Abstract group 1588.4: $D_{794}$

• Label: 1588.4
• Order: \( 2^{2} \cdot 397 \)
• Exponent: \( 2 \cdot 397 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{2} \cdot 11 \cdot 397 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{2} \cdot 11 \)
• Perm deg.: $399$
• Trans deg.: $794$
• Rank: $2$

Group information

Description:	$D_{794}$
Order:	\(1588\)\(\medspace = 2^{2} \cdot 397 \)
Exponent:	\(794\)\(\medspace = 2 \cdot 397 \)
Automorphism group:	$C_2\times F_{397}$, of order \(314424\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \cdot 397 \)
Composition factors:	$C_2$ x 2, $C_{397}$
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	397	794
Elements	1	795	396	396	1588
Conjugacy classes	1	3	198	198	400
Divisions	1	3	1	1	6
Autjugacy classes	1	2	1	1	5

Dimension	1	2	396
Irr. complex chars.	4	396	0	400
Irr. rational chars.	4	0	2	6

Minimal presentations

Permutation degree:	$399$
Transitive degree:	$794$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	2	396
Arbitrary	2	2	396

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{794}=1, b^{a}=b^{793} \rangle$
Permutation group:	Degree $399$ $\langle(2,5)(3,6)(4,7)(8,20)(9,21)(10,22)(11,15)(12,18)(13,23)(14,24)(16,19)(17,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 396 & 0 \\ 0 & 396 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 396 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{397})$
Direct product:	$C_2$ $\, \times\, $ $D_{397}$
Semidirect product:	$C_{794}$ $\,\rtimes\,$ $C_2$	$C_{397}$ $\,\rtimes\,$ $C_2^2$			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 1198 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:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{397}$
Commutator:	$G' \simeq$ $C_{397}$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_{794}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{794}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{794}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{794}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
397-Sylow subgroup:	$P_{ 397 } \simeq$ $C_{397}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 1588: $D_{794}$

Order 794: $D_{397}$ x 2, $C_{794}$

Order 397: $C_{397}$

Order 4: $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1588: $D_{794}$

Order 794: $C_{794}$, $D_{397}$

Order 397: $C_{397}$

Order 4: $C_2^2$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1588: $D_{794}$ ($C_1$)

Order 794: $D_{397}$ ($C_2$) x 2, $C_{794}$ ($C_2$)

Order 397: $C_{397}$ ($C_2^2$)

Order 2: $C_2$ ($D_{397}$)

Order 1: $C_1$ ($D_{794}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1588: $D_{794}$ ($C_1$)

Order 794: $C_{794}$ ($C_2$), $D_{397}$ ($C_2$)

Order 397: $C_{397}$ ($C_2^2$)

Order 2: $C_2$ ($D_{397}$)

Order 1: $C_1$ ($D_{794}$)

Series

Derived series	$D_{794}$	$\rhd$	$C_{397}$	$\rhd$	$C_1$
Chief series	$D_{794}$	$\rhd$	$D_{397}$	$\rhd$	$C_{397}$	$\rhd$	$C_1$
Lower central series	$D_{794}$	$\rhd$	$C_{397}$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $400 \times 400$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	397A	794A
	Size	1	1	397	397	396	396
	2 P	1A	1A	1A	1A	397A	397A
	397 P	1A	2A	2B	2C	397A	794A
1588.4.1a		$1$	$1$	$1$	$1$	$1$	$1$
1588.4.1b		$1$	$-1$	$-1$	$1$	$1$	$-1$
1588.4.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$
1588.4.1d		$1$	$1$	$-1$	$-1$	$1$	$1$
1588.4.2a		$396$	$396$	$0$	$0$	$-1$	$-1$
1588.4.2b		$396$	$-396$	$0$	$0$	$-1$	$1$