Abstract group 1914.3: $C_3\times D_{319}$

• Label: 1914.3
• Order: \( 2 \cdot 3 \cdot 11 \cdot 29 \)
• Exponent: \( 2 \cdot 3 \cdot 11 \cdot 29 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 5 \cdot 7 \cdot 11 \cdot 29 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 5 \cdot 7 \)
• Perm deg.: $43$
• Trans deg.: $957$
• Rank: $2$

Group information

Description:	$C_3\times D_{319}$
Order:	\(1914\)\(\medspace = 2 \cdot 3 \cdot 11 \cdot 29 \)
Exponent:	\(1914\)\(\medspace = 2 \cdot 3 \cdot 11 \cdot 29 \)
Automorphism group:	$C_{319}.C_{70}.C_2^3$, of order \(178640\)\(\medspace = 2^{4} \cdot 5 \cdot 7 \cdot 11 \cdot 29 \)
Composition factors:	$C_2$, $C_3$, $C_{11}$, $C_{29}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	11	29	33	87	319	957
Elements	1	319	2	638	10	28	20	56	280	560	1914
Conjugacy classes	1	1	2	2	5	14	10	28	140	280	483
Divisions	1	1	1	1	1	1	1	1	1	1	10
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	10

Minimal presentations

Permutation degree:	$43$
Transitive degree:	$957$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{957}=1, b^{a}=b^{637} \rangle$
Permutation group:	Degree $43$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $D_{319}$
Semidirect product:	$C_{87}$ $\,\rtimes\,$ $D_{11}$	$C_{33}$ $\,\rtimes\,$ $D_{29}$	$C_{957}$ $\,\rtimes\,$ $C_2$	$C_{319}$ $\,\rtimes\,$ $C_6$	all 6
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:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 728 subgroups in 16 conjugacy classes, 10 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $D_{319}$
Commutator:	$G' \simeq$ $C_{319}$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3\times D_{319}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{957}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_3\times D_{319}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{957}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
29-Sylow subgroup:	$P_{ 29 } \simeq$ $C_{29}$

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 1914: $C_3\times D_{319}$

Order 957: $C_{957}$

Order 638: $D_{319}$

Order 319: $C_{319}$

Order 174: $C_3\times D_{29}$

Order 87: $C_{87}$

Order 66: $C_3\times D_{11}$

Order 58: $D_{29}$

Order 33: $C_{33}$

Order 29: $C_{29}$

Order 22: $D_{11}$

Order 11: $C_{11}$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1914: $C_3\times D_{319}$

Order 957: $C_{957}$

Order 638: $D_{319}$

Order 319: $C_{319}$

Order 174: $C_3\times D_{29}$

Order 87: $C_{87}$

Order 66: $C_3\times D_{11}$

Order 58: $D_{29}$

Order 33: $C_{33}$

Order 29: $C_{29}$

Order 22: $D_{11}$

Order 11: $C_{11}$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1914: $C_3\times D_{319}$ ($C_1$)

Order 957: $C_{957}$ ($C_2$)

Order 638: $D_{319}$ ($C_3$)

Order 319: $C_{319}$ ($C_6$)

Order 87: $C_{87}$ ($D_{11}$)

Order 33: $C_{33}$ ($D_{29}$)

Order 29: $C_{29}$ ($C_3\times D_{11}$)

Order 11: $C_{11}$ ($C_3\times D_{29}$)

Order 3: $C_3$ ($D_{319}$)

Order 1: $C_1$ ($C_3\times D_{319}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1914: $C_3\times D_{319}$ ($C_1$)

Order 957: $C_{957}$ ($C_2$)

Order 638: $D_{319}$ ($C_3$)

Order 319: $C_{319}$ ($C_6$)

Order 87: $C_{87}$ ($D_{11}$)

Order 33: $C_{33}$ ($D_{29}$)

Order 29: $C_{29}$ ($C_3\times D_{11}$)

Order 11: $C_{11}$ ($C_3\times D_{29}$)

Order 3: $C_3$ ($D_{319}$)

Order 1: $C_1$ ($C_3\times D_{319}$)

Series

Derived series	$C_3\times D_{319}$	$\rhd$	$C_{319}$	$\rhd$	$C_1$
Chief series	$C_3\times D_{319}$	$\rhd$	$C_{957}$	$\rhd$	$C_{319}$	$\rhd$	$C_{29}$	$\rhd$	$C_1$
Lower central series	$C_3\times D_{319}$	$\rhd$	$C_{319}$
Upper central series	$C_1$	$\lhd$	$C_3$

Character theory

Complex character table

The $483 \times 483$ character table is not available for this group.

Rational character table

The $10 \times 10$ rational character table is not available for this group.