C₂×C₁₀₃⋊#564 #27209287

order	13184
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product	C₂×C₁₀₃⋊#564
unambiguous semidirect product	(C₂×C₁₀₃)⋊#564

Splittings

Sylow

C₁₀₃ ⋊ C₂×#564

Subgroups

	subgroup	G/H
center	C₂³	C₂³×D₁₀₃
commutator, cosocle	C₂²×C₁₀₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₃
Fitting	C₂×#125×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂⁴
hypercenter	C₂×#125	D₁₀₃
hypocenter	C₁₀₃	C₂×#564
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

896 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 469 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 137
transitive degree	μ_t(G) = 13184

A minimal faithful permutation representation — 5 generators on 137 points:

(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)(107,113)(111,120)(112,116)(119,121)(122,125,126,135)(123,128,129,133)(124,132,131,134)(127,136,130,137);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(106,109,110,118)(107,112,113,116)(108,114,115,117)(111,119,120,121)(124,131)(127,130)(132,134)(136,137);(106,108,110,115)(107,111,113,120)(109,117,118,114)(112,121,116,119)(122,124)(123,127)(125,134)(126,131)(128,137)(129,130)(132,135)(133,136);(106,107)(108,111)(109,116)(110,113)(112,118)(114,121)(115,120)(117,119)(122,123,126,129)(124,130,131,127)(125,133,135,128)(132,136,134,137);(104,105)

Conjugacy classes & character table

1466 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 18653);