Abstract group 1624.44: $D_{58}:C_{14}$

• Label: 1624.44
• Order: \( 2^{3} \cdot 7 \cdot 29 \)
• Exponent: \( 2 \cdot 7 \cdot 29 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 7 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 7 \cdot 29 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $33$
• Trans deg.: $116$
• Rank: $3$

Group information

Description:	$D_{58}:C_{14}$
Order:	\(1624\)\(\medspace = 2^{3} \cdot 7 \cdot 29 \)
Exponent:	\(406\)\(\medspace = 2 \cdot 7 \cdot 29 \)
Automorphism group:	$S_4\times F_{29}$, of order \(19488\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 29 \)
Composition factors:	$C_2$ x 3, $C_7$, $C_{29}$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Group statistics

Order	1	2	7	14	29	58
Elements	1	119	174	1218	28	84	1624
Conjugacy classes	1	7	6	42	2	6	64
Divisions	1	7	1	7	1	3	20
Autjugacy classes	1	2	6	12	1	1	23

Dimension	1	6	14	28
Irr. complex chars.	56	0	8	0	64
Irr. rational chars.	8	8	0	4	20

Minimal presentations

Permutation degree:	$33$
Transitive degree:	$116$
Rank:	$3$
Inequivalent generating triples:	$71820$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	15	15	29

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{14}=c^{58}=[a,b]=[a,c]=1, c^{b}=c^{51} \rangle$
Permutation group:	Degree $33$ $\langle(2,3)(4,6)(5,9)(7,12)(8,15)(10,16)(11,18)(13,22)(14,20)(17,25)(19,24)(21,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 57 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 29 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 57 & 0 \\ 0 & 57 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/58\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_{29}:C_{14})$
Semidirect product:	$D_{58}$ $\,\rtimes\,$ $C_{14}$ (6)	$(C_2\times D_{58})$ $\,\rtimes\,$ $C_7$	$D_{29}$ $\,\rtimes\,$ $(C_2\times C_{14})$ (4)	$(C_{58}:C_{14})$ $\,\rtimes\,$ $C_2$	all 9
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:	$C_{2}^{2} \times C_{14} \simeq C_{2}^{3} \times C_{7}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 820 subgroups in 64 conjugacy classes, 37 normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_{29}:C_{14}$
Commutator:	$G' \simeq$ $C_{29}$	$G/G' \simeq$ $C_2^2\times C_{14}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_{58}:C_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{58}$	$G/\operatorname{Fit} \simeq$ $C_{14}$
Radical:	$R \simeq$ $D_{58}:C_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{58}$	$G/\operatorname{soc} \simeq$ $C_{14}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
29-Sylow subgroup:	$P_{ 29 } \simeq$ $C_{29}$

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

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Classes of subgroups up to conjugation

Order 1624: $D_{58}:C_{14}$

Order 812: $C_{58}:C_{14}$ x 6, $C_{58}:C_{14}$

Order 406: $C_{29}:C_{14}$ x 4, $C_{29}:C_{14}$ x 3

Order 232: $C_2\times D_{58}$

Order 203: $C_{29}:C_7$

Order 116: $D_{58}$ x 6, $C_2\times C_{58}$

Order 58: $D_{29}$ x 4, $C_{58}$ x 3

Order 56: $C_2^2\times C_{14}$

Order 29: $C_{29}$

Order 28: $C_2\times C_{14}$ x 7

Order 14: $C_{14}$ x 7

Order 8: $C_2^3$

Order 7: $C_7$

Order 4: $C_2^2$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1624: $D_{58}:C_{14}$

Order 812: $C_{58}:C_{14}$, $C_{58}:C_{14}$

Order 406: $C_{29}:C_{14}$, $C_{29}:C_{14}$

Order 232: $C_2\times D_{58}$

Order 203: $C_{29}:C_7$

Order 116: $C_2\times C_{58}$, $D_{58}$

Order 58: $C_{58}$, $D_{29}$

Order 56: $C_2^2\times C_{14}$

Order 29: $C_{29}$

Order 28: $C_2\times C_{14}$ x 2

Order 14: $C_{14}$ x 2

Order 8: $C_2^3$

Order 7: $C_7$

Order 4: $C_2^2$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1624: $D_{58}:C_{14}$ ($C_1$)

Order 812: $C_{58}:C_{14}$ ($C_2$) x 6, $C_{58}:C_{14}$ ($C_2$)

Order 406: $C_{29}:C_{14}$ ($C_2^2$) x 4, $C_{29}:C_{14}$ ($C_2^2$) x 3

Order 232: $C_2\times D_{58}$ ($C_7$)

Order 203: $C_{29}:C_7$ ($C_2^3$)

Order 116: $D_{58}$ ($C_{14}$) x 6, $C_2\times C_{58}$ ($C_{14}$)

Order 58: $D_{29}$ ($C_2\times C_{14}$) x 4, $C_{58}$ ($C_2\times C_{14}$) x 3

Order 29: $C_{29}$ ($C_2^2\times C_{14}$)

Order 4: $C_2^2$ ($C_{29}:C_{14}$)

Order 2: $C_2$ ($C_{58}:C_{14}$) x 3

Order 1: $C_1$ ($D_{58}:C_{14}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1624: $D_{58}:C_{14}$ ($C_1$)

Order 812: $C_{58}:C_{14}$ ($C_2$), $C_{58}:C_{14}$ ($C_2$)

Order 406: $C_{29}:C_{14}$ ($C_2^2$), $C_{29}:C_{14}$ ($C_2^2$)

Order 232: $C_2\times D_{58}$ ($C_7$)

Order 203: $C_{29}:C_7$ ($C_2^3$)

Order 116: $C_2\times C_{58}$ ($C_{14}$), $D_{58}$ ($C_{14}$)

Order 58: $C_{58}$ ($C_2\times C_{14}$), $D_{29}$ ($C_2\times C_{14}$)

Order 29: $C_{29}$ ($C_2^2\times C_{14}$)

Order 4: $C_2^2$ ($C_{29}:C_{14}$)

Order 2: $C_2$ ($C_{58}:C_{14}$)

Order 1: $C_1$ ($D_{58}:C_{14}$)

Series

Derived series

$D_{58}:C_{14}$

$\rhd$

$C_{29}$

$\rhd$

$C_1$

Chief series

$D_{58}:C_{14}$

$\rhd$

$C_{58}:C_{14}$

$\rhd$

$C_{29}:C_{14}$

$\rhd$

$C_{58}$

$\rhd$

$C_{29}$

$\rhd$

$C_1$

Lower central series

$D_{58}:C_{14}$

$\rhd$

$C_{29}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $64 \times 64$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.