D₄⋊#13095486 #27139691

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

unambiguous semidirect product

D₄⋊#13095486

Splittings

Sylow

C₉₇ ⋊ D₄⋊C₁₆

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₉₇
commutator	C₄×C₉₇	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₉₇
Fitting	C₈×D₄×C₉₇	C₂
socle	C₂²×C₉₇	C₂²⋊C₈
prime radical	C₂×D₄×C₉₇	C₈
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	C₈×D₄	D₉₇
hypocenter	C₉₇	D₄⋊C₁₆
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×D₄

110 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 121
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 2 generators on 121 points:

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

Polycyclic presentation:

2^7,97|a^2=d,c^2=e,d^2=f,f^2=g,b^a=bc,c^a=ce,h^a=h^96,c^b=ce

Conjugacy classes & character table

1976 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 2465);