Abstract group 1568.464: $C_{28}.D_{28}$

• Label: 1568.464
• Order: \( 2^{5} \cdot 7^{2} \)
• Exponent: \( 2^{3} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \cdot 7^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{2} \)
• Perm deg.: $24$
• Trans deg.: $112$
• Rank: $3$

Group information

Description:	$C_{28}.D_{28}$
Order:	\(1568\)\(\medspace = 2^{5} \cdot 7^{2} \)
Exponent:	\(56\)\(\medspace = 2^{3} \cdot 7 \)
Automorphism group:	$C_{14}^2.C_6^2.C_2^5$, of order \(225792\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 7^{2} \)
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	395	60	48	56	144	528	336	1568
Conjugacy classes	1	5	4	15	4	45	66	24	164
Divisions	1	5	4	5	2	15	12	2	46
Autjugacy classes	1	3	3	3	1	6	7	1	25

Dimension	1	2	4	6	12	24
Irr. complex chars.	8	78	78	0	0	0	164
Irr. rational chars.	8	2	2	8	18	8	46

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$112$
Rank:	$3$
Inequivalent generating triples:	$5376$

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:	$\langle a, b, c \mid b^{56}=c^{14}=[a,c]=1, a^{2}=b^{28}, b^{a}=b^{27}, c^{b}=c^{13} \rangle$
Permutation group:	Degree $24$ $\langle(1,2,4,6)(3,5,7,8)(10,11)(12,13)(14,15)(23,24), (1,3,5,6,4,7,8,2)(17,18) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(C_{14}.D_{28})$
Semidirect product:	$(C_{14}:Q_8)$ $\,\rtimes\,$ $D_7$	$(C_7:Q_8)$ $\,\rtimes\,$ $D_{14}$	$C_{14}$ $\,\rtimes\,$ $(Q_8:D_7)$	$C_7$ $\,\rtimes\,$ $(Q_8:D_{14})$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{28}$ . $D_{28}$	$C_{14}^2$ . $D_4$	$C_4$ . $(D_7\times D_{14})$	$(C_7\times C_{28})$ . $D_4$	all 21

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 3486 subgroups in 217 conjugacy classes, 52 normal (30 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_7:D_{28}$
Commutator:	$G' \simeq$ $C_7\times C_{28}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $D_7\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{14}\times C_{28}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{28}.D_{28}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{14}^2$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times \SD_{16}$
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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1568: $C_{28}.D_{28}$

Order 784: $C_{14}.D_{28}$ x 4, $C_{14}:C_{56}$, $C_{14}:D_{28}$, $C_{28}.D_{14}$

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

Order 224: $C_8:D_{14}$, $Q_8:D_{14}$

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

Order 112: $C_2\times D_{28}$ x 5, $C_{56}:C_2$ x 4, $Q_8:D_7$ x 4, $C_{14}:C_8$, $Q_8\times C_{14}$, $C_{14}:Q_8$, $C_2\times C_{56}$

Order 98: $C_7\times C_{14}$ x 3, $C_7:D_7$ x 2

Order 56: $D_{28}$ x 18, $C_2\times D_{14}$ x 8, $C_2\times C_{28}$ x 6, $C_7:Q_8$ x 3, $C_7\times Q_8$ x 3, $C_{56}$ x 2, $C_7:C_8$ x 2, $C_{14}:C_4$

Order 49: $C_7^2$

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

Order 28: $D_{14}$ x 32, $C_{28}$ x 12, $C_2\times C_{14}$ x 5, $C_7:C_4$ x 2

Order 16: $\SD_{16}$ x 4, $C_2\times C_8$, $C_2\times Q_8$, $C_2\times D_4$

Order 14: $D_7$ x 16, $C_{14}$ x 15

Order 8: $Q_8$ x 3, $D_4$ x 3, $C_2\times C_4$ x 2, $C_8$ x 2, $C_2^3$

Order 7: $C_7$ x 5

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

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1568: $C_{28}.D_{28}$

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

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

Order 224: $C_8:D_{14}$, $Q_8:D_{14}$

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

Order 112: $C_2\times D_{28}$ x 3, $C_{14}:C_8$, $C_{56}:C_2$, $Q_8\times C_{14}$, $C_{14}:Q_8$, $C_2\times C_{56}$, $Q_8:D_7$

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

Order 56: $D_{28}$ x 6, $C_2\times C_{28}$ x 4, $C_2\times D_{14}$ x 3, $C_7:Q_8$ x 2, $C_7\times Q_8$ x 2, $C_{14}:C_4$, $C_{56}$, $C_7:C_8$

Order 49: $C_7^2$

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

Order 28: $C_{28}$ x 7, $D_{14}$ x 6, $C_2\times C_{14}$ x 3, $C_7:C_4$

Order 16: $\SD_{16}$, $C_2\times C_8$, $C_2\times Q_8$, $C_2\times D_4$

Order 14: $C_{14}$ x 6, $D_7$ x 3

Order 8: $Q_8$ x 2, $D_4$ x 2, $C_2\times C_4$ x 2, $C_2^3$, $C_8$

Order 7: $C_7$ x 3

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1568: $C_{28}.D_{28}$ ($C_1$)

Order 784: $C_{14}.D_{28}$ ($C_2$) x 4, $C_{14}:C_{56}$ ($C_2$), $C_{14}:D_{28}$ ($C_2$), $C_{28}.D_{14}$ ($C_2$)

Order 392: $C_7:D_{28}$ ($C_2^2$) x 2, $C_7^2:Q_8$ ($C_2^2$) x 2, $C_7:C_{56}$ ($C_2^2$) x 2, $C_{14}\times C_{28}$ ($C_2^2$)

Order 196: $C_7\times C_{28}$ ($C_2^3$), $C_7\times C_{28}$ ($D_4$), $C_{14}^2$ ($D_4$)

Order 112: $C_{14}:C_8$ ($D_7$), $C_{14}:Q_8$ ($D_7$)

Order 98: $C_7\times C_{14}$ ($\SD_{16}$) x 2, $C_7\times C_{14}$ ($C_2\times D_4$)

Order 56: $C_2\times C_{28}$ ($D_{14}$) x 2, $C_7:Q_8$ ($D_{14}$) x 2, $C_7:C_8$ ($D_{14}$) x 2

Order 49: $C_7^2$ ($C_2\times \SD_{16}$)

Order 28: $C_{28}$ ($C_2\times D_{14}$) x 2, $C_2\times C_{14}$ ($C_7:D_4$), $C_2\times C_{14}$ ($D_{28}$), $C_{28}$ ($C_7:D_4$), $C_{28}$ ($D_{28}$)

Order 14: $C_{14}$ ($C_{56}:C_2$) x 2, $C_{14}$ ($Q_8:D_7$) x 2, $C_{14}$ ($C_{14}:D_4$), $C_{14}$ ($C_2\times D_{28}$)

Order 8: $C_2\times C_4$ ($D_7^2$)

Order 7: $C_7$ ($C_8:D_{14}$), $C_7$ ($Q_8:D_{14}$)

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

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 1568: $C_{28}.D_{28}$ ($C_1$)

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

Order 392: $C_{14}\times C_{28}$ ($C_2^2$), $C_7:D_{28}$ ($C_2^2$), $C_7^2:Q_8$ ($C_2^2$), $C_7:C_{56}$ ($C_2^2$)

Order 196: $C_7\times C_{28}$ ($C_2^3$), $C_7\times C_{28}$ ($D_4$), $C_{14}^2$ ($D_4$)

Order 112: $C_{14}:C_8$ ($D_7$), $C_{14}:Q_8$ ($D_7$)

Order 98: $C_7\times C_{14}$ ($\SD_{16}$), $C_7\times C_{14}$ ($C_2\times D_4$)

Order 56: $C_2\times C_{28}$ ($D_{14}$) x 2, $C_7:Q_8$ ($D_{14}$), $C_7:C_8$ ($D_{14}$)

Order 49: $C_7^2$ ($C_2\times \SD_{16}$)

Order 28: $C_{28}$ ($C_2\times D_{14}$) x 2, $C_2\times C_{14}$ ($C_7:D_4$), $C_2\times C_{14}$ ($D_{28}$), $C_{28}$ ($C_7:D_4$), $C_{28}$ ($D_{28}$)

Order 14: $C_{14}$ ($C_{56}:C_2$), $C_{14}$ ($C_{14}:D_4$), $C_{14}$ ($C_2\times D_{28}$), $C_{14}$ ($Q_8:D_7$)

Order 8: $C_2\times C_4$ ($D_7^2$)

Order 7: $C_7$ ($C_8:D_{14}$), $C_7$ ($Q_8:D_{14}$)

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

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

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

Series

Derived series

$C_{28}.D_{28}$

$\rhd$

$C_7\times C_{28}$

$\rhd$

$C_1$

Chief series

$C_{28}.D_{28}$

$\rhd$

$C_{28}.D_{14}$

$\rhd$

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

$\rhd$

$C_{14}^2$

$\rhd$

$C_7\times C_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{28}.D_{28}$

$\rhd$

$C_7\times C_{28}$

$\rhd$

$C_7\times C_{14}$

$\rhd$

$C_7^2$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

Character theory

Complex character table

See the $164 \times 164$ 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 $46 \times 46$ rational character table.