Abstract group 14700.c: $C_{70}\times C_{210}$

• Label: 14700.c
• Order: \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 7 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{4} \cdot 5 \cdot 7 \)
• Perm deg.: $31$
• Trans deg.: $14700$
• Rank: $2$

Group information

Description:	$C_{70}\times C_{210}$
Order:	\(14700\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Exponent:	\(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_2\times S_3\times \GL(2,5)\times C_6.\SO(3,7)$, of order \(11612160\)\(\medspace = 2^{12} \cdot 3^{4} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 2, $C_3$, $C_5$ x 2, $C_7$ x 2
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and metacyclic.

Group statistics

Order	1	2	3	5	6	7	10	14	15	21	30	35	42	70	105	210
Elements	1	3	2	24	6	48	72	144	48	96	144	1152	288	3456	2304	6912	14700
Conjugacy classes	1	3	2	24	6	48	72	144	48	96	144	1152	288	3456	2304	6912	14700
Divisions	1	3	1	6	3	8	18	24	6	8	18	48	24	144	48	144	504
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	16

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$14700$
Rank:	$2$
Inequivalent generating pairs:	$4$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	2	not computed	not computed

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{70}=b^{210}=1 \rangle$
Permutation group:	Degree $31$ $\langle(3,4)(5,6,7)(13,14,15,16,17)(25,26,27,28,29,30,31), (1,2)(8,9,10,11,12)(18,19,20,21,22,23,24)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 2 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 132 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{211})$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $C_5$ ${}^2$ $\, \times\, $ $C_7$ ${}^2$
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:	$C_{2}^{2} \times C_{3} \times C_{5}^{2} \times C_{7}^{2}$
Schur multiplier:	$C_{70}$
Commutator length:	$0$

Subgroups

There are 800 subgroups, all normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{70}\times C_{210}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{70}\times C_{210}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{70}\times C_{210}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{70}\times C_{210}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{70}\times C_{210}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{70}\times C_{210}$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 14700: $C_{70}\times C_{210}$

Order 7350: $C_{35}\times C_{210}$ x 3

Order 4900: $C_{70}^2$

Order 3675: $C_{35}\times C_{105}$

Order 2940: $C_{14}\times C_{210}$ x 6

Order 2450: $C_{35}\times C_{70}$ x 3

Order 2100: $C_{10}\times C_{210}$ x 8

Order 1470: $C_7\times C_{210}$ x 18

Order 1225: $C_{35}^2$

Order 1050: $C_5\times C_{210}$ x 24

Order 980: $C_{14}\times C_{70}$ x 6

Order 735: $C_7\times C_{105}$ x 6

Order 700: $C_{10}\times C_{70}$ x 8

Order 588: $C_{14}\times C_{42}$

Order 525: $C_5\times C_{105}$ x 8

Order 490: $C_7\times C_{70}$ x 18

Order 420: $C_2\times C_{210}$ x 48

Order 350: $C_5\times C_{70}$ x 24

Order 300: $C_{10}\times C_{30}$

Order 294: $C_7\times C_{42}$ x 3

Order 245: $C_7\times C_{35}$ x 6

Order 210: $C_{210}$ x 144

Order 196: $C_{14}^2$

Order 175: $C_5\times C_{35}$ x 8

Order 150: $C_5\times C_{30}$ x 3

Order 147: $C_7\times C_{21}$

Order 140: $C_2\times C_{70}$ x 48

Order 105: $C_{105}$ x 48

Order 100: $C_{10}^2$

Order 98: $C_7\times C_{14}$ x 3

Order 84: $C_2\times C_{42}$ x 8

Order 75: $C_5\times C_{15}$

Order 70: $C_{70}$ x 144

Order 60: $C_2\times C_{30}$ x 6

Order 50: $C_5\times C_{10}$ x 3

Order 49: $C_7^2$

Order 42: $C_{42}$ x 24

Order 35: $C_{35}$ x 48

Order 30: $C_{30}$ x 18

Order 28: $C_2\times C_{14}$ x 8

Order 25: $C_5^2$

Order 21: $C_{21}$ x 8

Order 20: $C_2\times C_{10}$ x 6

Order 15: $C_{15}$ x 6

Order 14: $C_{14}$ x 24

Order 12: $C_2\times C_6$

Order 10: $C_{10}$ x 18

Order 7: $C_7$ x 8

Order 6: $C_6$ x 3

Order 5: $C_5$ x 6

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 14700: $C_{70}\times C_{210}$

Order 7350: $C_{35}\times C_{210}$

Order 4900: $C_{70}^2$

Order 3675: $C_{35}\times C_{105}$

Order 2940: $C_{14}\times C_{210}$

Order 2450: $C_{35}\times C_{70}$

Order 2100: $C_{10}\times C_{210}$

Order 1470: $C_7\times C_{210}$

Order 1225: $C_{35}^2$

Order 1050: $C_5\times C_{210}$

Order 980: $C_{14}\times C_{70}$

Order 735: $C_7\times C_{105}$

Order 700: $C_{10}\times C_{70}$

Order 588: $C_{14}\times C_{42}$

Order 525: $C_5\times C_{105}$

Order 490: $C_7\times C_{70}$

Order 420: $C_2\times C_{210}$

Order 350: $C_5\times C_{70}$

Order 300: $C_{10}\times C_{30}$

Order 294: $C_7\times C_{42}$

Order 245: $C_7\times C_{35}$

Order 210: $C_{210}$

Order 196: $C_{14}^2$

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

Order 150: $C_5\times C_{30}$

Order 147: $C_7\times C_{21}$

Order 140: $C_2\times C_{70}$

Order 105: $C_{105}$

Order 100: $C_{10}^2$

Order 98: $C_7\times C_{14}$

Order 84: $C_2\times C_{42}$

Order 75: $C_5\times C_{15}$

Order 70: $C_{70}$

Order 60: $C_2\times C_{30}$

Order 50: $C_5\times C_{10}$

Order 49: $C_7^2$

Order 42: $C_{42}$

Order 35: $C_{35}$

Order 30: $C_{30}$

Order 28: $C_2\times C_{14}$

Order 25: $C_5^2$

Order 21: $C_{21}$

Order 20: $C_2\times C_{10}$

Order 15: $C_{15}$

Order 14: $C_{14}$

Order 12: $C_2\times C_6$

Order 10: $C_{10}$

Order 7: $C_7$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 14700: $C_{70}\times C_{210}$ ($C_1$)

Order 7350: $C_{35}\times C_{210}$ ($C_2$) x 3

Order 4900: $C_{70}^2$ ($C_3$)

Order 3675: $C_{35}\times C_{105}$ ($C_2^2$)

Order 2940: $C_{14}\times C_{210}$ ($C_5$) x 6

Order 2450: $C_{35}\times C_{70}$ ($C_6$) x 3

Order 2100: $C_{10}\times C_{210}$ ($C_7$) x 8

Order 1470: $C_7\times C_{210}$ ($C_{10}$) x 18

Order 1225: $C_{35}^2$ ($C_2\times C_6$)

Order 1050: $C_5\times C_{210}$ ($C_{14}$) x 24

Order 980: $C_{14}\times C_{70}$ ($C_{15}$) x 6

Order 735: $C_7\times C_{105}$ ($C_2\times C_{10}$) x 6

Order 700: $C_{10}\times C_{70}$ ($C_{21}$) x 8

Order 588: $C_{14}\times C_{42}$ ($C_5^2$)

Order 525: $C_5\times C_{105}$ ($C_2\times C_{14}$) x 8

Order 490: $C_7\times C_{70}$ ($C_{30}$) x 18

Order 420: $C_2\times C_{210}$ ($C_{35}$) x 48

Order 350: $C_5\times C_{70}$ ($C_{42}$) x 24

Order 300: $C_{10}\times C_{30}$ ($C_7^2$)

Order 294: $C_7\times C_{42}$ ($C_5\times C_{10}$) x 3

Order 245: $C_7\times C_{35}$ ($C_2\times C_{30}$) x 6

Order 210: $C_{210}$ ($C_{70}$) x 144

Order 196: $C_{14}^2$ ($C_5\times C_{15}$)

Order 175: $C_5\times C_{35}$ ($C_2\times C_{42}$) x 8

Order 150: $C_5\times C_{30}$ ($C_7\times C_{14}$) x 3

Order 147: $C_7\times C_{21}$ ($C_{10}^2$)

Order 140: $C_2\times C_{70}$ ($C_{105}$) x 48

Order 105: $C_{105}$ ($C_2\times C_{70}$) x 48

Order 100: $C_{10}^2$ ($C_7\times C_{21}$)

Order 98: $C_7\times C_{14}$ ($C_5\times C_{30}$) x 3

Order 84: $C_2\times C_{42}$ ($C_5\times C_{35}$) x 8

Order 75: $C_5\times C_{15}$ ($C_{14}^2$)

Order 70: $C_{70}$ ($C_{210}$) x 144

Order 60: $C_2\times C_{30}$ ($C_7\times C_{35}$) x 6

Order 50: $C_5\times C_{10}$ ($C_7\times C_{42}$) x 3

Order 49: $C_7^2$ ($C_{10}\times C_{30}$)

Order 42: $C_{42}$ ($C_5\times C_{70}$) x 24

Order 35: $C_{35}$ ($C_2\times C_{210}$) x 48

Order 30: $C_{30}$ ($C_7\times C_{70}$) x 18

Order 28: $C_2\times C_{14}$ ($C_5\times C_{105}$) x 8

Order 25: $C_5^2$ ($C_{14}\times C_{42}$)

Order 21: $C_{21}$ ($C_{10}\times C_{70}$) x 8

Order 20: $C_2\times C_{10}$ ($C_7\times C_{105}$) x 6

Order 15: $C_{15}$ ($C_{14}\times C_{70}$) x 6

Order 14: $C_{14}$ ($C_5\times C_{210}$) x 24

Order 12: $C_2\times C_6$ ($C_{35}^2$)

Order 10: $C_{10}$ ($C_7\times C_{210}$) x 18

Order 7: $C_7$ ($C_{10}\times C_{210}$) x 8

Order 6: $C_6$ ($C_{35}\times C_{70}$) x 3

Order 5: $C_5$ ($C_{14}\times C_{210}$) x 6

Order 4: $C_2^2$ ($C_{35}\times C_{105}$)

Order 3: $C_3$ ($C_{70}^2$)

Order 2: $C_2$ ($C_{35}\times C_{210}$) x 3

Order 1: $C_1$ ($C_{70}\times C_{210}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 14700: $C_{70}\times C_{210}$ ($C_1$)

Order 7350: $C_{35}\times C_{210}$ ($C_2$)

Order 4900: $C_{70}^2$ ($C_3$)

Order 3675: $C_{35}\times C_{105}$ ($C_2^2$)

Order 2940: $C_{14}\times C_{210}$ ($C_5$)

Order 2450: $C_{35}\times C_{70}$ ($C_6$)

Order 2100: $C_{10}\times C_{210}$ ($C_7$)

Order 1470: $C_7\times C_{210}$ ($C_{10}$)

Order 1225: $C_{35}^2$ ($C_2\times C_6$)

Order 1050: $C_5\times C_{210}$ ($C_{14}$)

Order 980: $C_{14}\times C_{70}$ ($C_{15}$)

Order 735: $C_7\times C_{105}$ ($C_2\times C_{10}$)

Order 700: $C_{10}\times C_{70}$ ($C_{21}$)

Order 588: $C_{14}\times C_{42}$ ($C_5^2$)

Order 525: $C_5\times C_{105}$ ($C_2\times C_{14}$)

Order 490: $C_7\times C_{70}$ ($C_{30}$)

Order 420: $C_2\times C_{210}$ ($C_{35}$)

Order 350: $C_5\times C_{70}$ ($C_{42}$)

Order 300: $C_{10}\times C_{30}$ ($C_7^2$)

Order 294: $C_7\times C_{42}$ ($C_5\times C_{10}$)

Order 245: $C_7\times C_{35}$ ($C_2\times C_{30}$)

Order 210: $C_{210}$ ($C_{70}$)

Order 196: $C_{14}^2$ ($C_5\times C_{15}$)

Order 175: $C_5\times C_{35}$ ($C_2\times C_{42}$)

Order 150: $C_5\times C_{30}$ ($C_7\times C_{14}$)

Order 147: $C_7\times C_{21}$ ($C_{10}^2$)

Order 140: $C_2\times C_{70}$ ($C_{105}$)

Order 105: $C_{105}$ ($C_2\times C_{70}$)

Order 100: $C_{10}^2$ ($C_7\times C_{21}$)

Order 98: $C_7\times C_{14}$ ($C_5\times C_{30}$)

Order 84: $C_2\times C_{42}$ ($C_5\times C_{35}$)

Order 75: $C_5\times C_{15}$ ($C_{14}^2$)

Order 70: $C_{70}$ ($C_{210}$)

Order 60: $C_2\times C_{30}$ ($C_7\times C_{35}$)

Order 50: $C_5\times C_{10}$ ($C_7\times C_{42}$)

Order 49: $C_7^2$ ($C_{10}\times C_{30}$)

Order 42: $C_{42}$ ($C_5\times C_{70}$)

Order 35: $C_{35}$ ($C_2\times C_{210}$)

Order 30: $C_{30}$ ($C_7\times C_{70}$)

Order 28: $C_2\times C_{14}$ ($C_5\times C_{105}$)

Order 25: $C_5^2$ ($C_{14}\times C_{42}$)

Order 21: $C_{21}$ ($C_{10}\times C_{70}$)

Order 20: $C_2\times C_{10}$ ($C_7\times C_{105}$)

Order 15: $C_{15}$ ($C_{14}\times C_{70}$)

Order 14: $C_{14}$ ($C_5\times C_{210}$)

Order 12: $C_2\times C_6$ ($C_{35}^2$)

Order 10: $C_{10}$ ($C_7\times C_{210}$)

Order 7: $C_7$ ($C_{10}\times C_{210}$)

Order 6: $C_6$ ($C_{35}\times C_{70}$)

Order 5: $C_5$ ($C_{14}\times C_{210}$)

Order 4: $C_2^2$ ($C_{35}\times C_{105}$)

Order 3: $C_3$ ($C_{70}^2$)

Order 2: $C_2$ ($C_{35}\times C_{210}$)

Order 1: $C_1$ ($C_{70}\times C_{210}$)

Series

Derived series

$C_{70}\times C_{210}$

$\rhd$

$C_1$

Chief series

$C_{70}\times C_{210}$

$\rhd$

$C_{35}\times C_{210}$

$\rhd$

$C_7\times C_{210}$

$\rhd$

$C_{210}$

$\rhd$

$C_{105}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{70}\times C_{210}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{70}\times C_{210}$

Supergroups

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

This group is a maximal quotient of 5 larger groups in the database.

Character theory

Complex character table

The $14700 \times 14700$ character table is not available for this group.

Rational character table

The $504 \times 504$ rational character table is not available for this group.