Abstract group 4620.a: $C_{11}\times D_{210}$

• Label: 4620.a
• Order: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 11 \)
• $\card{Z(G)}$: \( 2 \cdot 11 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \cdot 5 \)
• Perm deg.: $28$
• Trans deg.: $2310$
• Rank: $2$

Group information

Description:	$C_{11}\times D_{210}$
Order:	\(4620\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Exponent:	\(2310\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Automorphism group:	$D_5:F_5^2$, of order \(100800\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Composition factors:	$C_2$ x 2, $C_3$, $C_5$, $C_7$, $C_{11}$
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	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	211	2	4	2	6	4	10	6	8	12	2110	8	20	24	12	40	20	24	60	48	40	60	80	48	120	80	240	120	240	480	480	4620
Conjugacy classes	1	3	1	2	1	3	2	10	3	4	6	30	4	10	12	6	20	10	12	30	24	20	30	40	24	60	40	120	60	120	240	240	1188
Divisions	1	3	1	1	1	1	1	1	1	1	1	3	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	36
Autjugacy classes	1	2	1	1	1	1	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	34

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$2310$
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:	$\langle a, b \mid a^{2}=b^{2310}=1, b^{a}=b^{419} \rangle$
Permutation group:	Degree $28$ $\langle(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,16,15)(17,18,19,21,23,22,20)(24,25,26)(27,28), (13,15)(14,16)(18,20)(19,22)(21,23)(25,26)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 43 & 135 \\ 277 & 43 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 418 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{419})$
Direct product:	$C_2$ $\, \times\, $ $C_{11}$ $\, \times\, $ $D_{105}$
Semidirect product:	$C_{770}$ $\,\rtimes\,$ $S_3$	$C_{462}$ $\,\rtimes\,$ $D_5$	$C_{385}$ $\,\rtimes\,$ $D_6$	$C_{330}$ $\,\rtimes\,$ $D_7$	all 28
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:	$C_{2} \times C_{22} \simeq C_{2}^{2} \times C_{11}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 1184 subgroups in 80 conjugacy classes, 38 normal (34 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{22}$	$G/Z \simeq$ $D_{105}$
Commutator:	$G' \simeq$ $C_{105}$	$G/G' \simeq$ $C_2\times C_{22}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{11}\times D_{210}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{2310}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{11}\times D_{210}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{2310}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 4620: $C_{11}\times D_{210}$

Order 2310: $C_{11}\times D_{105}$ x 2, $C_{2310}$

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

Order 1155: $C_{1155}$

Order 924: $C_{11}\times D_{42}$

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

Order 660: $C_{11}\times D_{30}$

Order 462: $C_{11}\times D_{21}$ x 2, $C_{462}$

Order 420: $D_{210}$

Order 385: $C_{385}$

Order 330: $C_{11}\times D_{15}$ x 2, $C_{330}$

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

Order 231: $C_{231}$

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

Order 210: $D_{105}$ x 2, $C_{210}$

Order 165: $C_{165}$

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

Order 140: $D_{70}$

Order 132: $S_3\times C_{22}$

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

Order 105: $C_{105}$

Order 84: $D_{42}$

Order 77: $C_{77}$

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

Order 66: $S_3\times C_{11}$ x 2, $C_{66}$

Order 60: $D_{30}$

Order 55: $C_{55}$

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

Order 42: $D_{21}$ x 2, $C_{42}$

Order 35: $C_{35}$

Order 33: $C_{33}$

Order 30: $D_{15}$ x 2, $C_{30}$

Order 28: $D_{14}$

Order 22: $C_{22}$ x 3

Order 21: $C_{21}$

Order 20: $D_{10}$

Order 15: $C_{15}$

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

Order 12: $D_6$

Order 11: $C_{11}$

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

Order 7: $C_7$

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4620: $C_{11}\times D_{210}$

Order 2310: $C_{11}\times D_{105}$, $C_{2310}$

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

Order 1155: $C_{1155}$

Order 924: $C_{11}\times D_{42}$

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

Order 660: $C_{11}\times D_{30}$

Order 462: $C_{462}$, $C_{11}\times D_{21}$

Order 420: $D_{210}$

Order 385: $C_{385}$

Order 330: $C_{330}$, $C_{11}\times D_{15}$

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

Order 231: $C_{231}$

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

Order 210: $C_{210}$, $D_{105}$

Order 165: $C_{165}$

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

Order 140: $D_{70}$

Order 132: $S_3\times C_{22}$

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

Order 105: $C_{105}$

Order 84: $D_{42}$

Order 77: $C_{77}$

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

Order 66: $C_{66}$, $S_3\times C_{11}$

Order 60: $D_{30}$

Order 55: $C_{55}$

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

Order 42: $C_{42}$, $D_{21}$

Order 35: $C_{35}$

Order 33: $C_{33}$

Order 30: $C_{30}$, $D_{15}$

Order 28: $D_{14}$

Order 22: $C_{22}$ x 2

Order 21: $C_{21}$

Order 20: $D_{10}$

Order 15: $C_{15}$

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

Order 12: $D_6$

Order 11: $C_{11}$

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

Order 7: $C_7$

Order 6: $C_6$, $S_3$

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4620: $C_{11}\times D_{210}$ ($C_1$)

Order 2310: $C_{11}\times D_{105}$ ($C_2$) x 2, $C_{2310}$ ($C_2$)

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

Order 770: $C_{770}$ ($S_3$)

Order 462: $C_{462}$ ($D_5$)

Order 420: $D_{210}$ ($C_{11}$)

Order 385: $C_{385}$ ($D_6$)

Order 330: $C_{330}$ ($D_7$)

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

Order 210: $D_{105}$ ($C_{22}$) x 2, $C_{210}$ ($C_{22}$)

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

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

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

Order 105: $C_{105}$ ($C_2\times C_{22}$)

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

Order 70: $C_{70}$ ($S_3\times C_{11}$)

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

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

Order 42: $C_{42}$ ($D_5\times C_{11}$)

Order 35: $C_{35}$ ($S_3\times C_{22}$)

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

Order 30: $C_{30}$ ($C_{11}\times D_7$)

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

Order 21: $C_{21}$ ($C_{11}\times D_{10}$)

Order 15: $C_{15}$ ($C_{11}\times D_{14}$)

Order 14: $C_{14}$ ($C_{11}\times D_{15}$)

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

Order 10: $C_{10}$ ($C_{11}\times D_{21}$)

Order 7: $C_7$ ($C_{11}\times D_{30}$)

Order 6: $C_6$ ($C_{11}\times D_{35}$)

Order 5: $C_5$ ($C_{11}\times D_{42}$)

Order 3: $C_3$ ($C_{11}\times D_{70}$)

Order 2: $C_2$ ($C_{11}\times D_{105}$)

Order 1: $C_1$ ($C_{11}\times D_{210}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4620: $C_{11}\times D_{210}$ ($C_1$)

Order 2310: $C_{11}\times D_{105}$ ($C_2$), $C_{2310}$ ($C_2$)

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

Order 770: $C_{770}$ ($S_3$)

Order 462: $C_{462}$ ($D_5$)

Order 420: $D_{210}$ ($C_{11}$)

Order 385: $C_{385}$ ($D_6$)

Order 330: $C_{330}$ ($D_7$)

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

Order 210: $C_{210}$ ($C_{22}$), $D_{105}$ ($C_{22}$)

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

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

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

Order 105: $C_{105}$ ($C_2\times C_{22}$)

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

Order 70: $C_{70}$ ($S_3\times C_{11}$)

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

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

Order 42: $C_{42}$ ($D_5\times C_{11}$)

Order 35: $C_{35}$ ($S_3\times C_{22}$)

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

Order 30: $C_{30}$ ($C_{11}\times D_7$)

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

Order 21: $C_{21}$ ($C_{11}\times D_{10}$)

Order 15: $C_{15}$ ($C_{11}\times D_{14}$)

Order 14: $C_{14}$ ($C_{11}\times D_{15}$)

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

Order 10: $C_{10}$ ($C_{11}\times D_{21}$)

Order 7: $C_7$ ($C_{11}\times D_{30}$)

Order 6: $C_6$ ($C_{11}\times D_{35}$)

Order 5: $C_5$ ($C_{11}\times D_{42}$)

Order 3: $C_3$ ($C_{11}\times D_{70}$)

Order 2: $C_2$ ($C_{11}\times D_{105}$)

Order 1: $C_1$ ($C_{11}\times D_{210}$)

Series

Derived series

$C_{11}\times D_{210}$

$\rhd$

$C_{105}$

$\rhd$

$C_1$

Chief series

$C_{11}\times D_{210}$

$\rhd$

$C_{11}\times D_{105}$

$\rhd$

$C_{1155}$

$\rhd$

$C_{385}$

$\rhd$

$C_{77}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{11}\times D_{210}$

$\rhd$

$C_{105}$

Upper central series

$C_1$

$\lhd$

$C_{22}$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $1188 \times 1188$ character table is not available for this group.

Rational character table

The $36 \times 36$ rational character table is not available for this group.