Abstract group 224.181: $Q_8\times D_{14}$

• Label: 224.181
• Order: \( 2^{5} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{2} \)
• Perm deg.: $17$
• Trans deg.: $112$
• Rank: $4$

Group information

Description:	$Q_8\times D_{14}$
Order:	\(224\)\(\medspace = 2^{5} \cdot 7 \)
Exponent:	\(28\)\(\medspace = 2^{2} \cdot 7 \)
Automorphism group:	$C_2\wr D_6.F_7$, of order \(32256\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	4	7	14	28
Elements	1	31	96	6	18	72	224
Conjugacy classes	1	7	12	3	9	18	50
Divisions	1	7	12	1	3	6	30
Autjugacy classes	1	3	2	1	2	1	10

Dimension	1	2	4	6	12
Irr. complex chars.	16	28	6	0	0	50
Irr. rational chars.	16	4	0	8	2	30

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$112$
Rank:	$4$
Inequivalent generating quadruples:	$23940$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid b^{2}=c^{4}=d^{14}=[a,b]=[a,d]=[b,c]=[c,d]=1, a^{2}=c^{2}, c^{a}=c^{3}, d^{b}=d^{13} \rangle$
Permutation group:	Degree $17$ $\langle(9,10)(12,13)(14,15)(16,17), (9,10), (1,2,4,6)(3,7,8,5), (1,3,4,8)(2,5,6,7)(9,10), (1,4)(2,6)(3,8)(5,7), (11,12,14,16,17,15,13)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 14 & 8 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 1 & 7 \\ 7 & 8 \end{array}\right), \left(\begin{array}{rr} 8 & 6 \\ 0 & 20 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/21\Z)$
Direct product:	$C_2$ $\, \times\, $ $Q_8$ $\, \times\, $ $D_7$
Semidirect product:	$(C_{14}:Q_8)$ $\,\rtimes\,$ $C_2$	$(Q_8\times C_{14})$ $\,\rtimes\,$ $C_2$	$C_{14}$ $\,\rtimes\,$ $(C_2\times Q_8)$	$(C_7:Q_8)$ $\,\rtimes\,$ $C_2^2$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{14}$ . $C_2^3$	$C_{28}$ . $C_2^3$	$C_{14}$ . $C_2^4$	$(C_4\times D_{14})$ . $C_2$	all 15

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 510 subgroups in 156 conjugacy classes, 97 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2\times D_{14}$
Commutator:	$G' \simeq$ $C_{14}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{14}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $Q_8\times D_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{14}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times Q_8$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

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 224: $Q_8\times D_{14}$

Order 112: $Q_8\times D_7$ x 8, $C_4\times D_{14}$ x 3, $C_{14}:Q_8$ x 3, $Q_8\times C_{14}$

Order 56: $C_4\times D_7$ x 12, $C_7:Q_8$ x 12, $C_7\times Q_8$ x 4, $C_2\times C_{28}$ x 3, $C_{14}:C_4$ x 3, $C_2\times D_{14}$

Order 32: $C_2^2\times Q_8$

Order 28: $D_{14}$ x 6, $C_{28}$ x 6, $C_7:C_4$ x 6, $C_2\times C_{14}$

Order 16: $C_2\times Q_8$ x 12, $C_2^2\times C_4$ x 3

Order 14: $D_7$ x 4, $C_{14}$ x 3

Order 8: $C_2\times C_4$ x 18, $Q_8$ x 16, $C_2^3$

Order 7: $C_7$

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 224: $Q_8\times D_{14}$

Order 112: $Q_8\times C_{14}$, $Q_8\times D_7$, $C_4\times D_{14}$, $C_{14}:Q_8$

Order 56: $C_2\times C_{28}$, $C_{14}:C_4$, $C_4\times D_7$, $C_7:Q_8$, $C_2\times D_{14}$, $C_7\times Q_8$

Order 32: $C_2^2\times Q_8$

Order 28: $D_{14}$ x 2, $C_2\times C_{14}$, $C_{28}$, $C_7:C_4$

Order 16: $C_2\times Q_8$ x 3, $C_2^2\times C_4$

Order 14: $C_{14}$ x 2, $D_7$

Order 8: $C_2\times C_4$ x 3, $Q_8$ x 2, $C_2^3$

Order 7: $C_7$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 224: $Q_8\times D_{14}$ ($C_1$)

Order 112: $Q_8\times D_7$ ($C_2$) x 8, $C_4\times D_{14}$ ($C_2$) x 3, $C_{14}:Q_8$ ($C_2$) x 3, $Q_8\times C_{14}$ ($C_2$)

Order 56: $C_4\times D_7$ ($C_2^2$) x 12, $C_7:Q_8$ ($C_2^2$) x 12, $C_7\times Q_8$ ($C_2^2$) x 4, $C_2\times C_{28}$ ($C_2^2$) x 3, $C_{14}:C_4$ ($C_2^2$) x 3, $C_2\times D_{14}$ ($C_2^2$)

Order 28: $C_{28}$ ($C_2^3$) x 6, $C_7:C_4$ ($C_2^3$) x 6, $D_{14}$ ($Q_8$) x 4, $D_{14}$ ($C_2^3$) x 2, $C_2\times C_{14}$ ($C_2^3$)

Order 16: $C_2\times Q_8$ ($D_7$)

Order 14: $D_7$ ($C_2\times Q_8$) x 4, $C_{14}$ ($C_2\times Q_8$) x 2, $C_{14}$ ($C_2^4$)

Order 8: $Q_8$ ($D_{14}$) x 4, $C_2\times C_4$ ($D_{14}$) x 3

Order 7: $C_7$ ($C_2^2\times Q_8$)

Order 4: $C_4$ ($C_2\times D_{14}$) x 6, $C_2^2$ ($C_2\times D_{14}$)

Order 2: $C_2$ ($Q_8\times D_7$) x 2, $C_2$ ($C_2^2\times D_{14}$)

Order 1: $C_1$ ($Q_8\times D_{14}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 224: $Q_8\times D_{14}$ ($C_1$)

Order 112: $Q_8\times C_{14}$ ($C_2$), $Q_8\times D_7$ ($C_2$), $C_4\times D_{14}$ ($C_2$), $C_{14}:Q_8$ ($C_2$)

Order 56: $C_2\times C_{28}$ ($C_2^2$), $C_{14}:C_4$ ($C_2^2$), $C_4\times D_7$ ($C_2^2$), $C_7:Q_8$ ($C_2^2$), $C_2\times D_{14}$ ($C_2^2$), $C_7\times Q_8$ ($C_2^2$)

Order 28: $C_2\times C_{14}$ ($C_2^3$), $D_{14}$ ($C_2^3$), $D_{14}$ ($Q_8$), $C_{28}$ ($C_2^3$), $C_7:C_4$ ($C_2^3$)

Order 16: $C_2\times Q_8$ ($D_7$)

Order 14: $C_{14}$ ($C_2^4$), $C_{14}$ ($C_2\times Q_8$), $D_7$ ($C_2\times Q_8$)

Order 8: $Q_8$ ($D_{14}$), $C_2\times C_4$ ($D_{14}$)

Order 7: $C_7$ ($C_2^2\times Q_8$)

Order 4: $C_2^2$ ($C_2\times D_{14}$), $C_4$ ($C_2\times D_{14}$)

Order 2: $C_2$ ($C_2^2\times D_{14}$), $C_2$ ($Q_8\times D_7$)

Order 1: $C_1$ ($Q_8\times D_{14}$)

Series

Derived series

$Q_8\times D_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_1$

Chief series

$Q_8\times D_{14}$

$\rhd$

$Q_8\times D_7$

$\rhd$

$C_4\times D_7$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$Q_8\times D_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times Q_8$

Supergroups

This group is a maximal subgroup of 52 larger groups in the database.

This group is a maximal quotient of 53 larger groups in the database.

Character theory

Complex character table

See the $50 \times 50$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.