Abstract group 1568.264: $C_{14}^2.D_4$

• Label: 1568.264
• Order: \( 2^{5} \cdot 7^{2} \)
• Exponent: \( 2^{3} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 7 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 7 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3^{2} \)
• Perm deg.: $26$
• Trans deg.: $224$
• Rank: $2$

Group information

Description:	$C_{14}^2.D_4$
Order:	\(1568\)\(\medspace = 2^{5} \cdot 7^{2} \)
Exponent:	\(56\)\(\medspace = 2^{3} \cdot 7 \)
Automorphism group:	$(C_6\times C_7:C_3).C_2^6.C_2$, of order \(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_7$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	4	7	8	14	28	56
Elements	1	3	116	48	8	144	864	384	1568
Conjugacy classes	1	3	6	27	4	81	120	192	434
Divisions	1	3	5	5	1	15	13	8	51
Autjugacy classes	1	3	4	3	1	9	8	3	32

Dimension	1	2	4	6	12	24	48
Irr. complex chars.	56	378	0	0	0	0	0	434
Irr. rational chars.	4	4	2	6	19	10	6	51

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$224$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	16

Constructions

Presentation:	$\langle a, b, c \mid b^{56}=c^{14}=[a,c]=[b,c]=1, a^{2}=b^{28}, b^{a}=b^{27}c^{7} \rangle$
Permutation group:	Degree $26$ $\langle(1,2)(3,5)(4,9)(6,8)(7,11)(10,14)(12,13)(15,16,19,20)(17,18,21,22), (15,17,18,20,19,21,22,16) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 57 & 29 \\ 58 & 57 \end{array}\right), \left(\begin{array}{rr} 70 & 58 \\ 29 & 70 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right), \left(\begin{array}{rr} 0 & 82 \\ 64 & 0 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 28 & 0 \\ 0 & 28 \end{array}\right), \left(\begin{array}{rr} 59 & 0 \\ 0 & 59 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/87\Z)$
Direct product:	$C_7$ $\, \times\, $ $(C_{14}.Q_{16})$
Semidirect product:	$(C_7:Q_8)$ $\,\rtimes\,$ $C_{28}$ (2)	$C_7$ $\,\rtimes\,$ $(Q_8:C_{28})$	$(C_7^2:Q_8)$ $\,\rtimes\,$ $C_4$ (2)	$C_7^2$ $\,\rtimes\,$ $(Q_8:C_4)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{14}^2$ . $D_4$	$(C_7\times C_{28})$ . $D_4$	$(C_2\times C_{56})$ . $D_7$	$C_{28}$ . $(C_4\times D_7)$	all 36

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

Homology

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

Subgroups

There are 414 subgroups in 117 conjugacy classes, 46 normal (42 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{14}$	$G/Z \simeq$ $D_{28}$
Commutator:	$G' \simeq$ $C_{28}$	$G/G' \simeq$ $C_2\times C_{28}$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_7\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{14}\times C_{56}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{14}^2.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{14}^2$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8:C_4$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1568: $C_{14}^2.D_4$

Order 784: $C_{14}\times C_{56}$, $C_{28}:C_{28}$, $C_{28}.D_{14}$

Order 392: $C_7^2:Q_8$ x 3, $C_{14}:C_{28}$ x 2, $C_{14}\times C_{28}$, $C_7\times C_{56}$

Order 224: $Q_8:C_{28}$, $C_{14}.Q_{16}$

Order 196: $C_7:C_{28}$ x 3, $C_7\times C_{28}$ x 2, $C_{14}^2$

Order 112: $C_2\times C_{56}$ x 5, $Q_8\times C_{14}$, $C_{14}:Q_8$, $C_4:C_{28}$, $C_{28}:C_4$

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

Order 56: $C_{56}$ x 8, $C_2\times C_{28}$ x 7, $C_7:Q_8$ x 3, $C_7\times Q_8$ x 3, $C_{14}:C_4$ x 2

Order 49: $C_7^2$

Order 32: $Q_8:C_4$

Order 28: $C_{28}$ x 13, $C_2\times C_{14}$ x 5, $C_7:C_4$ x 3

Order 16: $C_2\times C_8$, $C_4:C_4$, $C_2\times Q_8$

Order 14: $C_{14}$ x 15

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

Order 7: $C_7$ x 5

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1568: $C_{14}^2.D_4$

Order 784: $C_{14}\times C_{56}$, $C_{28}:C_{28}$, $C_{28}.D_{14}$

Order 392: $C_{14}:C_{28}$ x 2, $C_7^2:Q_8$ x 2, $C_{14}\times C_{28}$, $C_7\times C_{56}$

Order 224: $Q_8:C_{28}$, $C_{14}.Q_{16}$

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

Order 112: $C_2\times C_{56}$ x 3, $Q_8\times C_{14}$, $C_{14}:Q_8$, $C_4:C_{28}$, $C_{28}:C_4$

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

Order 56: $C_2\times C_{28}$ x 5, $C_{56}$ x 3, $C_{14}:C_4$ x 2, $C_7:Q_8$ x 2, $C_7\times Q_8$ x 2

Order 49: $C_7^2$

Order 32: $Q_8:C_4$

Order 28: $C_{28}$ x 8, $C_2\times C_{14}$ x 3, $C_7:C_4$ x 2

Order 16: $C_2\times C_8$, $C_4:C_4$, $C_2\times Q_8$

Order 14: $C_{14}$ x 9

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

Order 7: $C_7$ x 3

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1568: $C_{14}^2.D_4$ ($C_1$)

Order 784: $C_{14}\times C_{56}$ ($C_2$), $C_{28}:C_{28}$ ($C_2$), $C_{28}.D_{14}$ ($C_2$)

Order 392: $C_7^2:Q_8$ ($C_4$) x 2, $C_{14}\times C_{28}$ ($C_2^2$)

Order 224: $C_{14}.Q_{16}$ ($C_7$)

Order 196: $C_7\times C_{28}$ ($D_4$), $C_7\times C_{28}$ ($C_2\times C_4$), $C_{14}^2$ ($D_4$)

Order 112: $C_{14}:Q_8$ ($C_{14}$), $C_2\times C_{56}$ ($C_{14}$), $C_2\times C_{56}$ ($D_7$), $C_{28}:C_4$ ($C_{14}$)

Order 98: $C_7\times C_{14}$ ($Q_{16}$), $C_7\times C_{14}$ ($\SD_{16}$), $C_7\times C_{14}$ ($C_2^2:C_4$)

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

Order 49: $C_7^2$ ($Q_8:C_4$)

Order 28: $C_2\times C_{14}$ ($C_7\times D_4$), $C_2\times C_{14}$ ($D_{28}$), $C_{28}$ ($C_7\times D_4$), $C_{28}$ ($C_2\times C_{28}$), $C_{28}$ ($C_7:D_4$), $C_{28}$ ($C_4\times D_7$)

Order 16: $C_2\times C_8$ ($C_7\times D_7$)

Order 14: $C_{14}$ ($C_7:Q_{16}$), $C_{14}$ ($C_{56}:C_2$), $C_{14}$ ($C_7\times Q_{16}$), $C_{14}$ ($C_7\times \SD_{16}$), $C_{14}$ ($C_2^2:C_{28}$), $C_{14}$ ($D_{14}:C_4$)

Order 8: $C_2\times C_4$ ($C_7\times D_{14}$)

Order 7: $C_7$ ($Q_8:C_{28}$), $C_7$ ($C_{14}.Q_{16}$)

Order 4: $C_2^2$ ($C_7\times D_{28}$), $C_4$ ($C_{14}\wr C_2$), $C_4$ ($D_7\times C_{28}$)

Order 2: $C_2$ ($D_{14}:C_{28}$), $C_2$ ($C_7^2:Q_{16}$), $C_2$ ($C_{56}:C_{14}$)

Order 1: $C_1$ ($C_{14}^2.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1568: $C_{14}^2.D_4$ ($C_1$)

Order 784: $C_{14}\times C_{56}$ ($C_2$), $C_{28}:C_{28}$ ($C_2$), $C_{28}.D_{14}$ ($C_2$)

Order 392: $C_{14}\times C_{28}$ ($C_2^2$), $C_7^2:Q_8$ ($C_4$)

Order 224: $C_{14}.Q_{16}$ ($C_7$)

Order 196: $C_7\times C_{28}$ ($D_4$), $C_7\times C_{28}$ ($C_2\times C_4$), $C_{14}^2$ ($D_4$)

Order 112: $C_{14}:Q_8$ ($C_{14}$), $C_2\times C_{56}$ ($C_{14}$), $C_2\times C_{56}$ ($D_7$), $C_{28}:C_4$ ($C_{14}$)

Order 98: $C_7\times C_{14}$ ($Q_{16}$), $C_7\times C_{14}$ ($\SD_{16}$), $C_7\times C_{14}$ ($C_2^2:C_4$)

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

Order 49: $C_7^2$ ($Q_8:C_4$)

Order 28: $C_2\times C_{14}$ ($C_7\times D_4$), $C_2\times C_{14}$ ($D_{28}$), $C_{28}$ ($C_7\times D_4$), $C_{28}$ ($C_2\times C_{28}$), $C_{28}$ ($C_7:D_4$), $C_{28}$ ($C_4\times D_7$)

Order 16: $C_2\times C_8$ ($C_7\times D_7$)

Order 14: $C_{14}$ ($C_7:Q_{16}$), $C_{14}$ ($C_{56}:C_2$), $C_{14}$ ($C_7\times Q_{16}$), $C_{14}$ ($C_7\times \SD_{16}$), $C_{14}$ ($C_2^2:C_{28}$), $C_{14}$ ($D_{14}:C_4$)

Order 8: $C_2\times C_4$ ($C_7\times D_{14}$)

Order 7: $C_7$ ($Q_8:C_{28}$), $C_7$ ($C_{14}.Q_{16}$)

Order 4: $C_2^2$ ($C_7\times D_{28}$), $C_4$ ($C_{14}\wr C_2$), $C_4$ ($D_7\times C_{28}$)

Order 2: $C_2$ ($D_{14}:C_{28}$), $C_2$ ($C_7^2:Q_{16}$), $C_2$ ($C_{56}:C_{14}$)

Order 1: $C_1$ ($C_{14}^2.D_4$)

Series

Derived series

$C_{14}^2.D_4$

$\rhd$

$C_{28}$

$\rhd$

$C_1$

Chief series

$C_{14}^2.D_4$

$\rhd$

$C_{28}.D_{14}$

$\rhd$

$C_{14}\times C_{28}$

$\rhd$

$C_{14}^2$

$\rhd$

$C_7\times C_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{14}^2.D_4$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{14}$

$\lhd$

$C_2\times C_{28}$

$\lhd$

$C_2\times C_{56}$

Character theory

Complex character table

See the $434 \times 434$ 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 $51 \times 51$ rational character table.