Abstract group 9800.i: $D_{490}:C_{10}$

• Label: 9800.i
• Order: \( 2^{3} \cdot 5^{2} \cdot 7^{2} \)
• Exponent: \( 2^{2} \cdot 5 \cdot 7^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 5 \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \cdot 7 \)
• Perm deg.: $63$
• Trans deg.: $980$
• Rank: $2$

Group information

Description:	$D_{490}:C_{10}$
Order:	\(9800\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Exponent:	\(980\)\(\medspace = 2^{2} \cdot 5 \cdot 7^{2} \)
Automorphism group:	$C_2\times C_4\times C_{245}.C_{42}.C_2^3$, of order \(658560\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \cdot 7^{3} \)
Composition factors:	$C_2$ x 3, $C_5$ x 2, $C_7$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	4	5	7	10	14	20	35	49	70	98	245	490
Elements	1	493	490	24	6	2032	18	1960	144	42	432	126	1008	3024	9800
Conjugacy classes	1	3	1	14	3	42	9	4	72	21	216	63	504	1512	2465
Divisions	1	3	1	4	1	11	2	1	4	1	10	2	4	10	55
Autjugacy classes	1	3	1	3	1	7	2	1	3	1	6	2	3	6	40

Minimal presentations

Permutation degree:	$63$
Transitive degree:	$980$
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:	$\langle a, b, c \mid a^{2}=b^{490}=c^{10}=[a,c]=[b,c]=1, b^{a}=b^{489}c^{7} \rangle$
Permutation group:	Degree $63$ $\langle(2,4)(5,6,7,8,9)(10,11,12,14,13), (1,2)(3,4)(11,13)(12,14)(16,17)(18,19) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 390 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 2 & 0 \\ 0 & 246 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{491})$
Direct product:	$C_5$ $\, \times\, $ $(C_{245}:D_4)$
Semidirect product:	$D_{490}$ $\,\rtimes\,$ $C_{10}$	$C_{10}^2$ $\,\rtimes\,$ $D_{49}$	$(C_5\times D_{490})$ $\,\rtimes\,$ $C_2$	$(C_5\times C_{245})$ $\,\rtimes\,$ $D_4$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{70}$ . $D_{70}$	$C_{490}$ . $D_{10}$	$C_{10}$ . $D_{490}$	$(C_{10}\times C_{70})$ . $D_7$	all 20

Elements of the group are displayed as matrices in $\GL_{2}(\F_{491})$.

Homology

Abelianization:	$C_{2} \times C_{10} \simeq C_{2}^{2} \times C_{5}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 3540 subgroups in 126 conjugacy classes, 42 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $D_{490}$
Commutator:	$G' \simeq$ $C_{490}$	$G/G' \simeq$ $C_2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_{14}$	$G/\Phi \simeq$ $C_5\times D_{70}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{10}\times C_{490}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{490}:C_{10}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5\times C_{70}$	$G/\operatorname{soc} \simeq$ $D_{14}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_{49}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 9800: $D_{490}:C_{10}$

Order 4900: $C_{245}:C_{20}$, $C_{10}\times C_{490}$, $C_5\times D_{490}$

Order 2450: $C_5\times C_{490}$ x 2, $C_5\times D_{245}$

Order 1960: $C_{245}:D_4$, $C_{245}:D_4$

Order 1400: $D_{70}:C_{10}$

Order 1225: $C_5\times C_{245}$

Order 980: $C_2\times C_{490}$ x 4, $C_5\times D_{98}$, $C_{245}:C_4$, $C_{245}:C_4$, $D_{490}$

Order 700: $C_{10}\times C_{70}$, $C_5\times D_{70}$, $C_{35}:C_{20}$

Order 490: $C_{490}$ x 10, $D_{245}$, $C_5\times D_{49}$

Order 392: $C_{49}:D_4$

Order 350: $C_5\times C_{70}$ x 2, $C_5\times D_{35}$

Order 280: $C_{35}:D_4$, $C_{35}:D_4$

Order 245: $C_{245}$ x 4

Order 200: $C_{10}\wr C_2$

Order 196: $C_2\times C_{98}$, $D_{98}$, $C_{49}:C_4$

Order 175: $C_5\times C_{35}$

Order 140: $C_2\times C_{70}$ x 4, $C_5\times D_{14}$, $C_{35}:C_4$, $C_7:C_{20}$, $D_{70}$

Order 100: $C_5:C_{20}$, $C_{10}^2$, $C_5\times D_{10}$

Order 98: $C_{98}$ x 2, $D_{49}$

Order 70: $C_{70}$ x 10, $D_{35}$, $C_5\times D_7$

Order 56: $C_7:D_4$

Order 50: $C_5\times C_{10}$ x 2, $C_5\times D_5$

Order 49: $C_{49}$

Order 40: $C_5:D_4$, $C_5\times D_4$

Order 35: $C_{35}$ x 4

Order 28: $C_2\times C_{14}$, $D_{14}$, $C_7:C_4$

Order 25: $C_5^2$

Order 20: $C_2\times C_{10}$ x 5, $D_{10}$, $C_{20}$, $C_5:C_4$

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

Order 10: $C_{10}$ x 11, $D_5$

Order 8: $D_4$

Order 7: $C_7$

Order 5: $C_5$ x 4

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 9800: $D_{490}:C_{10}$

Order 4900: $C_{245}:C_{20}$, $C_{10}\times C_{490}$, $C_5\times D_{490}$

Order 2450: $C_5\times C_{490}$ x 2, $C_5\times D_{245}$

Order 1960: $C_{245}:D_4$, $C_{245}:D_4$

Order 1400: $D_{70}:C_{10}$

Order 1225: $C_5\times C_{245}$

Order 980: $C_2\times C_{490}$ x 3, $C_5\times D_{98}$, $C_{245}:C_4$, $C_{245}:C_4$, $D_{490}$

Order 700: $C_{10}\times C_{70}$, $C_5\times D_{70}$, $C_{35}:C_{20}$

Order 490: $C_{490}$ x 6, $D_{245}$, $C_5\times D_{49}$

Order 392: $C_{49}:D_4$

Order 350: $C_5\times C_{70}$ x 2, $C_5\times D_{35}$

Order 280: $C_{35}:D_4$, $C_{35}:D_4$

Order 245: $C_{245}$ x 3

Order 200: $C_{10}\wr C_2$

Order 196: $C_2\times C_{98}$, $D_{98}$, $C_{49}:C_4$

Order 175: $C_5\times C_{35}$

Order 140: $C_2\times C_{70}$ x 3, $C_5\times D_{14}$, $C_{35}:C_4$, $C_7:C_{20}$, $D_{70}$

Order 100: $C_5:C_{20}$, $C_{10}^2$, $C_5\times D_{10}$

Order 98: $C_{98}$ x 2, $D_{49}$

Order 70: $C_{70}$ x 6, $D_{35}$, $C_5\times D_7$

Order 56: $C_7:D_4$

Order 50: $C_5\times C_{10}$ x 2, $C_5\times D_5$

Order 49: $C_{49}$

Order 40: $C_5:D_4$, $C_5\times D_4$

Order 35: $C_{35}$ x 3

Order 28: $C_2\times C_{14}$, $D_{14}$, $C_7:C_4$

Order 25: $C_5^2$

Order 20: $C_2\times C_{10}$ x 4, $D_{10}$, $C_{20}$, $C_5:C_4$

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

Order 10: $C_{10}$ x 7, $D_5$

Order 8: $D_4$

Order 7: $C_7$

Order 5: $C_5$ x 3

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 9800: $D_{490}:C_{10}$ ($C_1$)

Order 4900: $C_{245}:C_{20}$ ($C_2$), $C_{10}\times C_{490}$ ($C_2$), $C_5\times D_{490}$ ($C_2$)

Order 2450: $C_5\times C_{490}$ ($C_2^2$)

Order 1960: $C_{245}:D_4$ ($C_5$)

Order 1225: $C_5\times C_{245}$ ($D_4$)

Order 980: $C_{245}:C_4$ ($C_{10}$), $C_2\times C_{490}$ ($C_{10}$), $C_2\times C_{490}$ ($D_5$), $D_{490}$ ($C_{10}$)

Order 700: $C_{10}\times C_{70}$ ($D_7$)

Order 490: $C_{490}$ ($C_2\times C_{10}$), $C_{490}$ ($D_{10}$)

Order 350: $C_5\times C_{70}$ ($D_{14}$)

Order 245: $C_{245}$ ($C_5:D_4$), $C_{245}$ ($C_5\times D_4$)

Order 196: $C_2\times C_{98}$ ($C_5\times D_5$)

Order 175: $C_5\times C_{35}$ ($C_7:D_4$)

Order 140: $C_2\times C_{70}$ ($D_{35}$), $C_2\times C_{70}$ ($C_5\times D_7$)

Order 100: $C_{10}^2$ ($D_{49}$)

Order 98: $C_{98}$ ($C_5\times D_{10}$)

Order 70: $C_{70}$ ($C_5\times D_{14}$), $C_{70}$ ($D_{70}$)

Order 50: $C_5\times C_{10}$ ($D_{98}$)

Order 49: $C_{49}$ ($C_{10}\wr C_2$)

Order 35: $C_{35}$ ($C_{35}:D_4$), $C_{35}$ ($C_{35}:D_4$)

Order 28: $C_2\times C_{14}$ ($C_5\times D_{35}$)

Order 25: $C_5^2$ ($C_{49}:D_4$)

Order 20: $C_2\times C_{10}$ ($D_{245}$), $C_2\times C_{10}$ ($C_5\times D_{49}$)

Order 14: $C_{14}$ ($C_5\times D_{70}$)

Order 10: $C_{10}$ ($C_5\times D_{98}$), $C_{10}$ ($D_{490}$)

Order 7: $C_7$ ($D_{70}:C_{10}$)

Order 5: $C_5$ ($C_{245}:D_4$), $C_5$ ($C_{245}:D_4$)

Order 4: $C_2^2$ ($C_5\times D_{245}$)

Order 2: $C_2$ ($C_5\times D_{490}$)

Order 1: $C_1$ ($D_{490}:C_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 9800: $D_{490}:C_{10}$ ($C_1$)

Order 4900: $C_{245}:C_{20}$ ($C_2$), $C_{10}\times C_{490}$ ($C_2$), $C_5\times D_{490}$ ($C_2$)

Order 2450: $C_5\times C_{490}$ ($C_2^2$)

Order 1960: $C_{245}:D_4$ ($C_5$)

Order 1225: $C_5\times C_{245}$ ($D_4$)

Order 980: $C_{245}:C_4$ ($C_{10}$), $C_2\times C_{490}$ ($C_{10}$), $C_2\times C_{490}$ ($D_5$), $D_{490}$ ($C_{10}$)

Order 700: $C_{10}\times C_{70}$ ($D_7$)

Order 490: $C_{490}$ ($C_2\times C_{10}$), $C_{490}$ ($D_{10}$)

Order 350: $C_5\times C_{70}$ ($D_{14}$)

Order 245: $C_{245}$ ($C_5:D_4$), $C_{245}$ ($C_5\times D_4$)

Order 196: $C_2\times C_{98}$ ($C_5\times D_5$)

Order 175: $C_5\times C_{35}$ ($C_7:D_4$)

Order 140: $C_2\times C_{70}$ ($D_{35}$), $C_2\times C_{70}$ ($C_5\times D_7$)

Order 100: $C_{10}^2$ ($D_{49}$)

Order 98: $C_{98}$ ($C_5\times D_{10}$)

Order 70: $C_{70}$ ($C_5\times D_{14}$), $C_{70}$ ($D_{70}$)

Order 50: $C_5\times C_{10}$ ($D_{98}$)

Order 49: $C_{49}$ ($C_{10}\wr C_2$)

Order 35: $C_{35}$ ($C_{35}:D_4$), $C_{35}$ ($C_{35}:D_4$)

Order 28: $C_2\times C_{14}$ ($C_5\times D_{35}$)

Order 25: $C_5^2$ ($C_{49}:D_4$)

Order 20: $C_2\times C_{10}$ ($D_{245}$), $C_2\times C_{10}$ ($C_5\times D_{49}$)

Order 14: $C_{14}$ ($C_5\times D_{70}$)

Order 10: $C_{10}$ ($C_5\times D_{98}$), $C_{10}$ ($D_{490}$)

Order 7: $C_7$ ($D_{70}:C_{10}$)

Order 5: $C_5$ ($C_{245}:D_4$), $C_5$ ($C_{245}:D_4$)

Order 4: $C_2^2$ ($C_5\times D_{245}$)

Order 2: $C_2$ ($C_5\times D_{490}$)

Order 1: $C_1$ ($D_{490}:C_{10}$)

Series

Derived series

$D_{490}:C_{10}$

$\rhd$

$C_{490}$

$\rhd$

$C_1$

Chief series

$D_{490}:C_{10}$

$\rhd$

$C_{10}\times C_{490}$

$\rhd$

$C_5\times C_{490}$

$\rhd$

$C_{490}$

$\rhd$

$C_{245}$

$\rhd$

$C_{49}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$D_{490}:C_{10}$

$\rhd$

$C_{490}$

$\rhd$

$C_{245}$

Upper central series

$C_1$

$\lhd$

$C_{10}$

$\lhd$

$C_2\times C_{10}$

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 $2465 \times 2465$ character table is not available for this group.

Rational character table

The $55 \times 55$ rational character table is not available for this group.