Abstract group 172.3: $D_{86}$

• Label: 172.3
• Order: \( 2^{2} \cdot 43 \)
• Exponent: \( 2 \cdot 43 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3 \cdot 7 \cdot 43 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \cdot 7 \)
• Perm deg.: $45$
• Trans deg.: $86$
• Rank: $2$

Group information

Description:	$D_{86}$
Order:	\(172\)\(\medspace = 2^{2} \cdot 43 \)
Exponent:	\(86\)\(\medspace = 2 \cdot 43 \)
Automorphism group:	$C_2\times F_{43}$, of order \(3612\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 43 \)
Composition factors:	$C_2$ x 2, $C_{43}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.

Group statistics

Order	1	2	43	86
Elements	1	87	42	42	172
Conjugacy classes	1	3	21	21	46
Divisions	1	3	1	1	6
Autjugacy classes	1	2	1	1	5

Dimension	1	2	42
Irr. complex chars.	4	42	0	46
Irr. rational chars.	4	0	2	6

Minimal presentations

Permutation degree:	$45$
Transitive degree:	$86$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	2	42
Arbitrary	2	2	42

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{86}=1, b^{a}=b^{85} \rangle$
Permutation group:	Degree $45$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 42 & 0 \\ 0 & 42 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 42 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{43})$
Direct product:	$C_2$ $\, \times\, $ $D_{43}$
Semidirect product:	$C_{86}$ $\,\rtimes\,$ $C_2$	$C_{43}$ $\,\rtimes\,$ $C_2^2$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 136 subgroups in 10 conjugacy classes, 7 normal (5 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{43}$
Commutator:	$G' \simeq$ $C_{43}$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_{86}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{86}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{86}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{86}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
43-Sylow subgroup:	$P_{ 43 } \simeq$ $C_{43}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 172: $D_{86}$

Order 86: $D_{43}$ x 2, $C_{86}$

Order 43: $C_{43}$

Order 4: $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 172: $D_{86}$

Order 86: $C_{86}$, $D_{43}$

Order 43: $C_{43}$

Order 4: $C_2^2$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 172: $D_{86}$ ($C_1$)

Order 86: $D_{43}$ ($C_2$) x 2, $C_{86}$ ($C_2$)

Order 43: $C_{43}$ ($C_2^2$)

Order 2: $C_2$ ($D_{43}$)

Order 1: $C_1$ ($D_{86}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 172: $D_{86}$ ($C_1$)

Order 86: $C_{86}$ ($C_2$), $D_{43}$ ($C_2$)

Order 43: $C_{43}$ ($C_2^2$)

Order 2: $C_2$ ($D_{43}$)

Order 1: $C_1$ ($D_{86}$)

Series

Derived series	$D_{86}$	$\rhd$	$C_{43}$	$\rhd$	$C_1$
Chief series	$D_{86}$	$\rhd$	$D_{43}$	$\rhd$	$C_{43}$	$\rhd$	$C_1$
Lower central series	$D_{86}$	$\rhd$	$C_{43}$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 21 larger groups in the database.

This group is a maximal quotient of 20 larger groups in the database.

Character theory

Complex character table

See the $46 \times 46$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	43A	86A
	Size	1	1	43	43	42	42
	2 P	1A	1A	1A	1A	43A	43A
	43 P	1A	2A	2B	2C	43A	86A
172.3.1a		$1$	$1$	$1$	$1$	$1$	$1$
172.3.1b		$1$	$-1$	$-1$	$1$	$1$	$-1$
172.3.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$
172.3.1d		$1$	$1$	$-1$	$-1$	$1$	$1$
172.3.2a		$42$	$42$	$0$	$0$	$-1$	$-1$
172.3.2b		$42$	$-42$	$0$	$0$	$-1$	$1$