Abstract group 344.6: $C_{86}:C_4$

• Label: 344.6
• Order: \( 2^{3} \cdot 43 \)
• Exponent: \( 2^{2} \cdot 43 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 7 \cdot 43 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \cdot 7 \)
• Perm deg.: $49$
• Trans deg.: $344$
• Rank: $2$

Group information

Description:	$C_{86}:C_4$
Order:	\(344\)\(\medspace = 2^{3} \cdot 43 \)
Exponent:	\(172\)\(\medspace = 2^{2} \cdot 43 \)
Automorphism group:	$C_{43}:(C_{42}\times D_4)$, of order \(14448\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \cdot 43 \)
Composition factors:	$C_2$ x 3, $C_{43}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	43	86
Elements	1	3	172	42	126	344
Conjugacy classes	1	3	4	21	63	92
Divisions	1	3	2	1	3	10
Autjugacy classes	1	2	1	1	2	7

Dimension	1	2	42
Irr. complex chars.	8	84	0	92
Irr. rational chars.	4	2	4	10

Minimal presentations

Permutation degree:	$49$
Transitive degree:	$344$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	4	44

Constructions

Presentation:	$\langle a, b \mid a^{4}=b^{86}=1, b^{a}=b^{85} \rangle$
Permutation group:	Degree $49$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(C_{43}:C_4)$
Semidirect product:	$C_{86}$ $\,\rtimes\,$ $C_4$ (2)	$C_{43}$ $\,\rtimes\,$ $(C_2\times C_4)$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $D_{86}$	$C_2^2$ . $D_{43}$	$C_{86}$ . $C_2^2$	$(C_2\times C_{86})$ . $C_2$	more information

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

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 142 subgroups in 16 conjugacy classes, 13 normal (7 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_{43}$
Commutator:	$G' \simeq$ $C_{43}$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_{86}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{86}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{86}:C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{86}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
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 344: $C_{86}:C_4$

Order 172: $C_{43}:C_4$ x 2, $C_2\times C_{86}$

Order 86: $C_{86}$ x 3

Order 43: $C_{43}$

Order 8: $C_2\times C_4$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 344: $C_{86}:C_4$

Order 172: $C_2\times C_{86}$, $C_{43}:C_4$

Order 86: $C_{86}$ x 2

Order 43: $C_{43}$

Order 8: $C_2\times C_4$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 344: $C_{86}:C_4$ ($C_1$)

Order 172: $C_{43}:C_4$ ($C_2$) x 2, $C_2\times C_{86}$ ($C_2$)

Order 86: $C_{86}$ ($C_4$) x 2, $C_{86}$ ($C_2^2$)

Order 43: $C_{43}$ ($C_2\times C_4$)

Order 4: $C_2^2$ ($D_{43}$)

Order 2: $C_2$ ($C_{43}:C_4$) x 2, $C_2$ ($D_{86}$)

Order 1: $C_1$ ($C_{86}:C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 344: $C_{86}:C_4$ ($C_1$)

Order 172: $C_2\times C_{86}$ ($C_2$), $C_{43}:C_4$ ($C_2$)

Order 86: $C_{86}$ ($C_2^2$), $C_{86}$ ($C_4$)

Order 43: $C_{43}$ ($C_2\times C_4$)

Order 4: $C_2^2$ ($D_{43}$)

Order 2: $C_2$ ($D_{86}$), $C_2$ ($C_{43}:C_4$)

Order 1: $C_1$ ($C_{86}:C_4$)

Series

Derived series	$C_{86}:C_4$	$\rhd$	$C_{43}$	$\rhd$	$C_1$
Chief series	$C_{86}:C_4$	$\rhd$	$C_{43}:C_4$	$\rhd$	$C_{86}$	$\rhd$	$C_{43}$	$\rhd$	$C_1$
Lower central series	$C_{86}:C_4$	$\rhd$	$C_{43}$
Upper central series	$C_1$	$\lhd$	$C_2^2$

Supergroups

This group is a maximal subgroup of 18 larger groups in the database.

This group is a maximal quotient of 13 larger groups in the database.

Character theory

Complex character table

See the $92 \times 92$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	4A	4B	43A	86A	86B	86C
	Size	1	1	1	1	86	86	42	42	42	42
	2 P	1A	1A	1A	1A	2B	2B	43A	43A	43A	43A
	43 P	1A	2A	2B	2C	4A	4B	43A	86A	86B	86C
	Schur
344.6.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
344.6.1b		$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$
344.6.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$
344.6.1d		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
344.6.1e		$2$	$-2$	$-2$	$2$	$0$	$0$	$2$	$-2$	$2$	$2$
344.6.1f		$2$	$2$	$-2$	$-2$	$0$	$0$	$2$	$2$	$-2$	$-2$
344.6.2a		$42$	$42$	$42$	$42$	$0$	$0$	$-1$	$-1$	$-1$	$-1$
344.6.2b		$42$	$-42$	$42$	$-42$	$0$	$0$	$-1$	$1$	$1$	$1$
344.6.2c	2	$42$	$-42$	$-42$	$42$	$0$	$0$	$-1$	$1$	$-1$	$-1$
344.6.2d	2	$42$	$42$	$-42$	$-42$	$0$	$0$	$-1$	$-1$	$1$	$1$