Abstract group 688.11: $C_{86}.D_4$

• Label: 688.11
• Order: \( 2^{4} \cdot 43 \)
• Exponent: \( 2^{2} \cdot 43 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 7 \cdot 43 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \cdot 7 \)
• Perm deg.: $51$
• Trans deg.: $688$
• Rank: $2$

Group information

Description:	$C_{86}.D_4$
Order:	\(688\)\(\medspace = 2^{4} \cdot 43 \)
Exponent:	\(172\)\(\medspace = 2^{2} \cdot 43 \)
Automorphism group:	$C_{43}.C_{42}.C_2^4$, of order \(28896\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 43 \)
Composition factors:	$C_2$ x 4, $C_{43}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	43	86	172
Elements	1	3	348	42	126	168	688
Conjugacy classes	1	3	6	21	63	84	178
Divisions	1	3	4	1	3	1	13
Autjugacy classes	1	3	3	1	3	1	12

Dimension	1	2	42	84
Irr. complex chars.	8	170	0	0	178
Irr. rational chars.	4	4	2	3	13

Minimal presentations

Permutation degree:	$51$
Transitive degree:	$688$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	46

Constructions

Presentation:	$\langle a, b, c \mid a^{4}=b^{4}=c^{43}=[a,c]=1, b^{a}=b^{3}, c^{b}=c^{42} \rangle$
Permutation group:	Degree $51$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{43}:C_4)$ $\,\rtimes\,$ $C_4$ (2)	$C_{43}$ $\,\rtimes\,$ $(C_4:C_4)$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{86}$ . $Q_8$	$C_{86}$ . $D_4$	$C_2^2$ . $D_{86}$	$C_2$ . $(C_{43}:Q_8)$	all 11

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 324 subgroups in 26 conjugacy classes, 17 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_{86}$
Commutator:	$G' \simeq$ $C_{86}$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $D_{86}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{172}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{86}.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{86}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4:C_4$
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

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 688: $C_{86}.D_4$

Order 344: $C_{86}:C_4$ x 2, $C_2\times C_{172}$

Order 172: $C_{43}:C_4$ x 3, $C_2\times C_{86}$, $C_{172}$

Order 86: $C_{86}$ x 3

Order 43: $C_{43}$

Order 16: $C_4:C_4$

Order 8: $C_2\times C_4$ x 3

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 688: $C_{86}.D_4$

Order 344: $C_{86}:C_4$ x 2, $C_2\times C_{172}$

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

Order 86: $C_{86}$ x 3

Order 43: $C_{43}$

Order 16: $C_4:C_4$

Order 8: $C_2\times C_4$ x 3

Order 4: $C_4$ x 3, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 688: $C_{86}.D_4$ ($C_1$)

Order 344: $C_{86}:C_4$ ($C_2$) x 2, $C_2\times C_{172}$ ($C_2$)

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

Order 86: $C_{86}$ ($Q_8$), $C_{86}$ ($D_4$), $C_{86}$ ($C_2\times C_4$)

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

Order 8: $C_2\times C_4$ ($D_{43}$)

Order 4: $C_2^2$ ($D_{86}$)

Order 2: $C_2$ ($C_{43}:D_4$), $C_2$ ($C_4\times D_{43}$), $C_2$ ($C_{43}:Q_8$)

Order 1: $C_1$ ($C_{86}.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 688: $C_{86}.D_4$ ($C_1$)

Order 344: $C_{86}:C_4$ ($C_2$) x 2, $C_2\times C_{172}$ ($C_2$)

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

Order 86: $C_{86}$ ($Q_8$), $C_{86}$ ($D_4$), $C_{86}$ ($C_2\times C_4$)

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

Order 8: $C_2\times C_4$ ($D_{43}$)

Order 4: $C_2^2$ ($D_{86}$)

Order 2: $C_2$ ($C_{43}:D_4$), $C_2$ ($C_4\times D_{43}$), $C_2$ ($C_{43}:Q_8$)

Order 1: $C_1$ ($C_{86}.D_4$)

Series

Derived series

$C_{86}.D_4$

$\rhd$

$C_{86}$

$\rhd$

$C_1$

Chief series

$C_{86}.D_4$

$\rhd$

$C_2\times C_{172}$

$\rhd$

$C_2\times C_{86}$

$\rhd$

$C_{86}$

$\rhd$

$C_{43}$

$\rhd$

$C_1$

Lower central series

$C_{86}.D_4$

$\rhd$

$C_{86}$

$\rhd$

$C_{43}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 26 larger groups in the database.

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

Character theory

Complex character table

See the $178 \times 178$ 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	2B	2C	4A	4B	4C	4D	43A	86A	86B	86C	172A
	Size	1	1	1	1	4	86	86	172	42	42	42	42	168
	2 P	1A	1A	1A	1A	2C	2A	2A	2C	43A	43A	43A	43A	86C
	43 P	1A	2A	2B	2C	4A	4B	4C	4D	43A	86A	86B	86C	172A
	Schur
688.11.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
688.11.1b		$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$
688.11.1c		$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$
688.11.1d		$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$
688.11.1e		$2$	$-2$	$2$	$-2$	$0$	$-2$	$2$	$0$	$2$	$2$	$-2$	$-2$	$0$
688.11.1f		$2$	$-2$	$2$	$-2$	$0$	$2$	$-2$	$0$	$2$	$2$	$-2$	$-2$	$0$
688.11.2a		$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$2$	$-2$	$-2$	$-2$	$0$
688.11.2b	2	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$2$	$-2$	$2$	$2$	$0$
688.11.2c		$42$	$42$	$42$	$42$	$42$	$0$	$0$	$0$	$-1$	$-1$	$-1$	$-1$	$-1$
688.11.2d		$42$	$42$	$42$	$42$	$-42$	$0$	$0$	$0$	$-1$	$-1$	$-1$	$-1$	$1$
688.11.2e		$84$	$-84$	$84$	$-84$	$0$	$0$	$0$	$0$	$-2$	$-2$	$2$	$2$	$0$
688.11.2f		$84$	$84$	$-84$	$-84$	$0$	$0$	$0$	$0$	$-2$	$2$	$2$	$2$	$0$
688.11.2g	2	$84$	$-84$	$-84$	$84$	$0$	$0$	$0$	$0$	$-2$	$2$	$-2$	$-2$	$0$