D₁₅₁₉₇ #44477255

order	30394
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₁₉₇
unambiguous semidirect product	C₇⋊D₂₁₇₁
unambiguous semidirect product	C₁₃⋊D₁₁₆₉
unambiguous semidirect product	C₁₆₇⋊D₉₁

Splittings

Fitting, socle, derived, Hall	C₇×C₁₃×C₁₆₇ ⋊ C₂
Hall	C₁₃×C₁₆₇ ⋊ D₇
Hall	C₇×C₁₆₇ ⋊ D₁₃
Sylow	C₁₆₇ ⋊ D₉₁
Hall	C₇×C₁₃ ⋊ D₁₆₇
Sylow	C₁₃ ⋊ D₁₁₆₉
Sylow	C₇ ⋊ D₂₁₇₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃×C₁₆₇	C₂
LCS	G ▷ C₇×C₁₃×C₁₆₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 187
transitive degree	μ_t(G) = 15197

A minimal faithful permutation representation — 2 generators on 187 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);(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(22,187)(23,186)(24,185)(25,184)(26,183)(27,182)(28,181)(29,180)(30,179)(31,178)(32,177)(33,176)(34,175)(35,174)(36,173)(37,172)(38,171)(39,170)(40,169)(41,168)(42,167)(43,166)(44,165)(45,164)(46,163)(47,162)(48,161)(49,160)(50,159)(51,158)(52,157)(53,156)(54,155)(55,154)(56,153)(57,152)(58,151)(59,150)(60,149)(61,148)(62,147)(63,146)(64,145)(65,144)(66,143)(67,142)(68,141)(69,140)(70,139)(71,138)(72,137)(73,136)(74,135)(75,134)(76,133)(77,132)(78,131)(79,130)(80,129)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,120)(90,119)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)

Polycyclic presentation:

2,7,13,167|b^a=b^6,c^a=c^12,d^a=d^166

Conjugacy classes & character table

7600 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₁₉₇
maximal normal subgroup of	C₂×D₁₅₁₉₇
maximal quotient of	Dic₁₅₁₉₇, C₂×D₁₅₁₉₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30394, 8);

D15197 #44477255