Abstract group 688.31: $D_4\times D_{43}$

• Label: 688.31
• Order: \( 2^{4} \cdot 43 \)
• Exponent: \( 2^{2} \cdot 43 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 7 \cdot 43 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \cdot 7 \)
• Perm deg.: $47$
• Trans deg.: $172$
• Rank: $3$

Group information

Description:	$D_4\times D_{43}$
Order:	\(688\)\(\medspace = 2^{4} \cdot 43 \)
Exponent:	\(172\)\(\medspace = 2^{2} \cdot 43 \)
Automorphism group:	$C_{86}.C_{42}.C_2^3$, 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	263	88	42	210	84	688
Conjugacy classes	1	7	2	21	63	21	115
Divisions	1	7	2	1	3	1	15
Autjugacy classes	1	4	2	1	2	1	11

Dimension	1	2	4	42	84
Irr. complex chars.	8	86	21	0	0	115
Irr. rational chars.	8	2	0	4	1	15

Minimal presentations

Permutation degree:	$47$
Transitive degree:	$172$
Rank:	$3$
Inequivalent generating triples:	$3696$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	4	84
Arbitrary	4	4	44

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{172}=[a,b]=1, c^{a}=c^{85}, c^{b}=c^{87} \rangle$
Permutation group:	Degree $47$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$
Direct product:	$D_4$ $\, \times\, $ $D_{43}$
Semidirect product:	$C_4$ $\,\rtimes\,$ $D_{86}$	$D_{172}$ $\,\rtimes\,$ $C_2$	$D_{86}$ $\,\rtimes\,$ $C_2^2$ (2)	$C_2^2$ $\,\rtimes\,$ $D_{86}$ (2)	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{86}$ . $C_2^2$	$C_{86}$ . $C_2^3$	$C_2$ . $(C_2\times D_{86})$		more information

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 1120 subgroups in 54 conjugacy classes, 25 normal (13 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2\times D_{86}$
Commutator:	$G' \simeq$ $C_{86}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times D_{86}$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_{43}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_4\times D_{43}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{86}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
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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Subgroup information

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

Classes of subgroups up to conjugation

Order 688: $D_4\times D_{43}$

Order 344: $C_{43}:D_4$ x 2, $C_2\times D_{86}$ x 2, $D_4\times C_{43}$, $D_{172}$, $C_4\times D_{43}$

Order 172: $D_{86}$ x 7, $C_2\times C_{86}$ x 2, $C_{172}$, $C_{43}:C_4$

Order 86: $D_{43}$ x 4, $C_{86}$ x 3

Order 43: $C_{43}$

Order 16: $C_2\times D_4$

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 688: $D_4\times D_{43}$

Order 344: $D_4\times C_{43}$, $C_{43}:D_4$, $D_{172}$, $C_4\times D_{43}$, $C_2\times D_{86}$

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

Order 86: $C_{86}$ x 2, $D_{43}$ x 2

Order 43: $C_{43}$

Order 16: $C_2\times D_4$

Order 8: $D_4$ x 3, $C_2^3$, $C_2\times C_4$

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

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 688: $D_4\times D_{43}$ ($C_1$)

Order 344: $C_{43}:D_4$ ($C_2$) x 2, $C_2\times D_{86}$ ($C_2$) x 2, $D_4\times C_{43}$ ($C_2$), $D_{172}$ ($C_2$), $C_4\times D_{43}$ ($C_2$)

Order 172: $D_{86}$ ($C_2^2$) x 3, $C_2\times C_{86}$ ($C_2^2$) x 2, $C_{172}$ ($C_2^2$), $C_{43}:C_4$ ($C_2^2$)

Order 86: $D_{43}$ ($D_4$) x 2, $C_{86}$ ($C_2^3$)

Order 43: $C_{43}$ ($C_2\times D_4$)

Order 8: $D_4$ ($D_{43}$)

Order 4: $C_2^2$ ($D_{86}$) x 2, $C_4$ ($D_{86}$)

Order 2: $C_2$ ($C_2\times D_{86}$)

Order 1: $C_1$ ($D_4\times D_{43}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 688: $D_4\times D_{43}$ ($C_1$)

Order 344: $D_4\times C_{43}$ ($C_2$), $C_{43}:D_4$ ($C_2$), $D_{172}$ ($C_2$), $C_4\times D_{43}$ ($C_2$), $C_2\times D_{86}$ ($C_2$)

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

Order 86: $C_{86}$ ($C_2^3$), $D_{43}$ ($D_4$)

Order 43: $C_{43}$ ($C_2\times D_4$)

Order 8: $D_4$ ($D_{43}$)

Order 4: $C_2^2$ ($D_{86}$), $C_4$ ($D_{86}$)

Order 2: $C_2$ ($C_2\times D_{86}$)

Order 1: $C_1$ ($D_4\times D_{43}$)

Series

Derived series

$D_4\times D_{43}$

$\rhd$

$C_{86}$

$\rhd$

$C_1$

Chief series

$D_4\times D_{43}$

$\rhd$

$C_2\times D_{86}$

$\rhd$

$D_{86}$

$\rhd$

$C_{86}$

$\rhd$

$C_{43}$

$\rhd$

$C_1$

Lower central series

$D_4\times D_{43}$

$\rhd$

$C_{86}$

$\rhd$

$C_{43}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$D_4$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 29 larger groups in the database.

Character theory

Complex character table

See the $115 \times 115$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	2D	2E	2F	2G	4A	4B	43A	86A	86B	86C	172A
	Size	1	1	2	2	43	43	86	86	2	86	42	42	84	84	84
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	2A	2A	43A	43A	43A	43A	86A
	43 P	1A	2A	2B	2C	2D	2E	2F	2G	4A	4B	43A	86A	86B	86C	172A
688.31.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
688.31.1b		$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$
688.31.1c		$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$
688.31.1d		$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$
688.31.1e		$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
688.31.1f		$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$
688.31.1g		$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
688.31.1h		$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$
688.31.2a		$2$	$-2$	$0$	$0$	$-2$	$2$	$0$	$0$	$0$	$0$	$2$	$-2$	$0$	$0$	$0$
688.31.2b		$2$	$-2$	$0$	$0$	$2$	$-2$	$0$	$0$	$0$	$0$	$2$	$-2$	$0$	$0$	$0$
688.31.2c		$42$	$42$	$42$	$42$	$0$	$0$	$0$	$0$	$42$	$0$	$-1$	$-1$	$-1$	$-1$	$-1$
688.31.2d		$42$	$42$	$-42$	$-42$	$0$	$0$	$0$	$0$	$42$	$0$	$-1$	$-1$	$1$	$1$	$-1$
688.31.2e		$42$	$42$	$-42$	$42$	$0$	$0$	$0$	$0$	$-42$	$0$	$-1$	$-1$	$-1$	$1$	$1$
688.31.2f		$42$	$42$	$42$	$-42$	$0$	$0$	$0$	$0$	$-42$	$0$	$-1$	$-1$	$1$	$-1$	$1$
688.31.4a		$84$	$-84$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$2$	$0$	$0$	$0$