Abstract group 148.4: $D_{74}$

• Label: 148.4
• Order: \( 2^{2} \cdot 37 \)
• Exponent: \( 2 \cdot 37 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{2} \cdot 37 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{2} \)
• Perm deg.: $39$
• Trans deg.: $74$
• Rank: $2$

Group information

Description:	$D_{74}$
Order:	\(148\)\(\medspace = 2^{2} \cdot 37 \)
Exponent:	\(74\)\(\medspace = 2 \cdot 37 \)
Automorphism group:	$C_2\times F_{37}$, of order \(2664\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 37 \)
Composition factors:	$C_2$ x 2, $C_{37}$
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	37	74
Elements	1	75	36	36	148
Conjugacy classes	1	3	18	18	40
Divisions	1	3	1	1	6
Autjugacy classes	1	2	1	1	5

Dimension	1	2	36
Irr. complex chars.	4	36	0	40
Irr. rational chars.	4	0	2	6

Minimal presentations

Permutation degree:	$39$
Transitive degree:	$74$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	2	36
Arbitrary	2	2	36

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{74}=1, b^{a}=b^{73} \rangle$
Permutation group:	Degree $39$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 36 \end{array}\right), \left(\begin{array}{rr} 36 & 0 \\ 0 & 36 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{37})$
Direct product:	$C_2$ $\, \times\, $ $D_{37}$
Semidirect product:	$C_{74}$ $\,\rtimes\,$ $C_2$	$C_{37}$ $\,\rtimes\,$ $C_2^2$			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 118 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:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{37}$
Commutator:	$G' \simeq$ $C_{37}$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_{74}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{74}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{74}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{74}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
37-Sylow subgroup:	$P_{ 37 } \simeq$ $C_{37}$

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 148: $D_{74}$

Order 74: $D_{37}$ x 2, $C_{74}$

Order 37: $C_{37}$

Order 4: $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 148: $D_{74}$

Order 74: $C_{74}$, $D_{37}$

Order 37: $C_{37}$

Order 4: $C_2^2$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 148: $D_{74}$ ($C_1$)

Order 74: $D_{37}$ ($C_2$) x 2, $C_{74}$ ($C_2$)

Order 37: $C_{37}$ ($C_2^2$)

Order 2: $C_2$ ($D_{37}$)

Order 1: $C_1$ ($D_{74}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 148: $D_{74}$ ($C_1$)

Order 74: $C_{74}$ ($C_2$), $D_{37}$ ($C_2$)

Order 37: $C_{37}$ ($C_2^2$)

Order 2: $C_2$ ($D_{37}$)

Order 1: $C_1$ ($D_{74}$)

Series

Derived series	$D_{74}$	$\rhd$	$C_{37}$	$\rhd$	$C_1$
Chief series	$D_{74}$	$\rhd$	$D_{37}$	$\rhd$	$C_{37}$	$\rhd$	$C_1$
Lower central series	$D_{74}$	$\rhd$	$C_{37}$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 25 larger groups in the database.

This group is a maximal quotient of 23 larger groups in the database.

Character theory

Complex character table

See the $40 \times 40$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	37A	74A
	Size	1	1	37	37	36	36
	2 P	1A	1A	1A	1A	37A	37A
	37 P	1A	2A	2B	2C	37A	74A
148.4.1a		$1$	$1$	$1$	$1$	$1$	$1$
148.4.1b		$1$	$-1$	$-1$	$1$	$1$	$-1$
148.4.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$
148.4.1d		$1$	$1$	$-1$	$-1$	$1$	$1$
148.4.2a		$36$	$36$	$0$	$0$	$-1$	$-1$
148.4.2b		$36$	$-36$	$0$	$0$	$-1$	$1$