Abstract group 602.4: $C_7\times D_{43}$

• Label: 602.4
• Order: \( 2 \cdot 7 \cdot 43 \)
• Exponent: \( 2 \cdot 7 \cdot 43 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 7 \)
• $\card{Z(G)}$: \( 7 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{2} \cdot 7 \cdot 43 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3^{2} \cdot 7 \)
• Perm deg.: $50$
• Trans deg.: $301$
• Rank: $2$

Group information

Description:	$C_7\times D_{43}$
Order:	\(602\)\(\medspace = 2 \cdot 7 \cdot 43 \)
Exponent:	\(602\)\(\medspace = 2 \cdot 7 \cdot 43 \)
Automorphism group:	$C_6\times F_{43}$, of order \(10836\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \cdot 43 \)
Composition factors:	$C_2$, $C_7$, $C_{43}$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.

Group statistics

Order	1	2	7	14	43	301
Elements	1	43	6	258	42	252	602
Conjugacy classes	1	1	6	6	21	126	161
Divisions	1	1	1	1	1	1	6
Autjugacy classes	1	1	1	1	1	1	6

Dimension	1	2	6	42	252
Irr. complex chars.	14	147	0	0	0	161
Irr. rational chars.	2	0	2	1	1	6

Minimal presentations

Permutation degree:	$50$
Transitive degree:	$301$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	252
Arbitrary	2	4	48

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{301}=1, b^{a}=b^{85} \rangle$
Permutation group:	Degree $50$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 39 & 0 \\ 0 & 4 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{43})$
Direct product:	$C_7$ $\, \times\, $ $D_{43}$
Semidirect product:	$C_{301}$ $\,\rtimes\,$ $C_2$	$C_{43}$ $\,\rtimes\,$ $C_{14}$			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:	$C_{14} \simeq C_{2} \times C_{7}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 92 subgroups in 8 conjugacy classes, 6 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_7$	$G/Z \simeq$ $D_{43}$
Commutator:	$G' \simeq$ $C_{43}$	$G/G' \simeq$ $C_{14}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_7\times D_{43}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{301}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_7\times D_{43}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{301}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
43-Sylow subgroup:	$P_{ 43 } \simeq$ $C_{43}$

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 602: $C_7\times D_{43}$

Order 301: $C_{301}$

Order 86: $D_{43}$

Order 43: $C_{43}$

Order 14: $C_{14}$

Order 7: $C_7$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 602: $C_7\times D_{43}$

Order 301: $C_{301}$

Order 86: $D_{43}$

Order 43: $C_{43}$

Order 14: $C_{14}$

Order 7: $C_7$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 602: $C_7\times D_{43}$ ($C_1$)

Order 301: $C_{301}$ ($C_2$)

Order 86: $D_{43}$ ($C_7$)

Order 43: $C_{43}$ ($C_{14}$)

Order 7: $C_7$ ($D_{43}$)

Order 1: $C_1$ ($C_7\times D_{43}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 602: $C_7\times D_{43}$ ($C_1$)

Order 301: $C_{301}$ ($C_2$)

Order 86: $D_{43}$ ($C_7$)

Order 43: $C_{43}$ ($C_{14}$)

Order 7: $C_7$ ($D_{43}$)

Order 1: $C_1$ ($C_7\times D_{43}$)

Series

Derived series	$C_7\times D_{43}$	$\rhd$	$C_{43}$	$\rhd$	$C_1$
Chief series	$C_7\times D_{43}$	$\rhd$	$C_{301}$	$\rhd$	$C_{43}$	$\rhd$	$C_1$
Lower central series	$C_7\times D_{43}$	$\rhd$	$C_{43}$
Upper central series	$C_1$	$\lhd$	$C_7$

Supergroups

This group is a maximal subgroup of 9 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $161 \times 161$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	7A	14A	43A	301A
	Size	1	43	6	258	42	252
	2 P	1A	1A	7A	7A	43A	301A
	7 P	1A	2A	7A	14A	43A	301A
	43 P	1A	2A	1A	2A	43A	43A
602.4.1a		$1$	$1$	$1$	$1$	$1$	$1$
602.4.1b		$1$	$-1$	$1$	$-1$	$1$	$1$
602.4.1c		$6$	$6$	$-1$	$-1$	$6$	$-1$
602.4.1d		$6$	$-6$	$-1$	$1$	$6$	$-1$
602.4.2a		$42$	$0$	$42$	$0$	$-1$	$-1$
602.4.2b		$252$	$0$	$-42$	$0$	$-6$	$1$