Abstract group 6560.c: $C_4\times F_{41}$

• Label: 6560.c
• Order: \( 2^{5} \cdot 5 \cdot 41 \)
• Exponent: \( 2^{3} \cdot 5 \cdot 41 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 5 \cdot 41 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $45$
• Trans deg.: $164$
• Rank: $2$

Group information

Description:	$C_4\times F_{41}$
Order:	\(6560\)\(\medspace = 2^{5} \cdot 5 \cdot 41 \)
Exponent:	\(1640\)\(\medspace = 2^{3} \cdot 5 \cdot 41 \)
Automorphism group:	$C_{82}.C_{40}.C_2^2$, of order \(13120\)\(\medspace = 2^{6} \cdot 5 \cdot 41 \)
Composition factors:	$C_2$ x 5, $C_5$, $C_{41}$
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	8	10	20	40	41	82	164
Elements	1	83	412	164	656	492	1968	2624	40	40	80	6560
Conjugacy classes	1	3	12	4	16	12	48	64	1	1	2	164
Divisions	1	3	6	1	4	3	6	4	1	1	1	31
Autjugacy classes	1	3	6	4	4	12	24	16	1	1	1	73

Dimension	1	2	4	8	16	40	80
Irr. complex chars.	160	0	0	0	0	4	0	164
Irr. rational chars.	4	6	8	6	4	2	1	31

Minimal presentations

Permutation degree:	$45$
Transitive degree:	$164$
Rank:	$2$
Inequivalent generating pairs:	$1152$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	40	80	80
Arbitrary	40	42	42

Constructions

Presentation:	$\langle a, b \mid a^{40}=b^{164}=1, b^{a}=b^{29} \rangle$
Permutation group:	Degree $45$ $\langle(1,2,3,6,11,19,23,15,27,39,12,22,35,40,34,31,32,4,9,16,28,26,29,17,10,18,30,20,33,38,36,24,37,41,7,14,25,21,13,8,5) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 15 \end{array}\right), \left(\begin{array}{rr} 32 & 0 \\ 0 & 9 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{41})$
Direct product:	$C_4$ $\, \times\, $ $F_{41}$
Semidirect product:	$C_{164}$ $\,\rtimes\,$ $C_{40}$	$(C_{164}:C_8)$ $\,\rtimes\,$ $C_5$	$(C_{164}:C_5)$ $\,\rtimes\,$ $C_8$	$(C_{41}:C_{20})$ $\,\rtimes\,$ $C_8$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times F_{41})$ . $C_2$ (2)	$C_2$ . $(C_2\times F_{41})$	$(C_4\times D_{41})$ . $C_{20}$	$D_{82}$ . $(C_2\times C_{20})$	all 21
Aut. group:	$\Aut(C_5\times D_{41})$	$\Aut(C_{41}:C_{16})$	$\Aut(C_{205}:C_4)$	$\Aut(C_{41}:C_{32})$	all 5

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

Homology

Abelianization:	$C_{4} \times C_{40} \simeq C_{4} \times C_{8} \times C_{5}$
Schur multiplier:	$C_{4}$
Commutator length:	$1$

Subgroups

There are 1728 subgroups in 88 conjugacy classes, 47 normal (27 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $F_{41}$
Commutator:	$G' \simeq$ $C_{41}$	$G/G' \simeq$ $C_4\times C_{40}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times F_{41}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{164}$	$G/\operatorname{Fit} \simeq$ $C_{40}$
Radical:	$R \simeq$ $C_4\times F_{41}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{82}$	$G/\operatorname{soc} \simeq$ $C_2\times C_{40}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times C_8$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
41-Sylow subgroup:	$P_{ 41 } \simeq$ $C_{41}$

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 6560: $C_4\times F_{41}$

Order 3280: $C_2\times F_{41}$ x 2, $C_{164}:C_{20}$

Order 1640: $F_{41}$ x 4, $C_{82}:C_{20}$ x 2, $C_{164}:C_{10}$

Order 1312: $C_{164}:C_8$

Order 820: $C_{41}:C_{20}$ x 4, $C_{82}:C_{10}$, $C_{164}:C_5$, $C_{41}:C_{20}$

Order 656: $C_{82}:C_8$ x 2, $C_{164}:C_4$

Order 410: $C_{41}:C_{10}$ x 2, $C_{41}:C_{10}$

Order 328: $C_{41}:C_8$ x 4, $C_{82}:C_4$ x 2, $C_4\times D_{41}$

Order 205: $C_{41}:C_5$

Order 164: $C_{41}:C_4$ x 4, $D_{82}$, $C_{164}$, $C_{41}:C_4$

Order 160: $C_4\times C_{40}$

Order 82: $D_{41}$ x 2, $C_{82}$

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

Order 41: $C_{41}$

Order 40: $C_{40}$ x 4, $C_2\times C_{20}$ x 3

Order 32: $C_4\times C_8$

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

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

Order 10: $C_{10}$ x 3

Order 8: $C_8$ x 4, $C_2\times C_4$ x 3

Order 5: $C_5$

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 6560: $C_4\times F_{41}$

Order 3280: $C_2\times F_{41}$, $C_{164}:C_{20}$

Order 1640: $C_{82}:C_{20}$ x 2, $C_{164}:C_{10}$, $F_{41}$

Order 1312: $C_{164}:C_8$

Order 820: $C_{41}:C_{20}$ x 2, $C_{82}:C_{10}$, $C_{164}:C_5$, $C_{41}:C_{20}$

Order 656: $C_{82}:C_8$, $C_{164}:C_4$

Order 410: $C_{41}:C_{10}$ x 2, $C_{41}:C_{10}$

Order 328: $C_{82}:C_4$ x 2, $C_4\times D_{41}$, $C_{41}:C_8$

Order 205: $C_{41}:C_5$

Order 164: $C_{41}:C_4$ x 2, $D_{82}$, $C_{164}$, $C_{41}:C_4$

Order 160: $C_4\times C_{40}$

Order 82: $D_{41}$ x 2, $C_{82}$

Order 80: $C_2\times C_{40}$, $C_4\times C_{20}$

Order 41: $C_{41}$

Order 40: $C_2\times C_{20}$ x 3, $C_{40}$

Order 32: $C_4\times C_8$

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

Order 16: $C_2\times C_8$, $C_4^2$

Order 10: $C_{10}$ x 3

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

Order 5: $C_5$

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 6560: $C_4\times F_{41}$ ($C_1$)

Order 3280: $C_2\times F_{41}$ ($C_2$) x 2, $C_{164}:C_{20}$ ($C_2$)

Order 1640: $F_{41}$ ($C_4$) x 4, $C_{164}:C_{10}$ ($C_4$), $C_{82}:C_{20}$ ($C_2^2$), $C_{82}:C_{20}$ ($C_4$)

Order 1312: $C_{164}:C_8$ ($C_5$)

Order 820: $C_{41}:C_{20}$ ($C_2\times C_4$) x 2, $C_{41}:C_{20}$ ($C_8$) x 2, $C_{82}:C_{10}$ ($C_2\times C_4$), $C_{164}:C_5$ ($C_8$), $C_{41}:C_{20}$ ($C_8$)

Order 656: $C_{82}:C_8$ ($C_{10}$) x 2, $C_{164}:C_4$ ($C_{10}$)

Order 410: $C_{41}:C_{10}$ ($C_2\times C_8$), $C_{41}:C_{10}$ ($C_2\times C_8$), $C_{41}:C_{10}$ ($C_4^2$)

Order 328: $C_{41}:C_8$ ($C_{20}$) x 4, $C_4\times D_{41}$ ($C_{20}$), $C_{82}:C_4$ ($C_2\times C_{10}$), $C_{82}:C_4$ ($C_{20}$)

Order 205: $C_{41}:C_5$ ($C_4\times C_8$)

Order 164: $C_{41}:C_4$ ($C_2\times C_{20}$) x 2, $C_{41}:C_4$ ($C_{40}$) x 2, $D_{82}$ ($C_2\times C_{20}$), $C_{164}$ ($C_{40}$), $C_{41}:C_4$ ($C_{40}$)

Order 82: $C_{82}$ ($C_2\times C_{40}$), $D_{41}$ ($C_2\times C_{40}$), $D_{41}$ ($C_4\times C_{20}$)

Order 41: $C_{41}$ ($C_4\times C_{40}$)

Order 4: $C_4$ ($F_{41}$)

Order 2: $C_2$ ($C_2\times F_{41}$)

Order 1: $C_1$ ($C_4\times F_{41}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6560: $C_4\times F_{41}$ ($C_1$)

Order 3280: $C_2\times F_{41}$ ($C_2$), $C_{164}:C_{20}$ ($C_2$)

Order 1640: $C_{164}:C_{10}$ ($C_4$), $C_{82}:C_{20}$ ($C_2^2$), $C_{82}:C_{20}$ ($C_4$), $F_{41}$ ($C_4$)

Order 1312: $C_{164}:C_8$ ($C_5$)

Order 820: $C_{82}:C_{10}$ ($C_2\times C_4$), $C_{41}:C_{20}$ ($C_2\times C_4$), $C_{41}:C_{20}$ ($C_8$), $C_{164}:C_5$ ($C_8$), $C_{41}:C_{20}$ ($C_8$)

Order 656: $C_{82}:C_8$ ($C_{10}$), $C_{164}:C_4$ ($C_{10}$)

Order 410: $C_{41}:C_{10}$ ($C_2\times C_8$), $C_{41}:C_{10}$ ($C_2\times C_8$), $C_{41}:C_{10}$ ($C_4^2$)

Order 328: $C_4\times D_{41}$ ($C_{20}$), $C_{82}:C_4$ ($C_2\times C_{10}$), $C_{82}:C_4$ ($C_{20}$), $C_{41}:C_8$ ($C_{20}$)

Order 205: $C_{41}:C_5$ ($C_4\times C_8$)

Order 164: $D_{82}$ ($C_2\times C_{20}$), $C_{41}:C_4$ ($C_2\times C_{20}$), $C_{41}:C_4$ ($C_{40}$), $C_{164}$ ($C_{40}$), $C_{41}:C_4$ ($C_{40}$)

Order 82: $C_{82}$ ($C_2\times C_{40}$), $D_{41}$ ($C_2\times C_{40}$), $D_{41}$ ($C_4\times C_{20}$)

Order 41: $C_{41}$ ($C_4\times C_{40}$)

Order 4: $C_4$ ($F_{41}$)

Order 2: $C_2$ ($C_2\times F_{41}$)

Order 1: $C_1$ ($C_4\times F_{41}$)

Series

Derived series

$C_4\times F_{41}$

$\rhd$

$C_{41}$

$\rhd$

$C_1$

Chief series

$C_4\times F_{41}$

$\rhd$

$C_{164}:C_{20}$

$\rhd$

$C_{164}:C_{10}$

$\rhd$

$C_{164}:C_5$

$\rhd$

$C_{164}$

$\rhd$

$C_{82}$

$\rhd$

$C_{41}$

$\rhd$

$C_1$

Lower central series

$C_4\times F_{41}$

$\rhd$

$C_{41}$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $164 \times 164$ 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 $31 \times 31$ rational character table.