Abstract group 1568.607: $C_{14}^2.C_2^3$

• Label: 1568.607
• Order: \( 2^{5} \cdot 7^{2} \)
• Exponent: \( 2^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 7 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 7 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{2} \)
• Perm deg.: $26$
• Trans deg.: $224$
• Rank: $3$

Group information

Description:	$C_{14}^2.C_2^3$
Order:	\(1568\)\(\medspace = 2^{5} \cdot 7^{2} \)
Exponent:	\(28\)\(\medspace = 2^{2} \cdot 7 \)
Automorphism group:	$C_{14}.(C_6^2\times D_4).C_2^3$, of order \(32256\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 7 \)
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	14	28
Elements	1	3	124	48	144	1248	1568
Conjugacy classes	1	3	16	27	81	222	350
Divisions	1	3	11	5	15	27	62
Autjugacy classes	1	3	5	3	9	9	30

Dimension	1	2	4	6	12	24
Irr. complex chars.	112	196	42	0	0	0	350
Irr. rational chars.	8	6	1	12	22	13	62

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$224$
Rank:	$3$
Inequivalent generating triples:	$38304$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	8	18

Constructions

Presentation:	$\langle a, b, c \mid b^{28}=c^{28}=[a,b]=1, a^{2}=c^{14}, c^{a}=c^{27}, c^{b}=c^{15} \rangle$
Permutation group:	Degree $26$ $\langle(1,2,5,4)(3,7,8,6)(17,18)(19,20)(21,22), (1,3,5,8)(2,6,4,7)(23,24)(25,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 29 \\ 29 & 59 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 28 & 0 \\ 0 & 28 \end{array}\right), \left(\begin{array}{rr} 0 & 34 \\ 35 & 0 \end{array}\right), \left(\begin{array}{rr} 59 & 0 \\ 0 & 59 \end{array}\right), \left(\begin{array}{rr} 41 & 29 \\ 29 & 70 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/87\Z)$
Direct product:	$C_7$ $\, \times\, $ $(C_7:(C_4\times Q_8))$
Semidirect product:	$(C_7:C_{28})$ $\,\rtimes\,$ $Q_8$ (2)	$(C_7:Q_8)$ $\,\rtimes\,$ $C_{28}$ (4)	$C_7$ $\,\rtimes\,$ $(Q_8\times C_{28})$	$(C_7^2:Q_8)$ $\,\rtimes\,$ $C_4$ (4)	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{14}^2$ . $C_2^3$	$C_{14}$ . $(Q_8\times D_7)$	$C_{14}$ . $(D_4:D_7)$	$(C_4:C_{28})$ . $D_7$	all 38

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:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 518 subgroups in 179 conjugacy classes, 86 normal (38 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{14}$	$G/Z \simeq$ $C_2\times D_{14}$
Commutator:	$G' \simeq$ $C_{14}$	$G/G' \simeq$ $C_2^2\times C_{28}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_{14}\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{28}:C_{28}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{14}^2.C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{14}^2$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times Q_8$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 1568: $C_{14}^2.C_2^3$

Order 784: $C_{28}:C_{28}$ x 3, $C_{14}^2.C_2^2$ x 2, $C_{28}:C_{28}$, $C_{28}.D_{14}$

Order 392: $C_{14}:C_{28}$ x 4, $C_7^2:Q_8$ x 4, $C_{14}\times C_{28}$ x 3

Order 224: $C_7:(C_4\times Q_8)$, $Q_8\times C_{28}$

Order 196: $C_7:C_{28}$ x 7, $C_7\times C_{28}$ x 4, $C_{14}^2$

Order 112: $C_4:C_{28}$ x 7, $C_4\times C_{28}$ x 3, $C_{28}:C_4$ x 3, $C_{14}.D_4$ x 2, $Q_8\times C_{14}$, $C_{14}:Q_8$

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

Order 56: $C_2\times C_{28}$ x 19, $C_{14}:C_4$ x 4, $C_7:Q_8$ x 4, $C_7\times Q_8$ x 4

Order 49: $C_7^2$

Order 32: $C_4\times Q_8$

Order 28: $C_{28}$ x 27, $C_7:C_4$ x 7, $C_2\times C_{14}$ x 5

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

Order 14: $C_{14}$ x 15

Order 8: $C_2\times C_4$ x 7, $Q_8$ x 4

Order 7: $C_7$ x 5

Order 4: $C_4$ x 11, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1568: $C_{14}^2.C_2^3$

Order 784: $C_{28}:C_{28}$ x 2, $C_{28}:C_{28}$, $C_{14}^2.C_2^2$, $C_{28}.D_{14}$

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

Order 224: $C_7:(C_4\times Q_8)$, $Q_8\times C_{28}$

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

Order 112: $C_4:C_{28}$ x 4, $C_4\times C_{28}$ x 2, $C_{28}:C_4$ x 2, $Q_8\times C_{14}$, $C_{14}:Q_8$, $C_{14}.D_4$

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

Order 56: $C_2\times C_{28}$ x 9, $C_{14}:C_4$ x 3, $C_7:Q_8$, $C_7\times Q_8$

Order 49: $C_7^2$

Order 32: $C_4\times Q_8$

Order 28: $C_{28}$ x 9, $C_2\times C_{14}$ x 3, $C_7:C_4$ x 3

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

Order 14: $C_{14}$ x 9

Order 8: $C_2\times C_4$ x 5, $Q_8$

Order 7: $C_7$ x 3

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1568: $C_{14}^2.C_2^3$ ($C_1$)

Order 784: $C_{28}:C_{28}$ ($C_2$) x 3, $C_{14}^2.C_2^2$ ($C_2$) x 2, $C_{28}:C_{28}$ ($C_2$), $C_{28}.D_{14}$ ($C_2$)

Order 392: $C_{14}:C_{28}$ ($C_2^2$) x 4, $C_7^2:Q_8$ ($C_4$) x 4, $C_{14}\times C_{28}$ ($C_2^2$) x 3

Order 224: $C_7:(C_4\times Q_8)$ ($C_7$)

Order 196: $C_7:C_{28}$ ($C_2\times C_4$) x 4, $C_7\times C_{28}$ ($C_2\times C_4$) x 2, $C_7:C_{28}$ ($Q_8$) x 2, $C_{14}^2$ ($C_2^3$)

Order 112: $C_{28}:C_4$ ($C_{14}$) x 3, $C_{14}.D_4$ ($C_{14}$) x 2, $C_{14}:Q_8$ ($C_{14}$), $C_4:C_{28}$ ($C_{14}$), $C_4:C_{28}$ ($D_7$)

Order 98: $C_7\times C_{14}$ ($D_4:C_2$), $C_7\times C_{14}$ ($C_2\times Q_8$), $C_7\times C_{14}$ ($C_2^2\times C_4$)

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

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

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

Order 16: $C_4:C_4$ ($C_7\times D_7$)

Order 14: $C_{14}$ ($D_4:C_{14}$), $C_{14}$ ($Q_8\times C_{14}$), $C_{14}$ ($C_2^2\times C_{28}$), $C_{14}$ ($Q_8\times D_7$), $C_{14}$ ($D_4:D_7$), $C_{14}$ ($C_4\times D_{14}$)

Order 8: $C_2\times C_4$ ($C_7\times D_{14}$) x 3

Order 7: $C_7$ ($C_7:(C_4\times Q_8)$), $C_7$ ($Q_8\times C_{28}$)

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

Order 2: $C_2$ ($C_{28}.D_{14}$), $C_2$ ($C_{28}.D_{14}$), $C_2$ ($C_{28}\times D_{14}$)

Order 1: $C_1$ ($C_{14}^2.C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1568: $C_{14}^2.C_2^3$ ($C_1$)

Order 784: $C_{28}:C_{28}$ ($C_2$) x 2, $C_{28}:C_{28}$ ($C_2$), $C_{14}^2.C_2^2$ ($C_2$), $C_{28}.D_{14}$ ($C_2$)

Order 392: $C_{14}:C_{28}$ ($C_2^2$) x 3, $C_{14}\times C_{28}$ ($C_2^2$) x 2, $C_7^2:Q_8$ ($C_4$)

Order 224: $C_7:(C_4\times Q_8)$ ($C_7$)

Order 196: $C_7\times C_{28}$ ($C_2\times C_4$), $C_7:C_{28}$ ($Q_8$), $C_7:C_{28}$ ($C_2\times C_4$), $C_{14}^2$ ($C_2^3$)

Order 112: $C_{28}:C_4$ ($C_{14}$) x 2, $C_{14}:Q_8$ ($C_{14}$), $C_4:C_{28}$ ($C_{14}$), $C_4:C_{28}$ ($D_7$), $C_{14}.D_4$ ($C_{14}$)

Order 98: $C_7\times C_{14}$ ($D_4:C_2$), $C_7\times C_{14}$ ($C_2\times Q_8$), $C_7\times C_{14}$ ($C_2^2\times C_4$)

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

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

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

Order 16: $C_4:C_4$ ($C_7\times D_7$)

Order 14: $C_{14}$ ($D_4:C_{14}$), $C_{14}$ ($Q_8\times C_{14}$), $C_{14}$ ($C_2^2\times C_{28}$), $C_{14}$ ($Q_8\times D_7$), $C_{14}$ ($D_4:D_7$), $C_{14}$ ($C_4\times D_{14}$)

Order 8: $C_2\times C_4$ ($C_7\times D_{14}$) x 2

Order 7: $C_7$ ($C_7:(C_4\times Q_8)$), $C_7$ ($Q_8\times C_{28}$)

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

Order 2: $C_2$ ($C_{28}.D_{14}$), $C_2$ ($C_{28}.D_{14}$), $C_2$ ($C_{28}\times D_{14}$)

Order 1: $C_1$ ($C_{14}^2.C_2^3$)

Series

Derived series

$C_{14}^2.C_2^3$

$\rhd$

$C_{14}$

$\rhd$

$C_1$

Chief series

$C_{14}^2.C_2^3$

$\rhd$

$C_{28}.D_{14}$

$\rhd$

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

$\rhd$

$C_7\times C_{28}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{14}^2.C_2^3$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{14}$

$\lhd$

$C_4:C_{28}$

Character theory

Complex character table

See the $350 \times 350$ 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 $62 \times 62$ rational character table.