4,5)

Conjugacy classes & character table

1285 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×D₇×C₂₅₇, D₃₅×C₂₅₇, D₄₉×C₂₅₇, …
maximal normal subgroup of	C₅×D₇×C₂₅₇, C₇×D₇×C₂₅₇
maximal quotient of	C₅×D₇×C₂₅₇, D₃₅×C₂₅₇, D₄₉×C₂₅₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3598, 1);

D7×C257 #16813340