Abstract group 840.47: $C_{21}:D_{20}$

• Label: 840.47
• Order: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $19$
• Trans deg.: $420$
• Rank: $2$

Group information

Description:	$C_{21}:D_{20}$
Order:	\(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_2^3\times F_5\times F_7$, of order \(6720\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_3$, $C_5$, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	7	10	12	14	15	20	21	30	35	42	60	70	105	210
Elements	1	81	2	14	4	162	6	4	28	66	8	56	12	8	24	132	112	24	48	48	840
Conjugacy classes	1	3	2	1	2	6	3	2	2	9	4	4	6	4	6	18	8	6	12	12	111
Divisions	1	3	1	1	1	3	1	1	1	2	1	1	1	1	1	2	1	1	1	1	26
Autjugacy classes	1	3	1	1	1	3	1	1	1	2	1	1	1	1	1	2	1	1	1	1	26

Dimension	1	2	4	6	8	12	16	24	48
Irr. complex chars.	12	63	36	0	0	0	0	0	0	111
Irr. rational chars.	4	5	3	2	3	3	1	3	2	26

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$420$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	96
Arbitrary	4	6	14

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{20}=c^{21}=[a,c]=1, b^{a}=b^{19}, c^{b}=c^{13} \rangle$
Permutation group:	Degree $19$ $\langle(2,3)(4,5)(6,7)(16,17,18,19), (12,13)(14,15)(17,19), (8,9,10), (16,18)(17,19), (11,12,14,15,13), (1,2,4,6,7,5,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 6 & 65 \\ 70 & 36 \end{array}\right), \left(\begin{array}{rr} 71 & 0 \\ 0 & 71 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right), \left(\begin{array}{rr} 36 & 41 \\ 35 & 69 \end{array}\right), \left(\begin{array}{rr} 1 & 21 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/105\Z)$
Direct product:	$C_3$ $\, \times\, $ $(C_7:D_{20})$
Semidirect product:	$D_{70}$ $\,\rtimes\,$ $C_6$	$C_{105}$ $\,\rtimes\,$ $D_4$	$C_{21}$ $\,\rtimes\,$ $D_{20}$	$(C_3\times D_{10})$ $\,\rtimes\,$ $D_7$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{42}$ . $D_{10}$	$C_{30}$ . $D_{14}$	$C_{210}$ . $C_2^2$	$C_6$ . $(D_5\times D_7)$	all 8

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

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 504 subgroups in 64 conjugacy classes, 28 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $D_5\times D_7$
Commutator:	$G' \simeq$ $C_{70}$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{15}:D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{210}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{21}:D_{20}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{210}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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 840: $C_{21}:D_{20}$

Order 420: $C_7:C_{60}$, $C_3\times D_{70}$, $D_5\times C_{42}$

Order 280: $C_7:D_{20}$

Order 210: $C_3\times D_{35}$, $D_5\times C_{21}$, $C_{210}$

Order 168: $C_{21}:D_4$

Order 140: $C_7\times D_{10}$, $C_7:C_{20}$, $D_{70}$

Order 120: $C_3\times D_{20}$

Order 105: $C_{105}$

Order 84: $C_7:C_{12}$, $C_2\times C_{42}$, $C_3\times D_{14}$

Order 70: $C_{70}$, $D_{35}$, $C_7\times D_5$

Order 60: $C_3\times D_{10}$ x 2, $C_{60}$

Order 56: $C_7:D_4$

Order 42: $C_{42}$ x 2, $C_3\times D_7$

Order 40: $D_{20}$

Order 35: $C_{35}$

Order 30: $C_3\times D_5$ x 2, $C_{30}$

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

Order 24: $C_3\times D_4$

Order 21: $C_{21}$

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

Order 15: $C_{15}$

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

Order 12: $C_2\times C_6$ x 2, $C_{12}$

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

Order 8: $D_4$

Order 7: $C_7$

Order 6: $C_6$ x 3

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 840: $C_{21}:D_{20}$

Order 420: $C_7:C_{60}$, $C_3\times D_{70}$, $D_5\times C_{42}$

Order 280: $C_7:D_{20}$

Order 210: $C_3\times D_{35}$, $D_5\times C_{21}$, $C_{210}$

Order 168: $C_{21}:D_4$

Order 140: $C_7\times D_{10}$, $C_7:C_{20}$, $D_{70}$

Order 120: $C_3\times D_{20}$

Order 105: $C_{105}$

Order 84: $C_7:C_{12}$, $C_2\times C_{42}$, $C_3\times D_{14}$

Order 70: $C_{70}$, $D_{35}$, $C_7\times D_5$

Order 60: $C_3\times D_{10}$ x 2, $C_{60}$

Order 56: $C_7:D_4$

Order 42: $C_{42}$ x 2, $C_3\times D_7$

Order 40: $D_{20}$

Order 35: $C_{35}$

Order 30: $C_3\times D_5$ x 2, $C_{30}$

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

Order 24: $C_3\times D_4$

Order 21: $C_{21}$

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

Order 15: $C_{15}$

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

Order 12: $C_2\times C_6$ x 2, $C_{12}$

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

Order 8: $D_4$

Order 7: $C_7$

Order 6: $C_6$ x 3

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 840: $C_{21}:D_{20}$ ($C_1$)

Order 420: $C_7:C_{60}$ ($C_2$), $C_3\times D_{70}$ ($C_2$), $D_5\times C_{42}$ ($C_2$)

Order 280: $C_7:D_{20}$ ($C_3$)

Order 210: $C_{210}$ ($C_2^2$)

Order 140: $C_7\times D_{10}$ ($C_6$), $C_7:C_{20}$ ($C_6$), $D_{70}$ ($C_6$)

Order 105: $C_{105}$ ($D_4$)

Order 84: $C_7:C_{12}$ ($D_5$)

Order 70: $C_{70}$ ($C_2\times C_6$)

Order 60: $C_3\times D_{10}$ ($D_7$)

Order 42: $C_{42}$ ($D_{10}$)

Order 35: $C_{35}$ ($C_3\times D_4$)

Order 30: $C_{30}$ ($D_{14}$)

Order 28: $C_7:C_4$ ($C_3\times D_5$)

Order 21: $C_{21}$ ($D_{20}$)

Order 20: $D_{10}$ ($C_3\times D_7$)

Order 15: $C_{15}$ ($C_7:D_4$)

Order 14: $C_{14}$ ($C_3\times D_{10}$)

Order 10: $C_{10}$ ($C_3\times D_{14}$)

Order 7: $C_7$ ($C_3\times D_{20}$)

Order 6: $C_6$ ($D_5\times D_7$)

Order 5: $C_5$ ($C_{21}:D_4$)

Order 3: $C_3$ ($C_7:D_{20}$)

Order 2: $C_2$ ($C_{15}:D_{14}$)

Order 1: $C_1$ ($C_{21}:D_{20}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 840: $C_{21}:D_{20}$ ($C_1$)

Order 420: $C_7:C_{60}$ ($C_2$), $C_3\times D_{70}$ ($C_2$), $D_5\times C_{42}$ ($C_2$)

Order 280: $C_7:D_{20}$ ($C_3$)

Order 210: $C_{210}$ ($C_2^2$)

Order 140: $C_7\times D_{10}$ ($C_6$), $C_7:C_{20}$ ($C_6$), $D_{70}$ ($C_6$)

Order 105: $C_{105}$ ($D_4$)

Order 84: $C_7:C_{12}$ ($D_5$)

Order 70: $C_{70}$ ($C_2\times C_6$)

Order 60: $C_3\times D_{10}$ ($D_7$)

Order 42: $C_{42}$ ($D_{10}$)

Order 35: $C_{35}$ ($C_3\times D_4$)

Order 30: $C_{30}$ ($D_{14}$)

Order 28: $C_7:C_4$ ($C_3\times D_5$)

Order 21: $C_{21}$ ($D_{20}$)

Order 20: $D_{10}$ ($C_3\times D_7$)

Order 15: $C_{15}$ ($C_7:D_4$)

Order 14: $C_{14}$ ($C_3\times D_{10}$)

Order 10: $C_{10}$ ($C_3\times D_{14}$)

Order 7: $C_7$ ($C_3\times D_{20}$)

Order 6: $C_6$ ($D_5\times D_7$)

Order 5: $C_5$ ($C_{21}:D_4$)

Order 3: $C_3$ ($C_7:D_{20}$)

Order 2: $C_2$ ($C_{15}:D_{14}$)

Order 1: $C_1$ ($C_{21}:D_{20}$)

Series

Derived series

$C_{21}:D_{20}$

$\rhd$

$C_{70}$

$\rhd$

$C_1$

Chief series

$C_{21}:D_{20}$

$\rhd$

$D_5\times C_{42}$

$\rhd$

$C_{210}$

$\rhd$

$C_{105}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{21}:D_{20}$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

Upper central series

$C_1$

$\lhd$

$C_6$

Supergroups

This group is a maximal subgroup of 16 larger groups in the database.

This group is a maximal quotient of 8 larger groups in the database.

Character theory

Complex character table

See the $111 \times 111$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $26 \times 26$ rational character table.