Abstract group 688.1: $C_{43}:C_{16}$

• Label: 688.1
• Order: \( 2^{4} \cdot 43 \)
• Exponent: \( 2^{4} \cdot 43 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 7 \cdot 43 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \cdot 7 \)
• Perm deg.: $59$
• Trans deg.: $688$
• Rank: $2$

Group information

Description:	$C_{43}:C_{16}$
Order:	\(688\)\(\medspace = 2^{4} \cdot 43 \)
Exponent:	\(688\)\(\medspace = 2^{4} \cdot 43 \)
Automorphism group:	$C_{43}:(C_2^2\times C_{84})$, of order \(14448\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \cdot 43 \)
Composition factors:	$C_2$ x 4, $C_{43}$
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	4	8	16	43	86	172	344
Elements	1	1	2	4	344	42	42	84	168	688
Conjugacy classes	1	1	2	4	8	21	21	42	84	184
Divisions	1	1	1	1	1	1	1	1	1	9
Autjugacy classes	1	1	1	1	1	1	1	1	1	9

Dimension	1	2	4	8	42	84	168
Irr. complex chars.	16	168	0	0	0	0	0	184
Irr. rational chars.	2	1	1	1	2	1	1	9

Minimal presentations

Permutation degree:	$59$
Transitive degree:	$688$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	336
Arbitrary	2	4	50

Constructions

Presentation:	$\langle a, b \mid a^{16}=b^{43}=1, b^{a}=b^{42} \rangle$
Permutation group:	Degree $59$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 128 & 70 \\ 109 & 128 \end{array}\right), \left(\begin{array}{rr} 202 & 141 \\ 210 & 55 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{257})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{43}$ $\,\rtimes\,$ $C_{16}$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_8$ . $D_{43}$	$C_{86}$ . $C_8$	$C_{344}$ . $C_2$	$C_{172}$ . $C_4$	all 6

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

Homology

Abelianization:	$C_{16} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 52 subgroups in 10 conjugacy classes, 9 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_8$	$G/Z \simeq$ $D_{43}$
Commutator:	$G' \simeq$ $C_{43}$	$G/G' \simeq$ $C_{16}$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $D_{43}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{344}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{43}:C_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{86}$	$G/\operatorname{soc} \simeq$ $C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{16}$
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 688: $C_{43}:C_{16}$

Order 344: $C_{344}$

Order 172: $C_{172}$

Order 86: $C_{86}$

Order 43: $C_{43}$

Order 16: $C_{16}$

Order 8: $C_8$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 688: $C_{43}:C_{16}$

Order 344: $C_{344}$

Order 172: $C_{172}$

Order 86: $C_{86}$

Order 43: $C_{43}$

Order 16: $C_{16}$

Order 8: $C_8$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 688: $C_{43}:C_{16}$ ($C_1$)

Order 344: $C_{344}$ ($C_2$)

Order 172: $C_{172}$ ($C_4$)

Order 86: $C_{86}$ ($C_8$)

Order 43: $C_{43}$ ($C_{16}$)

Order 8: $C_8$ ($D_{43}$)

Order 4: $C_4$ ($C_{43}:C_4$)

Order 2: $C_2$ ($C_{43}:C_8$)

Order 1: $C_1$ ($C_{43}:C_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 688: $C_{43}:C_{16}$ ($C_1$)

Order 344: $C_{344}$ ($C_2$)

Order 172: $C_{172}$ ($C_4$)

Order 86: $C_{86}$ ($C_8$)

Order 43: $C_{43}$ ($C_{16}$)

Order 8: $C_8$ ($D_{43}$)

Order 4: $C_4$ ($C_{43}:C_4$)

Order 2: $C_2$ ($C_{43}:C_8$)

Order 1: $C_1$ ($C_{43}:C_{16}$)

Series

Derived series

$C_{43}:C_{16}$

$\rhd$

$C_{43}$

$\rhd$

$C_1$

Chief series

$C_{43}:C_{16}$

$\rhd$

$C_{344}$

$\rhd$

$C_{172}$

$\rhd$

$C_{86}$

$\rhd$

$C_{43}$

$\rhd$

$C_1$

Lower central series

$C_{43}:C_{16}$

$\rhd$

$C_{43}$

Upper central series

$C_1$

$\lhd$

$C_8$

Supergroups

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

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $184 \times 184$ 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	4A	8A	16A	43A	86A	172A	344A
	Size	1	1	2	4	344	42	42	84	168
	2 P	1A	1A	2A	4A	8A	43A	43A	86A	172A
	43 P	1A	2A	4A	8A	16A	43A	86A	172A	344A
	Schur
688.1.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
688.1.1b		$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$
688.1.1c		$2$	$2$	$2$	$-2$	$0$	$2$	$2$	$2$	$-2$
688.1.1d		$4$	$4$	$-4$	$0$	$0$	$4$	$4$	$-4$	$0$
688.1.1e		$8$	$-8$	$0$	$0$	$0$	$8$	$-8$	$0$	$0$
688.1.2a		$42$	$42$	$42$	$42$	$0$	$-1$	$-1$	$-1$	$-1$
688.1.2b	2	$42$	$42$	$42$	$-42$	$0$	$-1$	$-1$	$-1$	$1$
688.1.2c		$84$	$84$	$-84$	$0$	$0$	$-2$	$-2$	$2$	$0$
688.1.2d	2	$168$	$-168$	$0$	$0$	$0$	$-4$	$4$	$0$	$0$