Abstract group 784.130: $C_{14}^2.C_2^2$

• Label: 784.130
• Order: \( 2^{4} \cdot 7^{2} \)
• Exponent: \( 2^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 7^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{2} \)
• Perm deg.: $20$
• Trans deg.: $56$
• Rank: $3$

Group information

Description:	$C_{14}^2.C_2^2$
Order:	\(784\)\(\medspace = 2^{4} \cdot 7^{2} \)
Exponent:	\(28\)\(\medspace = 2^{2} \cdot 7 \)
Automorphism group:	$C_7^2.(C_2^3\times C_6^2).C_2^3$, of order \(112896\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7^{2} \)
Composition factors:	$C_2$ x 4, $C_7$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	4	7	14	28
Elements	1	199	56	48	144	336	784
Conjugacy classes	1	7	8	15	45	24	100
Divisions	1	7	4	5	15	4	36
Autjugacy classes	1	3	1	2	4	1	12

Dimension	1	2	4	6	12
Irr. complex chars.	16	48	36	0	0	100
Irr. rational chars.	8	4	0	8	16	36

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$56$
Rank:	$3$
Inequivalent generating triples:	$1344$

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:	$\langle a, b, c \mid b^{28}=c^{14}=[a,c]=1, a^{2}=b^{14}, b^{a}=b^{13}, c^{b}=c^{13} \rangle$
Permutation group:	Degree $20$ $\langle(8,9,11,10)(13,14)(15,16)(17,18), (2,3)(4,5)(6,7)(8,10,11,9), (19,20), (8,11)(9,10), (1,2,4,6,7,5,3), (12,13,15,17,18,16,14)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 64 & 14 \\ 42 & 22 \end{array}\right), \left(\begin{array}{rr} 1 & 13 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 57 & 26 \\ 0 & 83 \end{array}\right), \left(\begin{array}{rr} 27 & 0 \\ 0 & 27 \end{array}\right), \left(\begin{array}{rr} 57 & 21 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 64 & 0 \\ 0 & 64 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/91\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_{14}.D_{14})$
Semidirect product:	$(C_7:D_{14})$ $\,\rtimes\,$ $C_4$	$(C_{14}:C_4)$ $\,\rtimes\,$ $D_7$	$(C_7:C_4)$ $\,\rtimes\,$ $D_{14}$	$C_{14}$ $\,\rtimes\,$ $(C_4\times D_7)$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^2$ . $D_7^2$	$C_{14}^2$ . $C_2^2$	$C_2$ . $(D_7\times D_{14})$	$(C_{14}:D_{14})$ . $C_2$	all 9

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

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 1632 subgroups in 156 conjugacy classes, 48 normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_7^2$
Commutator:	$G' \simeq$ $C_7^2$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_7\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{14}^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{14}^2.C_2^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{14}^2$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times C_4$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 784: $C_{14}^2.C_2^2$

Order 392: $C_{14}.D_{14}$ x 4, $C_{14}:C_{28}$ x 2, $C_{14}:D_{14}$

Order 196: $C_7:D_{14}$ x 6, $C_7:C_{28}$ x 4, $C_{14}^2$

Order 112: $C_4\times D_{14}$ x 2

Order 98: $C_7:D_7$ x 4, $C_7\times C_{14}$ x 3

Order 56: $C_4\times D_7$ x 8, $C_2\times D_{14}$ x 5, $C_2\times C_{28}$ x 2, $C_{14}:C_4$ x 2

Order 49: $C_7^2$

Order 28: $D_{14}$ x 30, $C_2\times C_{14}$ x 5, $C_{28}$ x 4, $C_7:C_4$ x 4

Order 16: $C_2^2\times C_4$

Order 14: $D_7$ x 20, $C_{14}$ x 15

Order 8: $C_2\times C_4$ x 6, $C_2^3$

Order 7: $C_7$ x 5

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

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 784: $C_{14}^2.C_2^2$

Order 392: $C_{14}:D_{14}$, $C_{14}:C_{28}$, $C_{14}.D_{14}$

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

Order 112: $C_4\times D_{14}$

Order 98: $C_7\times C_{14}$ x 2, $C_7:D_7$

Order 56: $C_2\times D_{14}$ x 2, $C_2\times C_{28}$, $C_{14}:C_4$, $C_4\times D_7$

Order 49: $C_7^2$

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

Order 16: $C_2^2\times C_4$

Order 14: $C_{14}$ x 4, $D_7$ x 2

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

Order 7: $C_7$ x 2

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 784: $C_{14}^2.C_2^2$ ($C_1$)

Order 392: $C_{14}.D_{14}$ ($C_2$) x 4, $C_{14}:C_{28}$ ($C_2$) x 2, $C_{14}:D_{14}$ ($C_2$)

Order 196: $C_7:C_{28}$ ($C_2^2$) x 4, $C_7:D_{14}$ ($C_4$) x 4, $C_7:D_{14}$ ($C_2^2$) x 2, $C_{14}^2$ ($C_2^2$)

Order 98: $C_7:D_7$ ($C_2\times C_4$) x 4, $C_7\times C_{14}$ ($C_2\times C_4$) x 2, $C_7\times C_{14}$ ($C_2^3$)

Order 56: $C_{14}:C_4$ ($D_7$) x 2

Order 49: $C_7^2$ ($C_2^2\times C_4$)

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

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

Order 7: $C_7$ ($C_4\times D_{14}$) x 2

Order 4: $C_2^2$ ($D_7^2$)

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 784: $C_{14}^2.C_2^2$ ($C_1$)

Order 392: $C_{14}:D_{14}$ ($C_2$), $C_{14}:C_{28}$ ($C_2$), $C_{14}.D_{14}$ ($C_2$)

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

Order 98: $C_7\times C_{14}$ ($C_2^3$), $C_7\times C_{14}$ ($C_2\times C_4$), $C_7:D_7$ ($C_2\times C_4$)

Order 56: $C_{14}:C_4$ ($D_7$)

Order 49: $C_7^2$ ($C_2^2\times C_4$)

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

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

Order 7: $C_7$ ($C_4\times D_{14}$)

Order 4: $C_2^2$ ($D_7^2$)

Order 2: $C_2$ ($D_7\times D_{14}$), $C_2$ ($C_{14}.D_{14}$)

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

Series

Derived series

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

$\rhd$

$C_7^2$

$\rhd$

$C_1$

Chief series

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

$\rhd$

$C_{14}:C_{28}$

$\rhd$

$C_{14}^2$

$\rhd$

$C_7\times C_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$C_7^2$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 28 larger groups in the database.

This group is a maximal quotient of 17 larger groups in the database.

Character theory

Complex character table

See the $100 \times 100$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $36 \times 36$ rational character table.