Abstract group 205.1: $C_{41}:C_5$

• Label: 205.1
• Order: \( 5 \cdot 41 \)
• Exponent: \( 5 \cdot 41 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 5 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 5 \cdot 41 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $41$
• Trans deg.: $41$
• Rank: $2$

Group information

Description:	$C_{41}:C_5$
Order:	\(205\)\(\medspace = 5 \cdot 41 \)
Exponent:	\(205\)\(\medspace = 5 \cdot 41 \)
Automorphism group:	$F_{41}$, of order \(1640\)\(\medspace = 2^{3} \cdot 5 \cdot 41 \)
Composition factors:	$C_5$, $C_{41}$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 5$.

Group statistics

Order	1	5	41
Elements	1	164	40	205
Conjugacy classes	1	4	8	13
Divisions	1	1	1	3
Autjugacy classes	1	4	1	6

Dimension	1	4	5	40
Irr. complex chars.	5	0	8	0	13
Irr. rational chars.	1	1	0	1	3

Minimal presentations

Permutation degree:	$41$
Transitive degree:	$41$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	5	10	40
Arbitrary	5	10	40

Constructions

Presentation:	$\langle a, b \mid a^{5}=b^{41}=1, b^{a}=b^{10} \rangle$
Permutation group:	Degree $41$ $\langle(2,11,19,17,38)(3,21,37,33,34)(4,31,14,8,30)(5,41,32,24,26)(6,10,9,40,22) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 18 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{41})$
Transitive group:	41T4				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{41}$ $\,\rtimes\,$ $C_5$				more information
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

Abelianization:	$C_{5} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 44 subgroups in 4 conjugacy classes, 3 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_{41}:C_5$
Commutator:	$G' \simeq$ $C_{41}$	$G/G' \simeq$ $C_5$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{41}:C_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_{41}$	$G/\operatorname{Fit} \simeq$ $C_5$
Radical:	$R \simeq$ $C_{41}:C_5$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{41}$	$G/\operatorname{soc} \simeq$ $C_5$
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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Classes of subgroups up to conjugation

Order 205: $C_{41}:C_5$

Order 41: $C_{41}$

Order 5: $C_5$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 205: $C_{41}:C_5$

Order 41: $C_{41}$

Order 5: $C_5$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 205: $C_{41}:C_5$ ($C_1$)

Order 41: $C_{41}$ ($C_5$)

Order 1: $C_1$ ($C_{41}:C_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 205: $C_{41}:C_5$ ($C_1$)

Order 41: $C_{41}$ ($C_5$)

Order 1: $C_1$ ($C_{41}:C_5$)

Series

Derived series	$C_{41}:C_5$	$\rhd$	$C_{41}$	$\rhd$	$C_1$
Chief series	$C_{41}:C_5$	$\rhd$	$C_{41}$	$\rhd$	$C_1$
Lower central series	$C_{41}:C_5$	$\rhd$	$C_{41}$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

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

Character theory

Complex character table

		1A	5A1	5A-1	5A2	5A-2	41A1	41A-1	41A2	41A-2	41A3	41A-3	41A6	41A-6
	Size	1	41	41	41	41	5	5	5	5	5	5	5	5
	5 P	1A	1A	1A	1A	1A	41A-2	41A2	41A1	41A-1	41A-6	41A6	41A3	41A-3
	41 P	1A	5A1	5A-1	5A2	5A-2	1A	1A	1A	1A	1A	1A	1A	1A
	Type
205.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
205.1.1b1	C	$1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5^{-1}$	${\zeta }_5$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
205.1.1b2	C	$1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5$	${\zeta }_5^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
205.1.1b3	C	$1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^2$	${\zeta }_5^{-2}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
205.1.1b4	C	$1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^{-2}$	${\zeta }_5^2$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
205.1.5a1	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$
205.1.5a2	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$
205.1.5a3	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$
205.1.5a4	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$
205.1.5a5	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$
205.1.5a6	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$
205.1.5a7	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$
205.1.5a8	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$

Rational character table

		1A	5A	41A
	Size	1	164	40
	5 P	1A	1A	41A
	41 P	1A	5A	1A
205.1.1a		$1$	$1$	$1$
205.1.1b		$4$	$-1$	$4$
205.1.5a		$40$	$0$	$-1$