Abstract group 784.136: $C_{28}.D_{14}$

• Label: 784.136
• Order: \( 2^{4} \cdot 7^{2} \)
• Exponent: \( 2^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 7 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 7 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{2} \)
• Perm deg.: $24$
• Trans deg.: $112$
• Rank: $3$

Group information

Description:	$C_{28}.D_{14}$
Order:	\(784\)\(\medspace = 2^{4} \cdot 7^{2} \)
Exponent:	\(28\)\(\medspace = 2^{2} \cdot 7 \)
Automorphism group:	$C_{14}.(C_6^2\times D_4).C_2^2$, of order \(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_7$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	4	7	14	28
Elements	1	3	60	48	144	528	784
Conjugacy classes	1	3	6	27	81	120	238
Divisions	1	3	6	5	15	14	44
Autjugacy classes	1	2	2	3	6	4	18

Dimension	1	2	6	12	24
Irr. complex chars.	56	182	0	0	0	238
Irr. rational chars.	8	2	12	16	6	44

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$112$
Rank:	$3$
Inequivalent generating triples:	$9576$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	5	16

Constructions

Presentation:	$\langle a, b, c \mid a^{14}=c^{28}=[a,b]=[a,c]=1, b^{2}=c^{14}, c^{b}=c^{27} \rangle$
Permutation group:	Degree $24$ $\langle(2,3)(4,5)(6,7)(8,9,10,12)(11,15,14,13), (8,10)(9,12)(11,14)(13,15)(23,24) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 29 & 22 \\ 54 & 29 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 29 & 50 \\ 40 & 29 \end{array}\right), \left(\begin{array}{rr} 1 & 29 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 57 & 0 \\ 0 & 57 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/58\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_7$ $\, \times\, $ $(C_7:Q_8)$
Semidirect product:	$(C_7\times C_{14})$ $\,\rtimes\,$ $Q_8$ (2)	$C_{14}$ $\,\rtimes\,$ $(C_7\times Q_8)$ (2)	$C_7$ $\,\rtimes\,$ $(Q_8\times C_{14})$	$C_7^2$ $\,\rtimes\,$ $(C_2\times Q_8)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{28}$ . $D_{14}$ (2)	$C_{14}^2$ . $C_2^2$	$(C_2\times C_{28})$ . $D_7$	$C_4$ . $(C_7\times D_{14})$ (2)	all 20

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

Homology

Abelianization:	$C_{2}^{2} \times C_{14} \simeq C_{2}^{3} \times C_{7}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 256 subgroups in 100 conjugacy classes, 54 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{14}$	$G/Z \simeq$ $D_{14}$
Commutator:	$G' \simeq$ $C_{14}$	$G/G' \simeq$ $C_2^2\times C_{14}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{14}\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{14}\times C_{28}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{28}.D_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{14}^2$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times Q_8$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$

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 784: $C_{28}.D_{14}$

Order 392: $C_7^2:Q_8$ x 4, $C_{14}:C_{28}$ x 2, $C_{14}\times C_{28}$

Order 196: $C_7:C_{28}$ x 4, $C_7\times C_{28}$ x 2, $C_{14}^2$

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

Order 98: $C_7\times C_{14}$ x 3

Order 56: $C_2\times C_{28}$ x 7, $C_7:Q_8$ x 4, $C_7\times Q_8$ x 4, $C_{14}:C_4$ x 2

Order 49: $C_7^2$

Order 28: $C_{28}$ x 14, $C_2\times C_{14}$ x 5, $C_7:C_4$ x 4

Order 16: $C_2\times Q_8$

Order 14: $C_{14}$ x 15

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 3

Order 7: $C_7$ x 5

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 784: $C_{28}.D_{14}$

Order 392: $C_{14}\times C_{28}$, $C_{14}:C_{28}$, $C_7^2:Q_8$

Order 196: $C_7\times C_{28}$, $C_7:C_{28}$, $C_{14}^2$

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

Order 98: $C_7\times C_{14}$ x 2

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

Order 49: $C_7^2$

Order 28: $C_{28}$ x 4, $C_2\times C_{14}$ x 3, $C_7:C_4$

Order 16: $C_2\times Q_8$

Order 14: $C_{14}$ x 6

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

Order 7: $C_7$ x 3

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 784: $C_{28}.D_{14}$ ($C_1$)

Order 392: $C_7^2:Q_8$ ($C_2$) x 4, $C_{14}:C_{28}$ ($C_2$) x 2, $C_{14}\times C_{28}$ ($C_2$)

Order 196: $C_7:C_{28}$ ($C_2^2$) x 4, $C_7\times C_{28}$ ($C_2^2$) x 2, $C_{14}^2$ ($C_2^2$)

Order 112: $C_{14}:Q_8$ ($C_7$)

Order 98: $C_7\times C_{14}$ ($Q_8$) x 2, $C_7\times C_{14}$ ($C_2^3$)

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

Order 49: $C_7^2$ ($C_2\times Q_8$)

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

Order 14: $C_{14}$ ($C_7:Q_8$) x 2, $C_{14}$ ($C_7\times Q_8$) x 2, $C_{14}$ ($C_2^2\times C_{14}$), $C_{14}$ ($C_2\times D_{14}$)

Order 8: $C_2\times C_4$ ($C_7\times D_7$)

Order 7: $C_7$ ($Q_8\times C_{14}$), $C_7$ ($C_{14}:Q_8$)

Order 4: $C_4$ ($C_7\times D_{14}$) x 2, $C_2^2$ ($C_7\times D_{14}$)

Order 2: $C_2$ ($C_7^2:Q_8$) x 2, $C_2$ ($C_{14}\times D_{14}$)

Order 1: $C_1$ ($C_{28}.D_{14}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 784: $C_{28}.D_{14}$ ($C_1$)

Order 392: $C_{14}\times C_{28}$ ($C_2$), $C_{14}:C_{28}$ ($C_2$), $C_7^2:Q_8$ ($C_2$)

Order 196: $C_7\times C_{28}$ ($C_2^2$), $C_7:C_{28}$ ($C_2^2$), $C_{14}^2$ ($C_2^2$)

Order 112: $C_{14}:Q_8$ ($C_7$)

Order 98: $C_7\times C_{14}$ ($C_2^3$), $C_7\times C_{14}$ ($Q_8$)

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

Order 49: $C_7^2$ ($C_2\times Q_8$)

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

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

Order 8: $C_2\times C_4$ ($C_7\times D_7$)

Order 7: $C_7$ ($Q_8\times C_{14}$), $C_7$ ($C_{14}:Q_8$)

Order 4: $C_2^2$ ($C_7\times D_{14}$), $C_4$ ($C_7\times D_{14}$)

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

Order 1: $C_1$ ($C_{28}.D_{14}$)

Series

Derived series

$C_{28}.D_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_1$

Chief series

$C_{28}.D_{14}$

$\rhd$

$C_7^2:Q_8$

$\rhd$

$C_7:Q_8$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{28}.D_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{14}$

$\lhd$

$C_2\times C_{28}$

Supergroups

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

This group is a maximal quotient of 10 larger groups in the database.

Character theory

Complex character table

See the $238 \times 238$ 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 $44 \times 44$ rational character table.