Abstract group 6496.d: $D_4\times F_{29}$

• Label: 6496.d
• Order: \( 2^{5} \cdot 7 \cdot 29 \)
• Exponent: \( 2^{2} \cdot 7 \cdot 29 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 7 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 7 \cdot 29 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $33$
• Trans deg.: $116$
• Rank: $3$

Group information

Description:	$D_4\times F_{29}$
Order:	\(6496\)\(\medspace = 2^{5} \cdot 7 \cdot 29 \)
Exponent:	\(812\)\(\medspace = 2^{2} \cdot 7 \cdot 29 \)
Automorphism group:	$C_{58}.C_{14}.C_2^4$, of order \(12992\)\(\medspace = 2^{6} \cdot 7 \cdot 29 \)
Composition factors:	$C_2$ x 5, $C_7$, $C_{29}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	7	14	28	29	58	116
Elements	1	179	524	174	1914	3480	28	140	56	6496
Conjugacy classes	1	7	12	6	42	72	1	3	1	145
Divisions	1	7	7	1	7	7	1	3	1	35
Autjugacy classes	1	5	8	6	30	48	1	2	1	102

Dimension	1	2	4	6	12	24	28	56
Irr. complex chars.	112	28	0	0	0	0	4	1	145
Irr. rational chars.	8	6	1	8	6	1	4	1	35

Minimal presentations

Permutation degree:	$33$
Transitive degree:	$116$
Rank:	$3$
Inequivalent generating triples:	$6894720$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	56	56	56
Arbitrary	30	30	30

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{28}=c^{116}=[a,b]=1, c^{a}=c^{115}, c^{b}=c^{77} \rangle$
Permutation group:	Degree $33$ $\langle(2,3)(4,6)(5,9)(7,12)(8,10)(11,17)(13,20)(14,15)(16,25)(18,27)(19,22)(21,28) \!\cdots\! \rangle$
Direct product:	$D_4$ $\, \times\, $ $F_{29}$
Semidirect product:	$D_{116}$ $\,\rtimes\,$ $C_{28}$	$(C_4\times F_{29})$ $\,\rtimes\,$ $C_2$	$C_4$ $\,\rtimes\,$ $(C_2\times F_{29})$	$(D_{58}:C_{14})$ $\,\rtimes\,$ $C_4$ (2)	all 26
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_4\times D_{29})$ . $C_{14}$	$D_{29}$ . $(D_4\times C_{14})$	$D_{29}$ . $(D_4:C_{14})$	$(C_2\times F_{29})$ . $C_2^2$ (4)	all 17
Aut. group:	$\Aut(C_{29}:Q_8)$	$\Aut(D_{116})$	$\Aut(C_{58}:C_4)$	$\Aut(D_{29}:C_8)$	all 20

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

Homology

Abelianization:	$C_{2}^{2} \times C_{28} \simeq C_{2}^{2} \times C_{4} \times C_{7}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 3440 subgroups in 188 conjugacy classes, 70 normal (36 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^2\times F_{29}$
Commutator:	$G' \simeq$ $C_{58}$	$G/G' \simeq$ $C_2^2\times C_{28}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times F_{29}$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_{29}$	$G/\operatorname{Fit} \simeq$ $C_{28}$
Radical:	$R \simeq$ $D_4\times F_{29}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{58}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_{28}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times D_4$
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

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

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.

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

Subgroup information

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

Classes of subgroups up to conjugation

Order 6496: $D_4\times F_{29}$

Order 3248: $C_{29}:(C_7\times C_2^2:C_4)$ x 2, $C_2^2\times F_{29}$ x 2, $C_{29}:(C_7\times C_4:C_4)$, $C_{29}:(C_{14}\times D_4)$, $C_4\times F_{29}$

Order 1624: $C_2\times F_{29}$ x 8, $D_{58}:C_{14}$ x 2, $D_{58}:C_{14}$ x 2, $C_{116}:C_{14}$, $C_{116}:C_{14}$, $C_{116}:C_{14}$

Order 928: $D_{116}:C_4$

Order 812: $C_{58}:C_{14}$ x 7, $F_{29}$ x 5, $C_{58}:C_{14}$ x 2, $C_{116}:C_7$, $C_{29}:C_{28}$

Order 464: $D_{58}:C_4$ x 2, $D_{58}:C_4$ x 2, $D_4\times D_{29}$, $C_{116}:C_4$, $C_{116}:C_4$

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

Order 232: $C_{58}:C_4$ x 8, $C_{29}:D_4$ x 2, $C_2\times D_{58}$ x 2, $D_{116}$, $C_4\times D_{29}$, $D_4\times C_{29}$

Order 224: $D_4\times C_{28}$

Order 203: $C_{29}:C_7$

Order 116: $D_{58}$ x 7, $C_{29}:C_4$ x 5, $C_2\times C_{58}$ x 2, $C_{116}$, $C_{29}:C_4$

Order 112: $C_2^2\times C_{28}$ x 2, $C_2^2:C_{28}$ x 2, $D_4\times C_{14}$, $C_4:C_{28}$, $C_4\times C_{28}$

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

Order 56: $C_2\times C_{28}$ x 9, $C_7\times D_4$ x 4, $C_2^2\times C_{14}$ x 2

Order 32: $C_4\times D_4$

Order 29: $C_{29}$

Order 28: $C_2\times C_{14}$ x 9, $C_{28}$ x 7

Order 16: $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2\times D_4$

Order 14: $C_{14}$ x 7

Order 8: $C_2\times C_4$ x 9, $D_4$ x 4, $C_2^3$ x 2

Order 7: $C_7$

Order 4: $C_2^2$ x 9, $C_4$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 6496: $D_4\times F_{29}$

Order 3248: $C_{29}:(C_7\times C_2^2:C_4)$, $C_{29}:(C_7\times C_4:C_4)$, $C_{29}:(C_{14}\times D_4)$, $C_2^2\times F_{29}$, $C_4\times F_{29}$

Order 1624: $C_2\times F_{29}$ x 4, $C_{116}:C_{14}$, $D_{58}:C_{14}$, $C_{116}:C_{14}$, $D_{58}:C_{14}$, $C_{116}:C_{14}$

Order 928: $D_{116}:C_4$

Order 812: $C_{58}:C_{14}$ x 4, $F_{29}$ x 3, $C_{58}:C_{14}$, $C_{116}:C_7$, $C_{29}:C_{28}$

Order 464: $D_{58}:C_4$, $D_4\times D_{29}$, $D_{58}:C_4$, $C_{116}:C_4$, $C_{116}:C_4$

Order 406: $C_{29}:C_{14}$ x 3, $C_{29}:C_{14}$ x 2

Order 232: $C_{58}:C_4$ x 4, $C_{29}:D_4$, $D_{116}$, $C_4\times D_{29}$, $C_2\times D_{58}$, $D_4\times C_{29}$

Order 224: $D_4\times C_{28}$

Order 203: $C_{29}:C_7$

Order 116: $D_{58}$ x 4, $C_{29}:C_4$ x 3, $C_2\times C_{58}$, $C_{116}$, $C_{29}:C_4$

Order 112: $D_4\times C_{14}$, $C_2^2\times C_{28}$, $C_4:C_{28}$, $C_2^2:C_{28}$, $C_4\times C_{28}$

Order 58: $D_{29}$ x 3, $C_{58}$ x 2

Order 56: $C_2\times C_{28}$ x 5, $C_7\times D_4$ x 3, $C_2^2\times C_{14}$

Order 32: $C_4\times D_4$

Order 29: $C_{29}$

Order 28: $C_2\times C_{14}$ x 5, $C_{28}$ x 5

Order 16: $C_4:C_4$, $C_2^2:C_4$, $C_4^2$, $C_2\times D_4$, $C_2^2\times C_4$

Order 14: $C_{14}$ x 5

Order 8: $C_2\times C_4$ x 5, $D_4$ x 3, $C_2^3$

Order 7: $C_7$

Order 4: $C_2^2$ x 5, $C_4$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6496: $D_4\times F_{29}$ ($C_1$)

Order 3248: $C_{29}:(C_7\times C_2^2:C_4)$ ($C_2$) x 2, $C_2^2\times F_{29}$ ($C_2$) x 2, $C_{29}:(C_7\times C_4:C_4)$ ($C_2$), $C_{29}:(C_{14}\times D_4)$ ($C_2$), $C_4\times F_{29}$ ($C_2$)

Order 1624: $C_2\times F_{29}$ ($C_2^2$) x 4, $D_{58}:C_{14}$ ($C_2^2$) x 2, $D_{58}:C_{14}$ ($C_4$) x 2, $C_{116}:C_{14}$ ($C_2^2$), $C_{116}:C_{14}$ ($C_4$), $C_{116}:C_{14}$ ($C_4$)

Order 928: $D_{116}:C_4$ ($C_7$)

Order 812: $C_{58}:C_{14}$ ($C_2\times C_4$) x 2, $C_{58}:C_{14}$ ($C_2\times C_4$) x 2, $F_{29}$ ($D_4$) x 2, $C_{58}:C_{14}$ ($C_2^3$), $C_{116}:C_7$ ($C_2\times C_4$), $C_{29}:C_{28}$ ($C_2\times C_4$)

Order 464: $D_{58}:C_4$ ($C_{14}$) x 2, $D_{58}:C_4$ ($C_{14}$) x 2, $D_4\times D_{29}$ ($C_{14}$), $C_{116}:C_4$ ($C_{14}$), $C_{116}:C_4$ ($C_{14}$)

Order 406: $C_{29}:C_{14}$ ($C_2^2\times C_4$), $C_{29}:C_{14}$ ($D_4:C_2$), $C_{29}:C_{14}$ ($C_2\times D_4$)

Order 232: $C_{58}:C_4$ ($C_2\times C_{14}$) x 4, $C_{29}:D_4$ ($C_{28}$) x 2, $C_2\times D_{58}$ ($C_2\times C_{14}$) x 2, $D_{116}$ ($C_{28}$), $C_4\times D_{29}$ ($C_2\times C_{14}$), $D_4\times C_{29}$ ($C_{28}$)

Order 203: $C_{29}:C_7$ ($C_4\times D_4$)

Order 116: $C_2\times C_{58}$ ($C_2\times C_{28}$) x 2, $D_{58}$ ($C_2\times C_{28}$) x 2, $C_{29}:C_4$ ($C_7\times D_4$) x 2, $D_{58}$ ($C_2^2\times C_{14}$), $C_{116}$ ($C_2\times C_{28}$), $C_{29}:C_4$ ($C_2\times C_{28}$)

Order 58: $C_{58}$ ($C_2^2\times C_{28}$), $D_{29}$ ($D_4:C_{14}$), $D_{29}$ ($D_4\times C_{14}$)

Order 29: $C_{29}$ ($D_4\times C_{28}$)

Order 8: $D_4$ ($F_{29}$)

Order 4: $C_2^2$ ($C_2\times F_{29}$) x 2, $C_4$ ($C_2\times F_{29}$)

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

Order 1: $C_1$ ($D_4\times F_{29}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6496: $D_4\times F_{29}$ ($C_1$)

Order 3248: $C_{29}:(C_7\times C_2^2:C_4)$ ($C_2$), $C_{29}:(C_7\times C_4:C_4)$ ($C_2$), $C_{29}:(C_{14}\times D_4)$ ($C_2$), $C_2^2\times F_{29}$ ($C_2$), $C_4\times F_{29}$ ($C_2$)

Order 1624: $C_2\times F_{29}$ ($C_2^2$) x 3, $C_{116}:C_{14}$ ($C_2^2$), $D_{58}:C_{14}$ ($C_2^2$), $C_{116}:C_{14}$ ($C_4$), $D_{58}:C_{14}$ ($C_4$), $C_{116}:C_{14}$ ($C_4$)

Order 928: $D_{116}:C_4$ ($C_7$)

Order 812: $C_{58}:C_{14}$ ($C_2\times C_4$), $C_{58}:C_{14}$ ($C_2^3$), $C_{58}:C_{14}$ ($C_2\times C_4$), $F_{29}$ ($D_4$), $C_{116}:C_7$ ($C_2\times C_4$), $C_{29}:C_{28}$ ($C_2\times C_4$)

Order 464: $D_{58}:C_4$ ($C_{14}$), $D_4\times D_{29}$ ($C_{14}$), $D_{58}:C_4$ ($C_{14}$), $C_{116}:C_4$ ($C_{14}$), $C_{116}:C_4$ ($C_{14}$)

Order 406: $C_{29}:C_{14}$ ($C_2^2\times C_4$), $C_{29}:C_{14}$ ($D_4:C_2$), $C_{29}:C_{14}$ ($C_2\times D_4$)

Order 232: $C_{58}:C_4$ ($C_2\times C_{14}$) x 3, $C_{29}:D_4$ ($C_{28}$), $D_{116}$ ($C_{28}$), $C_4\times D_{29}$ ($C_2\times C_{14}$), $C_2\times D_{58}$ ($C_2\times C_{14}$), $D_4\times C_{29}$ ($C_{28}$)

Order 203: $C_{29}:C_7$ ($C_4\times D_4$)

Order 116: $C_2\times C_{58}$ ($C_2\times C_{28}$), $D_{58}$ ($C_2\times C_{28}$), $D_{58}$ ($C_2^2\times C_{14}$), $C_{29}:C_4$ ($C_7\times D_4$), $C_{116}$ ($C_2\times C_{28}$), $C_{29}:C_4$ ($C_2\times C_{28}$)

Order 58: $C_{58}$ ($C_2^2\times C_{28}$), $D_{29}$ ($D_4:C_{14}$), $D_{29}$ ($D_4\times C_{14}$)

Order 29: $C_{29}$ ($D_4\times C_{28}$)

Order 8: $D_4$ ($F_{29}$)

Order 4: $C_2^2$ ($C_2\times F_{29}$), $C_4$ ($C_2\times F_{29}$)

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

Order 1: $C_1$ ($D_4\times F_{29}$)

Series

Derived series

$D_4\times F_{29}$

$\rhd$

$C_{58}$

$\rhd$

$C_1$

Chief series

$D_4\times F_{29}$

$\rhd$

$C_{29}:(C_{14}\times D_4)$

$\rhd$

$C_{116}:C_{14}$

$\rhd$

$C_{116}:C_7$

$\rhd$

$C_{116}$

$\rhd$

$C_{58}$

$\rhd$

$C_{29}$

$\rhd$

$C_1$

Lower central series

$D_4\times F_{29}$

$\rhd$

$C_{58}$

$\rhd$

$C_{29}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$D_4$

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 $145 \times 145$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $35 \times 35$ rational character table.