C₂×#359×C₁₃₇ #31777153

order	17536
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#359×C₁₃₇

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃₇	D₈
commutator	C₈	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₈	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#102
prime radical	C₂×#135×C₁₃₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ C₂²×C₈×C₁₃₇ ◁ G

224 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 171
transitive degree	μ_t(G) = 8768

A minimal faithful permutation representation — 3 generators on 171 points:

(1,2)(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);(3,4,7,12,9,13,8,23)(5,14,16,30,17,31,24,28)(6,18,20,29,22,33,21,11)(10,19,25,32,27,34,26,15);(3,5)(4,10)(6,19)(7,16)(8,24)(9,17)(11,28)(12,25)(13,27)(14,18)(15,21)(20,32)(22,34)(23,26)(29,30)(31,33)

Conjugacy classes & character table

6028 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄×#359×C₁₃₇, C₁₃₇×#12640, C₁₃₇×#12641, …
maximal normal subgroup of	C₄×#359×C₁₃₇, C₁₃₇×#12640, C₁₃₇×#12641, …
maximal quotient of	C₂×#1331×C₁₃₇, C₂×#1332×C₁₃₇, C₁₃₇×#11591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17536, 874);