Abstract group 3080.a: $C_{11}\times D_{140}$

• Label: 3080.a
• Order: \( 2^{3} \cdot 5 \cdot 7 \cdot 11 \)
• Exponent: \( 2^{2} \cdot 5 \cdot 7 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 11 \)
• $\card{Z(G)}$: \( 2 \cdot 11 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 5^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \cdot 5 \)
• Perm deg.: $27$
• Trans deg.: $1540$
• Rank: $2$

Group information

Description:	$C_{11}\times D_{140}$
Order:	\(3080\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \cdot 11 \)
Exponent:	\(1540\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \cdot 11 \)
Automorphism group:	$C_{70}.C_{30}.C_2^5$, of order \(67200\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_5$, $C_7$, $C_{11}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	7	10	11	14	20	22	28	35	44	55	70	77	110	140	154	220	308	385	770	1540
Elements	1	141	2	4	6	4	10	6	8	1410	12	24	20	40	24	60	40	48	60	80	120	240	240	480	3080
Conjugacy classes	1	3	1	2	3	2	10	3	4	30	6	12	10	20	12	30	20	24	30	40	60	120	120	240	803
Divisions	1	3	1	1	1	1	1	1	1	3	1	1	1	1	1	1	1	1	1	1	1	1	1	1	28
Autjugacy classes	1	2	1	1	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	26

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$1540$
Rank:	$2$
Inequivalent generating pairs:	$36$

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:	$\langle a, b \mid a^{2}=b^{1540}=1, b^{a}=b^{419} \rangle$
Permutation group:	Degree $27$ $\langle(13,14)(17,18)(19,20)(21,22)(24,25)(26,27), (1,2,3,4,5,6,7,8,9,10,11)(12,13,15,14)(16,17,19,21,22,20,18)(23,24,26,27,25)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 418 \end{array}\right), \left(\begin{array}{rr} 235 & 121 \\ 270 & 235 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{419})$
Direct product:	$C_{11}$ $\, \times\, $ $D_{140}$
Semidirect product:	$C_{385}$ $\,\rtimes\,$ $D_4$	$C_{308}$ $\,\rtimes\,$ $D_5$	$C_{220}$ $\,\rtimes\,$ $D_7$	$D_{70}$ $\,\rtimes\,$ $C_{22}$ (2)	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{22}$ . $D_{70}$	$C_{154}$ . $D_{10}$	$C_{110}$ . $D_{14}$	$C_{770}$ . $C_2^2$	all 8

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

Homology

Abelianization:	$C_{2} \times C_{22} \simeq C_{2}^{2} \times C_{11}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 696 subgroups in 64 conjugacy classes, 30 normal (26 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{22}$	$G/Z \simeq$ $D_{70}$
Commutator:	$G' \simeq$ $C_{70}$	$G/G' \simeq$ $C_2\times C_{22}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{11}\times D_{70}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{1540}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{11}\times D_{140}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{770}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
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

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.

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

Subgroup information

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

Classes of subgroups up to conjugation

Order 3080: $C_{11}\times D_{140}$

Order 1540: $C_{11}\times D_{70}$ x 2, $C_{1540}$

Order 770: $C_{11}\times D_{35}$ x 2, $C_{770}$

Order 616: $C_{11}\times D_{28}$

Order 440: $C_{11}\times D_{20}$

Order 385: $C_{385}$

Order 308: $C_{11}\times D_{14}$ x 2, $C_{308}$

Order 280: $D_{140}$

Order 220: $C_{11}\times D_{10}$ x 2, $C_{220}$

Order 154: $C_{11}\times D_7$ x 2, $C_{154}$

Order 140: $D_{70}$ x 2, $C_{140}$

Order 110: $D_5\times C_{11}$ x 2, $C_{110}$

Order 88: $D_4\times C_{11}$

Order 77: $C_{77}$

Order 70: $D_{35}$ x 2, $C_{70}$

Order 56: $D_{28}$

Order 55: $C_{55}$

Order 44: $C_2\times C_{22}$ x 2, $C_{44}$

Order 40: $D_{20}$

Order 35: $C_{35}$

Order 28: $D_{14}$ x 2, $C_{28}$

Order 22: $C_{22}$ x 3

Order 20: $D_{10}$ x 2, $C_{20}$

Order 14: $D_7$ x 2, $C_{14}$

Order 11: $C_{11}$

Order 10: $D_5$ x 2, $C_{10}$

Order 8: $D_4$

Order 7: $C_7$

Order 5: $C_5$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3080: $C_{11}\times D_{140}$

Order 1540: $C_{11}\times D_{70}$, $C_{1540}$

Order 770: $C_{770}$, $C_{11}\times D_{35}$

Order 616: $C_{11}\times D_{28}$

Order 440: $C_{11}\times D_{20}$

Order 385: $C_{385}$

Order 308: $C_{11}\times D_{14}$, $C_{308}$

Order 280: $D_{140}$

Order 220: $C_{220}$, $C_{11}\times D_{10}$

Order 154: $C_{154}$, $C_{11}\times D_7$

Order 140: $C_{140}$, $D_{70}$

Order 110: $C_{110}$, $D_5\times C_{11}$

Order 88: $D_4\times C_{11}$

Order 77: $C_{77}$

Order 70: $C_{70}$, $D_{35}$

Order 56: $D_{28}$

Order 55: $C_{55}$

Order 44: $C_2\times C_{22}$, $C_{44}$

Order 40: $D_{20}$

Order 35: $C_{35}$

Order 28: $D_{14}$, $C_{28}$

Order 22: $C_{22}$ x 2

Order 20: $D_{10}$, $C_{20}$

Order 14: $C_{14}$, $D_7$

Order 11: $C_{11}$

Order 10: $C_{10}$, $D_5$

Order 8: $D_4$

Order 7: $C_7$

Order 5: $C_5$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3080: $C_{11}\times D_{140}$ ($C_1$)

Order 1540: $C_{11}\times D_{70}$ ($C_2$) x 2, $C_{1540}$ ($C_2$)

Order 770: $C_{770}$ ($C_2^2$)

Order 385: $C_{385}$ ($D_4$)

Order 308: $C_{308}$ ($D_5$)

Order 280: $D_{140}$ ($C_{11}$)

Order 220: $C_{220}$ ($D_7$)

Order 154: $C_{154}$ ($D_{10}$)

Order 140: $D_{70}$ ($C_{22}$) x 2, $C_{140}$ ($C_{22}$)

Order 110: $C_{110}$ ($D_{14}$)

Order 77: $C_{77}$ ($D_{20}$)

Order 70: $C_{70}$ ($C_2\times C_{22}$)

Order 55: $C_{55}$ ($D_{28}$)

Order 44: $C_{44}$ ($D_{35}$)

Order 35: $C_{35}$ ($D_4\times C_{11}$)

Order 28: $C_{28}$ ($D_5\times C_{11}$)

Order 22: $C_{22}$ ($D_{70}$)

Order 20: $C_{20}$ ($C_{11}\times D_7$)

Order 14: $C_{14}$ ($C_{11}\times D_{10}$)

Order 11: $C_{11}$ ($D_{140}$)

Order 10: $C_{10}$ ($C_{11}\times D_{14}$)

Order 7: $C_7$ ($C_{11}\times D_{20}$)

Order 5: $C_5$ ($C_{11}\times D_{28}$)

Order 4: $C_4$ ($C_{11}\times D_{35}$)

Order 2: $C_2$ ($C_{11}\times D_{70}$)

Order 1: $C_1$ ($C_{11}\times D_{140}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3080: $C_{11}\times D_{140}$ ($C_1$)

Order 1540: $C_{11}\times D_{70}$ ($C_2$), $C_{1540}$ ($C_2$)

Order 770: $C_{770}$ ($C_2^2$)

Order 385: $C_{385}$ ($D_4$)

Order 308: $C_{308}$ ($D_5$)

Order 280: $D_{140}$ ($C_{11}$)

Order 220: $C_{220}$ ($D_7$)

Order 154: $C_{154}$ ($D_{10}$)

Order 140: $C_{140}$ ($C_{22}$), $D_{70}$ ($C_{22}$)

Order 110: $C_{110}$ ($D_{14}$)

Order 77: $C_{77}$ ($D_{20}$)

Order 70: $C_{70}$ ($C_2\times C_{22}$)

Order 55: $C_{55}$ ($D_{28}$)

Order 44: $C_{44}$ ($D_{35}$)

Order 35: $C_{35}$ ($D_4\times C_{11}$)

Order 28: $C_{28}$ ($D_5\times C_{11}$)

Order 22: $C_{22}$ ($D_{70}$)

Order 20: $C_{20}$ ($C_{11}\times D_7$)

Order 14: $C_{14}$ ($C_{11}\times D_{10}$)

Order 11: $C_{11}$ ($D_{140}$)

Order 10: $C_{10}$ ($C_{11}\times D_{14}$)

Order 7: $C_7$ ($C_{11}\times D_{20}$)

Order 5: $C_5$ ($C_{11}\times D_{28}$)

Order 4: $C_4$ ($C_{11}\times D_{35}$)

Order 2: $C_2$ ($C_{11}\times D_{70}$)

Order 1: $C_1$ ($C_{11}\times D_{140}$)

Series

Derived series

$C_{11}\times D_{140}$

$\rhd$

$C_{70}$

$\rhd$

$C_1$

Chief series

$C_{11}\times D_{140}$

$\rhd$

$C_{11}\times D_{70}$

$\rhd$

$C_{770}$

$\rhd$

$C_{385}$

$\rhd$

$C_{77}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{11}\times D_{140}$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

Upper central series

$C_1$

$\lhd$

$C_{22}$

$\lhd$

$C_{44}$

Supergroups

Character theory

Complex character table

The $803 \times 803$ character table is not available for this group.

Rational character table

The $28 \times 28$ rational character table is not available for this group.