Abstract group 2664.b: $C_2\times F_{37}$

• Label: 2664.b
• Order: \( 2^{3} \cdot 3^{2} \cdot 37 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 37 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{2} \cdot 37 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $39$
• Trans deg.: $74$
• Rank: $2$

Group information

Description:	$C_2\times F_{37}$
Order:	\(2664\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 37 \)
Exponent:	\(1332\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \)
Automorphism group:	$C_2\times F_{37}$, of order \(2664\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 37 \)
Composition factors:	$C_2$ x 3, $C_3$ x 2, $C_{37}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.

Group statistics

Order	1	2	3	4	6	9	12	18	36	37	74
Elements	1	75	74	148	222	222	296	666	888	36	36	2664
Conjugacy classes	1	3	2	4	6	6	8	18	24	1	1	74
Divisions	1	3	1	2	3	1	2	3	2	1	1	20
Autjugacy classes	1	3	2	2	6	6	4	18	12	1	1	56

Dimension	1	2	4	6	12	36
Irr. complex chars.	72	0	0	0	0	2	74
Irr. rational chars.	4	6	2	4	2	2	20

Minimal presentations

Permutation degree:	$39$
Transitive degree:	$74$
Rank:	$2$
Inequivalent generating pairs:	$864$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b \mid a^{36}=b^{74}=1, b^{a}=b^{5} \rangle$
Permutation group:	Degree $39$ $\langle(1,5,9,13,17,2,6,10,14,18,21,24,27,30,33,37,3,7,11,15,19,22,25,28,31,34,39,4,8,12,16,20,23,26,29,32,35) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 32 & 0 \\ 0 & 28 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 27 & 0 \\ 0 & 11 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{37})$
Direct product:	$C_2$ $\, \times\, $ $F_{37}$
Semidirect product:	$D_{37}$ $\,\rtimes\,$ $C_{36}$	$C_{74}$ $\,\rtimes\,$ $C_{36}$	$(C_{74}:C_4)$ $\,\rtimes\,$ $C_9$	$(C_{37}:C_{18})$ $\,\rtimes\,$ $C_4$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{74}$ . $C_{18}$	$(C_{74}:C_6)$ . $C_6$	$D_{37}$ . $(C_2\times C_{18})$	$(C_{74}:C_{18})$ . $C_2$	all 11
Aut. group:	$\Aut(C_{37}:C_4)$	$\Aut(D_{74})$	$\Aut(C_3\times D_{37})$	$\Aut(C_{37}:C_8)$	all 14

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

Homology

Abelianization:	$C_{2} \times C_{36} \simeq C_{2} \times C_{4} \times C_{9}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 840 subgroups in 48 conjugacy classes, 26 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $F_{37}$
Commutator:	$G' \simeq$ $C_{37}$	$G/G' \simeq$ $C_2\times C_{36}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2\times F_{37}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{74}$	$G/\operatorname{Fit} \simeq$ $C_{36}$
Radical:	$R \simeq$ $C_2\times F_{37}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{74}$	$G/\operatorname{soc} \simeq$ $C_{36}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
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

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Classes of subgroups up to conjugation

Order 2664: $C_2\times F_{37}$

Order 1332: $F_{37}$ x 2, $C_{74}:C_{18}$

Order 888: $C_{74}:C_{12}$

Order 666: $C_{37}:C_{18}$ x 2, $C_{37}:C_{18}$

Order 444: $C_{37}:C_{12}$ x 2, $C_{74}:C_6$

Order 333: $C_{37}:C_9$

Order 296: $C_{74}:C_4$

Order 222: $C_{37}:C_6$ x 2, $C_{37}:C_6$

Order 148: $C_{37}:C_4$ x 2, $D_{74}$

Order 111: $C_{37}:C_3$

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

Order 72: $C_2\times C_{36}$

Order 37: $C_{37}$

Order 36: $C_{36}$ x 2, $C_2\times C_{18}$

Order 24: $C_2\times C_{12}$

Order 18: $C_{18}$ x 3

Order 12: $C_{12}$ x 2, $C_2\times C_6$

Order 9: $C_9$

Order 8: $C_2\times C_4$

Order 6: $C_6$ x 3

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2664: $C_2\times F_{37}$

Order 1332: $C_{74}:C_{18}$, $F_{37}$

Order 888: $C_{74}:C_{12}$

Order 666: $C_{37}:C_{18}$ x 2, $C_{37}:C_{18}$

Order 444: $C_{74}:C_6$, $C_{37}:C_{12}$

Order 333: $C_{37}:C_9$

Order 296: $C_{74}:C_4$

Order 222: $C_{37}:C_6$ x 2, $C_{37}:C_6$

Order 148: $D_{74}$, $C_{37}:C_4$

Order 111: $C_{37}:C_3$

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

Order 72: $C_2\times C_{36}$

Order 37: $C_{37}$

Order 36: $C_2\times C_{18}$, $C_{36}$

Order 24: $C_2\times C_{12}$

Order 18: $C_{18}$ x 3

Order 12: $C_2\times C_6$, $C_{12}$

Order 9: $C_9$

Order 8: $C_2\times C_4$

Order 6: $C_6$ x 3

Order 4: $C_2^2$, $C_4$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2664: $C_2\times F_{37}$ ($C_1$)

Order 1332: $F_{37}$ ($C_2$) x 2, $C_{74}:C_{18}$ ($C_2$)

Order 888: $C_{74}:C_{12}$ ($C_3$)

Order 666: $C_{37}:C_{18}$ ($C_4$), $C_{37}:C_{18}$ ($C_2^2$), $C_{37}:C_{18}$ ($C_4$)

Order 444: $C_{37}:C_{12}$ ($C_6$) x 2, $C_{74}:C_6$ ($C_6$)

Order 333: $C_{37}:C_9$ ($C_2\times C_4$)

Order 296: $C_{74}:C_4$ ($C_9$)

Order 222: $C_{37}:C_6$ ($C_{12}$), $C_{37}:C_6$ ($C_2\times C_6$), $C_{37}:C_6$ ($C_{12}$)

Order 148: $C_{37}:C_4$ ($C_{18}$) x 2, $D_{74}$ ($C_{18}$)

Order 111: $C_{37}:C_3$ ($C_2\times C_{12}$)

Order 74: $C_{74}$ ($C_{36}$), $D_{37}$ ($C_2\times C_{18}$), $D_{37}$ ($C_{36}$)

Order 37: $C_{37}$ ($C_2\times C_{36}$)

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

Order 1: $C_1$ ($C_2\times F_{37}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2664: $C_2\times F_{37}$ ($C_1$)

Order 1332: $C_{74}:C_{18}$ ($C_2$), $F_{37}$ ($C_2$)

Order 888: $C_{74}:C_{12}$ ($C_3$)

Order 666: $C_{37}:C_{18}$ ($C_4$), $C_{37}:C_{18}$ ($C_2^2$), $C_{37}:C_{18}$ ($C_4$)

Order 444: $C_{74}:C_6$ ($C_6$), $C_{37}:C_{12}$ ($C_6$)

Order 333: $C_{37}:C_9$ ($C_2\times C_4$)

Order 296: $C_{74}:C_4$ ($C_9$)

Order 222: $C_{37}:C_6$ ($C_{12}$), $C_{37}:C_6$ ($C_2\times C_6$), $C_{37}:C_6$ ($C_{12}$)

Order 148: $D_{74}$ ($C_{18}$), $C_{37}:C_4$ ($C_{18}$)

Order 111: $C_{37}:C_3$ ($C_2\times C_{12}$)

Order 74: $C_{74}$ ($C_{36}$), $D_{37}$ ($C_2\times C_{18}$), $D_{37}$ ($C_{36}$)

Order 37: $C_{37}$ ($C_2\times C_{36}$)

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

Order 1: $C_1$ ($C_2\times F_{37}$)

Series

Derived series

$C_2\times F_{37}$

$\rhd$

$C_{37}$

$\rhd$

$C_1$

Chief series

$C_2\times F_{37}$

$\rhd$

$C_{74}:C_{18}$

$\rhd$

$C_{37}:C_{18}$

$\rhd$

$C_{37}:C_6$

$\rhd$

$C_{74}$

$\rhd$

$C_{37}$

$\rhd$

$C_1$

Lower central series

$C_2\times F_{37}$

$\rhd$

$C_{37}$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 3 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 $74 \times 74$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.