Abstract group 1600.1724: $C_{20}:C_{80}$

• Label: 1600.1724
• Order: \( 2^{6} \cdot 5^{2} \)
• Exponent: \( 2^{4} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{4} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $30$
• Trans deg.: $320$
• Rank: $2$

Group information

Description:	$C_{20}:C_{80}$
Order:	\(1600\)\(\medspace = 2^{6} \cdot 5^{2} \)
Exponent:	\(80\)\(\medspace = 2^{4} \cdot 5 \)
Automorphism group:	$C_5:(C_2^4.C_2^6.C_2)$, of order \(10240\)\(\medspace = 2^{11} \cdot 5 \)
Composition factors:	$C_2$ x 6, $C_5$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	8	10	16	20	40	80
Elements	1	3	12	24	16	72	160	288	384	640	1600
Conjugacy classes	1	3	8	14	12	42	16	152	208	64	520
Divisions	1	3	5	4	3	12	2	22	14	2	68
Autjugacy classes	1	3	4	3	3	9	1	12	9	1	46

Dimension	1	2	4	8	16	32
Irr. complex chars.	160	360	0	0	0	0	520
Irr. rational chars.	4	4	11	19	18	12	68

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$320$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b \mid a^{80}=b^{20}=1, b^{a}=b^{19} \rangle$
Permutation group:	Degree $30$ $\langle(1,2,4,7,5,8,11,14,3,6,9,12,10,13,15,16)(17,18)(19,20)(21,22)(23,25)(24,27) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 79 & 0 \\ 0 & 79 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right), \left(\begin{array}{rr} 40 & 0 \\ 0 & 40 \end{array}\right), \left(\begin{array}{rr} 91 & 0 \\ 0 & 91 \end{array}\right), \left(\begin{array}{rr} 0 & 97 \\ 49 & 0 \end{array}\right), \left(\begin{array}{rr} 82 & 105 \\ 60 & 41 \end{array}\right), \left(\begin{array}{rr} 50 & 0 \\ 0 & 50 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/123\Z)$
Direct product:	$C_5$ $\, \times\, $ $(C_{20}:C_{16})$
Semidirect product:	$C_{20}$ $\,\rtimes\,$ $C_{80}$ (2)	$C_5$ $\,\rtimes\,$ $(C_4:C_{80})$	$C_4$ $\,\rtimes\,$ $(C_5:C_{80})$ (2)	$(C_5\times C_{20})$ $\,\rtimes\,$ $C_{16}$ (2)	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{40}$ . $D_{20}$	$C_{20}^2$ . $C_4$	$(C_5\times C_{40})$ . $Q_8$	$C_{40}$ . $(C_5:Q_8)$	all 58

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

Homology

Abelianization:	$C_{2} \times C_{80} \simeq C_{2} \times C_{16} \times C_{5}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 260 subgroups in 144 conjugacy classes, 78 normal (66 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{40}$	$G/Z \simeq$ $D_{10}$
Commutator:	$G' \simeq$ $C_{10}$	$G/G' \simeq$ $C_2\times C_{80}$
Frattini:	$\Phi \simeq$ $C_2\times C_8$	$G/\Phi \simeq$ $C_5\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{20}\times C_{40}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{20}:C_{80}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{10}^2$	$G/\operatorname{soc} \simeq$ $C_2\times C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4:C_{16}$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$

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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Subgroup information

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Classes of subgroups up to conjugation

Order 1600: $C_{20}:C_{80}$

Order 800: $C_{10}:C_{80}$ x 2, $C_{20}\times C_{40}$

Order 400: $C_5:C_{80}$ x 2, $C_{10}\times C_{40}$ x 2, $C_{20}^2$

Order 320: $C_{20}:C_{16}$, $C_4:C_{80}$

Order 200: $C_{10}\times C_{20}$ x 3, $C_5\times C_{40}$ x 3

Order 160: $C_4\times C_{40}$ x 4, $C_2\times C_{80}$ x 2, $C_{10}:C_{16}$ x 2

Order 100: $C_5\times C_{20}$ x 5, $C_{10}^2$

Order 80: $C_2\times C_{40}$ x 8, $C_4\times C_{20}$ x 4, $C_{80}$ x 2, $C_5:C_{16}$ x 2

Order 64: $C_4:C_{16}$

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

Order 40: $C_{40}$ x 14, $C_2\times C_{20}$ x 12

Order 32: $C_2\times C_{16}$ x 2, $C_4\times C_8$

Order 25: $C_5^2$

Order 20: $C_{20}$ x 22, $C_2\times C_{10}$ x 4

Order 16: $C_2\times C_8$ x 2, $C_{16}$ x 2, $C_4^2$

Order 10: $C_{10}$ x 12

Order 8: $C_2\times C_4$ x 3, $C_8$ x 3

Order 5: $C_5$ x 4

Order 4: $C_4$ x 5, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1600: $C_{20}:C_{80}$

Order 800: $C_{20}\times C_{40}$, $C_{10}:C_{80}$

Order 400: $C_{10}\times C_{40}$ x 2, $C_5:C_{80}$, $C_{20}^2$

Order 320: $C_{20}:C_{16}$, $C_4:C_{80}$

Order 200: $C_{10}\times C_{20}$ x 3, $C_5\times C_{40}$ x 3

Order 160: $C_4\times C_{40}$ x 3, $C_2\times C_{80}$, $C_{10}:C_{16}$

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

Order 80: $C_2\times C_{40}$ x 6, $C_4\times C_{20}$ x 3, $C_{80}$, $C_5:C_{16}$

Order 64: $C_4:C_{16}$

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

Order 40: $C_2\times C_{20}$ x 9, $C_{40}$ x 9

Order 32: $C_4\times C_8$, $C_2\times C_{16}$

Order 25: $C_5^2$

Order 20: $C_{20}$ x 12, $C_2\times C_{10}$ x 3

Order 16: $C_2\times C_8$ x 2, $C_4^2$, $C_{16}$

Order 10: $C_{10}$ x 9

Order 8: $C_2\times C_4$ x 3, $C_8$ x 3

Order 5: $C_5$ x 3

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1600: $C_{20}:C_{80}$ ($C_1$)

Order 800: $C_{10}:C_{80}$ ($C_2$) x 2, $C_{20}\times C_{40}$ ($C_2$)

Order 400: $C_{10}\times C_{40}$ ($C_2^2$), $C_{10}\times C_{40}$ ($C_4$), $C_{20}^2$ ($C_4$)

Order 320: $C_{20}:C_{16}$ ($C_5$)

Order 200: $C_{10}\times C_{20}$ ($C_8$) x 2, $C_{10}\times C_{20}$ ($C_2\times C_4$), $C_5\times C_{40}$ ($Q_8$), $C_5\times C_{40}$ ($D_4$)

Order 160: $C_{10}:C_{16}$ ($C_{10}$) x 2, $C_4\times C_{40}$ ($C_{10}$), $C_4\times C_{40}$ ($D_5$)

Order 100: $C_5\times C_{20}$ ($C_{16}$) x 2, $C_5\times C_{20}$ ($\OD_{16}$), $C_5\times C_{20}$ ($C_4:C_4$), $C_{10}^2$ ($C_2\times C_8$)

Order 80: $C_2\times C_{40}$ ($C_2\times C_{10}$), $C_2\times C_{40}$ ($D_{10}$), $C_2\times C_{40}$ ($C_{20}$), $C_2\times C_{40}$ ($C_5:C_4$), $C_4\times C_{20}$ ($C_{20}$), $C_4\times C_{20}$ ($C_5:C_4$)

Order 50: $C_5\times C_{10}$ ($\OD_{32}$), $C_5\times C_{10}$ ($C_2\times C_{16}$), $C_5\times C_{10}$ ($C_4:C_8$)

Order 40: $C_2\times C_{20}$ ($C_{40}$) x 2, $C_2\times C_{20}$ ($C_5:C_8$) x 2, $C_2\times C_{20}$ ($C_2\times C_{20}$), $C_2\times C_{20}$ ($C_{10}:C_4$), $C_{40}$ ($D_{20}$), $C_{40}$ ($C_5:Q_8$), $C_{40}$ ($C_5\times Q_8$), $C_{40}$ ($C_5\times D_4$)

Order 32: $C_4\times C_8$ ($C_5\times D_5$)

Order 25: $C_5^2$ ($C_4:C_{16}$)

Order 20: $C_{20}$ ($C_{80}$) x 2, $C_{20}$ ($C_5:C_{16}$) x 2, $C_2\times C_{10}$ ($C_{10}:C_8$), $C_2\times C_{10}$ ($C_2\times C_{40}$), $C_{20}$ ($C_5\times \OD_{16}$), $C_{20}$ ($C_4:C_{20}$), $C_{20}$ ($C_{20}:C_4$), $C_{20}$ ($C_5:\OD_{16}$)

Order 16: $C_2\times C_8$ ($C_5:C_{20}$), $C_2\times C_8$ ($C_5\times D_{10}$), $C_4^2$ ($C_5:C_{20}$)

Order 10: $C_{10}$ ($C_5\times \OD_{32}$), $C_{10}$ ($C_2\times C_{80}$), $C_{10}$ ($C_4:C_{40}$), $C_{10}$ ($C_5:\OD_{32}$), $C_{10}$ ($C_{10}:C_{16}$), $C_{10}$ ($C_{20}:C_8$)

Order 8: $C_2\times C_4$ ($C_5:C_{40}$) x 2, $C_2\times C_4$ ($C_{10}:C_{20}$), $C_8$ ($C_5\times D_{20}$), $C_8$ ($C_5^2:Q_8$)

Order 5: $C_5$ ($C_{20}:C_{16}$), $C_5$ ($C_4:C_{80}$)

Order 4: $C_4$ ($C_5:C_{80}$) x 2, $C_2^2$ ($C_{10}:C_{40}$), $C_4$ ($C_{20}:C_{20}$), $C_4$ ($C_{20}.C_{20}$)

Order 2: $C_2$ ($C_{20}.C_{40}$), $C_2$ ($C_{10}:C_{80}$), $C_2$ ($C_{20}:C_{40}$)

Order 1: $C_1$ ($C_{20}:C_{80}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1600: $C_{20}:C_{80}$ ($C_1$)

Order 800: $C_{20}\times C_{40}$ ($C_2$), $C_{10}:C_{80}$ ($C_2$)

Order 400: $C_{10}\times C_{40}$ ($C_2^2$), $C_{10}\times C_{40}$ ($C_4$), $C_{20}^2$ ($C_4$)

Order 320: $C_{20}:C_{16}$ ($C_5$)

Order 200: $C_{10}\times C_{20}$ ($C_8$) x 2, $C_{10}\times C_{20}$ ($C_2\times C_4$), $C_5\times C_{40}$ ($Q_8$), $C_5\times C_{40}$ ($D_4$)

Order 160: $C_4\times C_{40}$ ($C_{10}$), $C_4\times C_{40}$ ($D_5$), $C_{10}:C_{16}$ ($C_{10}$)

Order 100: $C_5\times C_{20}$ ($\OD_{16}$), $C_5\times C_{20}$ ($C_4:C_4$), $C_5\times C_{20}$ ($C_{16}$), $C_{10}^2$ ($C_2\times C_8$)

Order 80: $C_2\times C_{40}$ ($C_2\times C_{10}$), $C_2\times C_{40}$ ($D_{10}$), $C_2\times C_{40}$ ($C_{20}$), $C_2\times C_{40}$ ($C_5:C_4$), $C_4\times C_{20}$ ($C_{20}$), $C_4\times C_{20}$ ($C_5:C_4$)

Order 50: $C_5\times C_{10}$ ($\OD_{32}$), $C_5\times C_{10}$ ($C_2\times C_{16}$), $C_5\times C_{10}$ ($C_4:C_8$)

Order 40: $C_2\times C_{20}$ ($C_{40}$) x 2, $C_2\times C_{20}$ ($C_5:C_8$) x 2, $C_2\times C_{20}$ ($C_2\times C_{20}$), $C_2\times C_{20}$ ($C_{10}:C_4$), $C_{40}$ ($D_{20}$), $C_{40}$ ($C_5:Q_8$), $C_{40}$ ($C_5\times Q_8$), $C_{40}$ ($C_5\times D_4$)

Order 32: $C_4\times C_8$ ($C_5\times D_5$)

Order 25: $C_5^2$ ($C_4:C_{16}$)

Order 20: $C_2\times C_{10}$ ($C_{10}:C_8$), $C_2\times C_{10}$ ($C_2\times C_{40}$), $C_{20}$ ($C_5\times \OD_{16}$), $C_{20}$ ($C_4:C_{20}$), $C_{20}$ ($C_{80}$), $C_{20}$ ($C_{20}:C_4$), $C_{20}$ ($C_5:\OD_{16}$), $C_{20}$ ($C_5:C_{16}$)

Order 16: $C_2\times C_8$ ($C_5:C_{20}$), $C_2\times C_8$ ($C_5\times D_{10}$), $C_4^2$ ($C_5:C_{20}$)

Order 10: $C_{10}$ ($C_5\times \OD_{32}$), $C_{10}$ ($C_2\times C_{80}$), $C_{10}$ ($C_4:C_{40}$), $C_{10}$ ($C_5:\OD_{32}$), $C_{10}$ ($C_{10}:C_{16}$), $C_{10}$ ($C_{20}:C_8$)

Order 8: $C_2\times C_4$ ($C_5:C_{40}$) x 2, $C_2\times C_4$ ($C_{10}:C_{20}$), $C_8$ ($C_5\times D_{20}$), $C_8$ ($C_5^2:Q_8$)

Order 5: $C_5$ ($C_{20}:C_{16}$), $C_5$ ($C_4:C_{80}$)

Order 4: $C_2^2$ ($C_{10}:C_{40}$), $C_4$ ($C_{20}:C_{20}$), $C_4$ ($C_{20}.C_{20}$), $C_4$ ($C_5:C_{80}$)

Order 2: $C_2$ ($C_{20}.C_{40}$), $C_2$ ($C_{10}:C_{80}$), $C_2$ ($C_{20}:C_{40}$)

Order 1: $C_1$ ($C_{20}:C_{80}$)

Series

Derived series

$C_{20}:C_{80}$

$\rhd$

$C_{10}$

$\rhd$

$C_1$

Chief series

$C_{20}:C_{80}$

$\rhd$

$C_{20}\times C_{40}$

$\rhd$

$C_{20}^2$

$\rhd$

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

$\rhd$

$C_5\times C_{20}$

$\rhd$

$C_{20}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{20}:C_{80}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{40}$

$\lhd$

$C_4\times C_{40}$

Supergroups

Character theory

Complex character table

The $520 \times 520$ character table is not available for this group.

Rational character table

See the $68 \times 68$ rational character table.