Abstract group 224.100: $C_{14}:Q_{16}$

• Label: 224.100
• Order: \( 2^{5} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $25$
• Trans deg.: $224$
• Rank: $3$

Group information

Description:	$C_{14}:Q_{16}$
Order:	\(224\)\(\medspace = 2^{5} \cdot 7 \)
Exponent:	\(56\)\(\medspace = 2^{3} \cdot 7 \)
Automorphism group:	$C_{28}.(C_6\times D_4^2)$, of order \(10752\)\(\medspace = 2^{9} \cdot 3 \cdot 7 \)
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	8	14	28	56
Elements	1	3	116	6	8	18	24	48	224
Conjugacy classes	1	3	6	3	4	9	12	24	62
Divisions	1	3	6	1	2	3	2	2	20
Autjugacy classes	1	2	3	1	1	2	2	1	13

Dimension	1	2	4	6	12	24
Irr. complex chars.	8	54	0	0	0	0	62
Irr. rational chars.	8	2	2	4	2	2	20

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$224$
Rank:	$3$
Inequivalent generating triples:	$336$

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:	$\langle a, b, c \mid b^{56}=c^{2}=[a,c]=[b,c]=1, a^{2}=b^{28}, b^{a}=b^{55} \rangle$
Permutation group:	Degree $25$ $\langle(2,3)(4,5)(6,7)(10,11,12,15)(13,20,18,25)(14,22,19,17)(16,24,23,21), (8,9) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 64 & 0 \\ 0 & 64 \end{array}\right), \left(\begin{array}{rr} 66 & 13 \\ 52 & 53 \end{array}\right), \left(\begin{array}{rr} 64 & 14 \\ 42 & 22 \end{array}\right), \left(\begin{array}{rr} 46 & 60 \\ 85 & 45 \end{array}\right), \left(\begin{array}{rr} 90 & 65 \\ 78 & 25 \end{array}\right), \left(\begin{array}{rr} 27 & 0 \\ 0 & 27 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/91\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_7:Q_{16})$
Semidirect product:	$C_{14}$ $\,\rtimes\,$ $Q_{16}$ (2)	$C_7$ $\,\rtimes\,$ $(C_2\times Q_{16})$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{28}$ . $D_4$	$C_8$ . $D_{14}$ (2)	$C_4$ . $D_{28}$	$C_2^2$ . $D_{28}$	all 16

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

Homology

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

Subgroups

There are 238 subgroups in 60 conjugacy classes, 33 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_{28}$
Commutator:	$G' \simeq$ $C_{28}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{56}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{14}:Q_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{14}$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times Q_{16}$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 224: $C_{14}:Q_{16}$

Order 112: $C_7:Q_{16}$ x 4, $C_{14}:Q_8$ x 2, $C_2\times C_{56}$

Order 56: $C_7:Q_8$ x 6, $C_{14}:C_4$ x 2, $C_{56}$ x 2, $C_2\times C_{28}$

Order 32: $C_2\times Q_{16}$

Order 28: $C_7:C_4$ x 4, $C_{28}$ x 2, $C_2\times C_{14}$

Order 16: $Q_{16}$ x 4, $C_2\times Q_8$ x 2, $C_2\times C_8$

Order 14: $C_{14}$ x 3

Order 8: $Q_8$ x 6, $C_2\times C_4$ x 3, $C_8$ x 2

Order 7: $C_7$

Order 4: $C_4$ x 6, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 224: $C_{14}:Q_{16}$

Order 112: $C_7:Q_{16}$, $C_{14}:Q_8$, $C_2\times C_{56}$

Order 56: $C_7:Q_8$ x 2, $C_2\times C_{28}$, $C_{14}:C_4$, $C_{56}$

Order 32: $C_2\times Q_{16}$

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

Order 16: $Q_{16}$, $C_2\times C_8$, $C_2\times Q_8$

Order 14: $C_{14}$ x 2

Order 8: $Q_8$ x 2, $C_2\times C_4$ x 2, $C_8$

Order 7: $C_7$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 224: $C_{14}:Q_{16}$ ($C_1$)

Order 112: $C_7:Q_{16}$ ($C_2$) x 4, $C_{14}:Q_8$ ($C_2$) x 2, $C_2\times C_{56}$ ($C_2$)

Order 56: $C_7:Q_8$ ($C_2^2$) x 4, $C_{56}$ ($C_2^2$) x 2, $C_2\times C_{28}$ ($C_2^2$)

Order 28: $C_2\times C_{14}$ ($D_4$), $C_{28}$ ($C_2^3$), $C_{28}$ ($D_4$)

Order 16: $C_2\times C_8$ ($D_7$)

Order 14: $C_{14}$ ($Q_{16}$) x 2, $C_{14}$ ($C_2\times D_4$)

Order 8: $C_8$ ($D_{14}$) x 2, $C_2\times C_4$ ($D_{14}$)

Order 7: $C_7$ ($C_2\times Q_{16}$)

Order 4: $C_2^2$ ($D_{28}$), $C_4$ ($D_{28}$), $C_4$ ($C_2\times D_{14}$)

Order 2: $C_2$ ($C_7:Q_{16}$) x 2, $C_2$ ($C_2\times D_{28}$)

Order 1: $C_1$ ($C_{14}:Q_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 224: $C_{14}:Q_{16}$ ($C_1$)

Order 112: $C_7:Q_{16}$ ($C_2$), $C_{14}:Q_8$ ($C_2$), $C_2\times C_{56}$ ($C_2$)

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

Order 28: $C_2\times C_{14}$ ($D_4$), $C_{28}$ ($C_2^3$), $C_{28}$ ($D_4$)

Order 16: $C_2\times C_8$ ($D_7$)

Order 14: $C_{14}$ ($Q_{16}$), $C_{14}$ ($C_2\times D_4$)

Order 8: $C_2\times C_4$ ($D_{14}$), $C_8$ ($D_{14}$)

Order 7: $C_7$ ($C_2\times Q_{16}$)

Order 4: $C_2^2$ ($D_{28}$), $C_4$ ($D_{28}$), $C_4$ ($C_2\times D_{14}$)

Order 2: $C_2$ ($C_7:Q_{16}$), $C_2$ ($C_2\times D_{28}$)

Order 1: $C_1$ ($C_{14}:Q_{16}$)

Series

Derived series

$C_{14}:Q_{16}$

$\rhd$

$C_{28}$

$\rhd$

$C_1$

Chief series

$C_{14}:Q_{16}$

$\rhd$

$C_{14}:Q_8$

$\rhd$

$C_2\times C_{28}$

$\rhd$

$C_2\times C_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_{14}:Q_{16}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2\times C_8$

Supergroups

This group is a maximal subgroup of 56 larger groups in the database.

This group is a maximal quotient of 25 larger groups in the database.

Character theory

Complex character table

See the $62 \times 62$ 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.