Abstract group 2310.1: $C_{2310}$

• Label: 2310.1
• Order: \( 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 5 \)
• Perm deg.: $28$
• Trans deg.: $2310$
• Rank: $1$

Group information

Description:	$C_{2310}$
Order:	\(2310\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Exponent:	\(2310\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Automorphism group:	$C_2^3\times C_{60}$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$, $C_3$, $C_5$, $C_7$, $C_{11}$
Nilpotency class:	$1$
Derived length:	$1$

This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,5,7,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Group statistics

Order	1	2	3	5	6	7	10	11	14	15	21	22	30	33	35	42	55	66	70	77	105	110	154	165	210	231	330	385	462	770	1155	2310
Elements	1	1	2	4	2	6	4	10	6	8	12	10	8	20	24	12	40	20	24	60	48	40	60	80	48	120	80	240	120	240	480	480	2310
Conjugacy classes	1	1	2	4	2	6	4	10	6	8	12	10	8	20	24	12	40	20	24	60	48	40	60	80	48	120	80	240	120	240	480	480	2310
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	32
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	32

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$2310$
Rank:	$1$
Inequivalent generators:	$1$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	1	not computed	not computed
Arbitrary	1	not computed	not computed

Constructions

Presentation:	$\langle a \mid a^{2310}=1 \rangle$
Permutation group:	Degree $28$ $\langle(1,2)(3,4,5)(6,10,9,8,7)(11,14,17,13,16,12,15)(18,28,27,26,25,24,23,22,21,20,19) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 170 & 329 \\ 374 & 170 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{419})$
Direct product:	$C_2$ $\, \times\, $ $C_3$ $\, \times\, $ $C_5$ $\, \times\, $ $C_7$ $\, \times\, $ $C_{11}$
Semidirect product:	not isomorphic to a non-trivial semidirect product
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

Primary decomposition:	$C_{2} \times C_{3} \times C_{5} \times C_{7} \times C_{11}$
Schur multiplier:	$C_1$
Commutator length:	$0$

Subgroups

There are 32 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{2310}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{2310}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{2310}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{2310}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{2310}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{2310}$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

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 2310: $C_{2310}$

Order 1155: $C_{1155}$

Order 770: $C_{770}$

Order 462: $C_{462}$

Order 385: $C_{385}$

Order 330: $C_{330}$

Order 231: $C_{231}$

Order 210: $C_{210}$

Order 165: $C_{165}$

Order 154: $C_{154}$

Order 110: $C_{110}$

Order 105: $C_{105}$

Order 77: $C_{77}$

Order 70: $C_{70}$

Order 66: $C_{66}$

Order 55: $C_{55}$

Order 42: $C_{42}$

Order 35: $C_{35}$

Order 33: $C_{33}$

Order 30: $C_{30}$

Order 22: $C_{22}$

Order 21: $C_{21}$

Order 15: $C_{15}$

Order 14: $C_{14}$

Order 11: $C_{11}$

Order 10: $C_{10}$

Order 7: $C_7$

Order 6: $C_6$

Order 5: $C_5$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2310: $C_{2310}$

Order 1155: $C_{1155}$

Order 770: $C_{770}$

Order 462: $C_{462}$

Order 385: $C_{385}$

Order 330: $C_{330}$

Order 231: $C_{231}$

Order 210: $C_{210}$

Order 165: $C_{165}$

Order 154: $C_{154}$

Order 110: $C_{110}$

Order 105: $C_{105}$

Order 77: $C_{77}$

Order 70: $C_{70}$

Order 66: $C_{66}$

Order 55: $C_{55}$

Order 42: $C_{42}$

Order 35: $C_{35}$

Order 33: $C_{33}$

Order 30: $C_{30}$

Order 22: $C_{22}$

Order 21: $C_{21}$

Order 15: $C_{15}$

Order 14: $C_{14}$

Order 11: $C_{11}$

Order 10: $C_{10}$

Order 7: $C_7$

Order 6: $C_6$

Order 5: $C_5$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2310: $C_{2310}$ ($C_1$)

Order 1155: $C_{1155}$ ($C_2$)

Order 770: $C_{770}$ ($C_3$)

Order 462: $C_{462}$ ($C_5$)

Order 385: $C_{385}$ ($C_6$)

Order 330: $C_{330}$ ($C_7$)

Order 231: $C_{231}$ ($C_{10}$)

Order 210: $C_{210}$ ($C_{11}$)

Order 165: $C_{165}$ ($C_{14}$)

Order 154: $C_{154}$ ($C_{15}$)

Order 110: $C_{110}$ ($C_{21}$)

Order 105: $C_{105}$ ($C_{22}$)

Order 77: $C_{77}$ ($C_{30}$)

Order 70: $C_{70}$ ($C_{33}$)

Order 66: $C_{66}$ ($C_{35}$)

Order 55: $C_{55}$ ($C_{42}$)

Order 42: $C_{42}$ ($C_{55}$)

Order 35: $C_{35}$ ($C_{66}$)

Order 33: $C_{33}$ ($C_{70}$)

Order 30: $C_{30}$ ($C_{77}$)

Order 22: $C_{22}$ ($C_{105}$)

Order 21: $C_{21}$ ($C_{110}$)

Order 15: $C_{15}$ ($C_{154}$)

Order 14: $C_{14}$ ($C_{165}$)

Order 11: $C_{11}$ ($C_{210}$)

Order 10: $C_{10}$ ($C_{231}$)

Order 7: $C_7$ ($C_{330}$)

Order 6: $C_6$ ($C_{385}$)

Order 5: $C_5$ ($C_{462}$)

Order 3: $C_3$ ($C_{770}$)

Order 2: $C_2$ ($C_{1155}$)

Order 1: $C_1$ ($C_{2310}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2310: $C_{2310}$ ($C_1$)

Order 1155: $C_{1155}$ ($C_2$)

Order 770: $C_{770}$ ($C_3$)

Order 462: $C_{462}$ ($C_5$)

Order 385: $C_{385}$ ($C_6$)

Order 330: $C_{330}$ ($C_7$)

Order 231: $C_{231}$ ($C_{10}$)

Order 210: $C_{210}$ ($C_{11}$)

Order 165: $C_{165}$ ($C_{14}$)

Order 154: $C_{154}$ ($C_{15}$)

Order 110: $C_{110}$ ($C_{21}$)

Order 105: $C_{105}$ ($C_{22}$)

Order 77: $C_{77}$ ($C_{30}$)

Order 70: $C_{70}$ ($C_{33}$)

Order 66: $C_{66}$ ($C_{35}$)

Order 55: $C_{55}$ ($C_{42}$)

Order 42: $C_{42}$ ($C_{55}$)

Order 35: $C_{35}$ ($C_{66}$)

Order 33: $C_{33}$ ($C_{70}$)

Order 30: $C_{30}$ ($C_{77}$)

Order 22: $C_{22}$ ($C_{105}$)

Order 21: $C_{21}$ ($C_{110}$)

Order 15: $C_{15}$ ($C_{154}$)

Order 14: $C_{14}$ ($C_{165}$)

Order 11: $C_{11}$ ($C_{210}$)

Order 10: $C_{10}$ ($C_{231}$)

Order 7: $C_7$ ($C_{330}$)

Order 6: $C_6$ ($C_{385}$)

Order 5: $C_5$ ($C_{462}$)

Order 3: $C_3$ ($C_{770}$)

Order 2: $C_2$ ($C_{1155}$)

Order 1: $C_1$ ($C_{2310}$)

Series

Derived series

$C_{2310}$

$\rhd$

$C_1$

Chief series

$C_{2310}$

$\rhd$

$C_{1155}$

$\rhd$

$C_{385}$

$\rhd$

$C_{77}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{2310}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{2310}$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

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

Character theory

Complex character table

The $2310 \times 2310$ character table is not available for this group.

Rational character table

The $32 \times 32$ rational character table is not available for this group.