Abstract group 400.83: $C_{20}:C_{20}$

• Label: 400.83
• Order: \( 2^{4} \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $18$
• Trans deg.: $80$
• Rank: $2$

Group information

Description:	$C_{20}:C_{20}$
Order:	\(400\)\(\medspace = 2^{4} \cdot 5^{2} \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$C_2^{16}.\PSL(2,7)$, of order \(2560\)\(\medspace = 2^{9} \cdot 5 \)
Composition factors:	$C_2$ x 4, $C_5$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	10	20
Elements	1	3	44	24	72	256	400
Conjugacy classes	1	3	12	14	42	88	160
Divisions	1	3	6	4	12	12	38
Autjugacy classes	1	2	2	3	6	4	18

Dimension	1	2	4	8	16
Irr. complex chars.	80	80	0	0	0	160
Irr. rational chars.	4	6	8	16	4	38

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$80$
Rank:	$2$
Inequivalent generating pairs:	$18$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	10

Constructions

Presentation:	$\langle a, b \mid a^{20}=b^{20}=1, b^{a}=b^{9} \rangle$
Permutation group:	Degree $18$ $\langle(6,7,8,9)(11,12)(13,14)(15,16,17,18), (6,7,8,9)(15,17)(16,18), (1,2,3,4,5), (6,8)(7,9)(15,17)(16,18), (6,8)(7,9), (10,11,13,14,12)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 4 & 3 & 4 & 0 \\ 0 & 2 & 4 & 0 \\ 0 & 4 & 1 & 0 \\ 0 & 2 & 1 & 4 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 3 & 4 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 3 & 4 & 2 \\ 2 & 2 & 3 & 4 \\ 1 & 4 & 3 & 2 \\ 3 & 2 & 3 & 3 \end{array}\right), \left(\begin{array}{rrrr} 2 & 4 & 2 & 3 \\ 3 & 3 & 3 & 1 \\ 4 & 2 & 3 & 3 \\ 2 & 1 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 4 & 4 & 0 & 2 \\ 3 & 0 & 0 & 2 \\ 4 & 2 & 1 & 1 \\ 2 & 1 & 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
Direct product:	$C_4$ $\, \times\, $ $C_5$ $\, \times\, $ $(C_5:C_4)$
Semidirect product:	$C_{20}$ $\,\rtimes\,$ $C_{20}$ (2)	$C_5^2$ $\,\rtimes\,$ $C_4^2$	$(C_5\times C_{20})$ $\,\rtimes\,$ $C_4$ (2)	$C_5$ $\,\rtimes\,$ $(C_4\times C_{20})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{10}^2$ . $C_2^2$	$(C_2\times C_{20})$ . $D_5$	$C_{10}$ . $(C_4\times D_5)$ (2)	$C_2$ . $(D_5\times C_{20})$ (2)	all 16

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

Homology

Abelianization:	$C_{4} \times C_{20} \simeq C_{4}^{2} \times C_{5}$
Schur multiplier:	$C_{4}$
Commutator length:	$1$

Subgroups

There are 148 subgroups in 76 conjugacy classes, 46 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{20}$	$G/Z \simeq$ $D_5$
Commutator:	$G' \simeq$ $C_5$	$G/G' \simeq$ $C_4\times C_{20}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_5\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{10}\times C_{20}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{20}:C_{20}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{10}^2$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$

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 400: $C_{20}:C_{20}$

Order 200: $C_{10}:C_{20}$ x 2, $C_{10}\times C_{20}$

Order 100: $C_5:C_{20}$ x 4, $C_5\times C_{20}$ x 2, $C_{10}^2$

Order 80: $C_4\times C_{20}$, $C_{20}:C_4$

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

Order 40: $C_2\times C_{20}$ x 6, $C_{10}:C_4$ x 2

Order 25: $C_5^2$

Order 20: $C_{20}$ x 12, $C_2\times C_{10}$ x 4, $C_5:C_4$ x 4

Order 16: $C_4^2$

Order 10: $C_{10}$ x 12

Order 8: $C_2\times C_4$ x 3

Order 5: $C_5$ x 4

Order 4: $C_4$ x 6, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 400: $C_{20}:C_{20}$

Order 200: $C_{10}\times C_{20}$, $C_{10}:C_{20}$

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

Order 80: $C_4\times C_{20}$, $C_{20}:C_4$

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

Order 40: $C_2\times C_{20}$ x 4, $C_{10}:C_4$

Order 25: $C_5^2$

Order 20: $C_{20}$ x 4, $C_2\times C_{10}$ x 3, $C_5:C_4$

Order 16: $C_4^2$

Order 10: $C_{10}$ x 6

Order 8: $C_2\times C_4$ x 2

Order 5: $C_5$ x 3

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 400: $C_{20}:C_{20}$ ($C_1$)

Order 200: $C_{10}:C_{20}$ ($C_2$) x 2, $C_{10}\times C_{20}$ ($C_2$)

Order 100: $C_5:C_{20}$ ($C_4$) x 4, $C_5\times C_{20}$ ($C_4$) x 2, $C_{10}^2$ ($C_2^2$)

Order 80: $C_{20}:C_4$ ($C_5$)

Order 50: $C_5\times C_{10}$ ($C_2\times C_4$) x 3

Order 40: $C_{10}:C_4$ ($C_{10}$) x 2, $C_2\times C_{20}$ ($C_{10}$), $C_2\times C_{20}$ ($D_5$)

Order 25: $C_5^2$ ($C_4^2$)

Order 20: $C_5:C_4$ ($C_{20}$) x 4, $C_{20}$ ($C_{20}$) x 2, $C_{20}$ ($C_5:C_4$) x 2, $C_2\times C_{10}$ ($C_2\times C_{10}$), $C_2\times C_{10}$ ($D_{10}$)

Order 10: $C_{10}$ ($C_2\times C_{20}$) x 3, $C_{10}$ ($C_4\times D_5$) x 2, $C_{10}$ ($C_{10}:C_4$)

Order 8: $C_2\times C_4$ ($C_5\times D_5$)

Order 5: $C_5$ ($C_4\times C_{20}$), $C_5$ ($C_{20}:C_4$)

Order 4: $C_4$ ($C_5:C_{20}$) x 2, $C_2^2$ ($C_5\times D_{10}$)

Order 2: $C_2$ ($D_5\times C_{20}$) x 2, $C_2$ ($C_{10}:C_{20}$)

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 400: $C_{20}:C_{20}$ ($C_1$)

Order 200: $C_{10}\times C_{20}$ ($C_2$), $C_{10}:C_{20}$ ($C_2$)

Order 100: $C_5\times C_{20}$ ($C_4$), $C_5:C_{20}$ ($C_4$), $C_{10}^2$ ($C_2^2$)

Order 80: $C_{20}:C_4$ ($C_5$)

Order 50: $C_5\times C_{10}$ ($C_2\times C_4$) x 2

Order 40: $C_2\times C_{20}$ ($C_{10}$), $C_2\times C_{20}$ ($D_5$), $C_{10}:C_4$ ($C_{10}$)

Order 25: $C_5^2$ ($C_4^2$)

Order 20: $C_2\times C_{10}$ ($C_2\times C_{10}$), $C_2\times C_{10}$ ($D_{10}$), $C_{20}$ ($C_{20}$), $C_{20}$ ($C_5:C_4$), $C_5:C_4$ ($C_{20}$)

Order 10: $C_{10}$ ($C_2\times C_{20}$) x 2, $C_{10}$ ($C_{10}:C_4$), $C_{10}$ ($C_4\times D_5$)

Order 8: $C_2\times C_4$ ($C_5\times D_5$)

Order 5: $C_5$ ($C_4\times C_{20}$), $C_5$ ($C_{20}:C_4$)

Order 4: $C_2^2$ ($C_5\times D_{10}$), $C_4$ ($C_5:C_{20}$)

Order 2: $C_2$ ($C_{10}:C_{20}$), $C_2$ ($D_5\times C_{20}$)

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

Series

Derived series

$C_{20}:C_{20}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Chief series

$C_{20}:C_{20}$

$\rhd$

$C_{10}\times C_{20}$

$\rhd$

$C_5\times C_{20}$

$\rhd$

$C_{20}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{20}:C_{20}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{20}$

Supergroups

This group is a maximal subgroup of 80 larger groups in the database.

This group is a maximal quotient of 18 larger groups in the database.

Character theory

Complex character table

See the $160 \times 160$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.