Abstract group 896.1576: $C_{16}.D_{28}$

• Label: 896.1576
• Order: \( 2^{7} \cdot 7 \)
• Exponent: \( 2^{4} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $39$
• Trans deg.: $224$
• Rank: $3$

Group information

Description:	$C_{16}.D_{28}$
Order:	\(896\)\(\medspace = 2^{7} \cdot 7 \)
Exponent:	\(112\)\(\medspace = 2^{4} \cdot 7 \)
Automorphism group:	$C_{28}.(C_2^4\times C_{12}).C_2$, of order \(10752\)\(\medspace = 2^{9} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 7, $C_7$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	4	7	8	14	16	28	56	112
Elements	1	59	68	6	128	18	256	72	96	192	896
Conjugacy classes	1	4	9	3	14	9	28	36	48	96	248
Divisions	1	4	6	1	6	2	5	4	4	3	36
Autjugacy classes	1	3	4	1	5	2	4	3	4	3	30

Dimension	1	2	4	6	8	12	16	24	48	96
Irr. complex chars.	32	216	0	0	0	0	0	0	0	0	248
Irr. rational chars.	8	6	5	4	1	4	2	3	1	2	36

Minimal presentations

Permutation degree:	$39$
Transitive degree:	$224$
Rank:	$3$
Inequivalent generating triples:	$21504$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	96
Arbitrary	2	4	22

Constructions

Presentation:	$\langle a, b, c \mid b^{16}=c^{28}=[b,c]=1, a^{2}=b^{10}c^{14}, b^{a}=b^{13}c^{14}, c^{a}=b^{12}c^{27} \rangle$
Permutation group:	Degree $39$ $\langle(1,2,6,10,5,11,20,28,7,12,22,29,21,9,25,27)(3,13,15,18,16,24,30,19,17,26,31,32,14,8,23,4) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 6 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 88 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{113})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_8.D_{28})$ $\,\rtimes\,$ $C_2$ (2)	$(C_4\times C_{16})$ $\,\rtimes\,$ $D_7$	$C_7$ $\,\rtimes\,$ $(C_{16}.D_4)$	$(C_{56}.C_8)$ $\,\rtimes\,$ $C_2$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{56}$ . $C_8$	$C_{112}$ . $D_4$ (2)	$C_{16}$ . $D_{28}$ (2)	$C_8$ . $(C_8\times D_7)$ (2)	all 41

Elements of the group are displayed as matrices in $\GL_{2}(\F_{113})$.

Homology

Abelianization:	$C_{2}^{2} \times C_{8} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 658 subgroups in 146 conjugacy classes, 65 normal (45 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{16}$	$G/Z \simeq$ $D_{28}$
Commutator:	$G' \simeq$ $C_{28}$	$G/G' \simeq$ $C_2^2\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_8$	$G/\Phi \simeq$ $C_2\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_4\times C_{112}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{16}.D_{28}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{14}$	$G/\operatorname{soc} \simeq$ $C_8\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{16}.D_4$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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 896: $C_{16}.D_{28}$

Order 448: $C_8.D_{28}$ x 2, $C_{16}.D_{14}$ x 2, $D_{56}:C_4$, $C_{56}.C_8$, $C_4\times C_{112}$

Order 224: $C_8.D_{14}$ x 2, $C_2\times C_{112}$ x 2, $C_{112}:C_2$ x 2, $D_7\times C_{16}$ x 2, $C_7:\OD_{32}$ x 2, $D_{28}:C_4$ x 2, $D_{56}:C_2$, $C_4\times C_{56}$, $C_{56}.C_4$

Order 128: $C_{16}.D_4$

Order 112: $C_{112}$ x 3, $C_7:\OD_{16}$ x 2, $C_{56}:C_2$ x 2, $C_{56}:C_2$ x 2, $D_{28}:C_2$ x 2, $C_8\times D_7$ x 2, $C_2\times C_{56}$ x 2, $C_7:C_{16}$ x 2, $C_7:Q_{16}$, $D_{56}$, $C_4\times C_{28}$

Order 64: $\OD_{32}:C_2$ x 2, $\OD_{32}:C_2$ x 2, $C_8.C_8$, $C_4\times C_{16}$, $D_8:C_4$

Order 56: $C_{56}$ x 4, $C_2\times C_{28}$ x 2, $C_7:D_4$ x 2, $D_{28}$ x 2, $C_4\times D_7$ x 2, $C_7:Q_8$ x 2, $C_7:C_8$ x 2

Order 32: $\OD_{32}$ x 4, $C_2\times C_{16}$ x 4, $\OD_{16}:C_2$ x 2, $C_4\wr C_2$ x 2, $D_8:C_2$, $C_4\times C_8$, $C_8.C_4$

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

Order 16: $C_{16}$ x 5, $\OD_{16}$ x 4, $C_2\times C_8$ x 4, $\SD_{16}$ x 2, $D_4:C_2$ x 2, $Q_{16}$, $D_8$, $C_4^2$

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

Order 8: $C_8$ x 6, $D_4$ x 4, $C_2\times C_4$ x 4, $Q_8$ x 2

Order 7: $C_7$

Order 4: $C_4$ x 6, $C_2^2$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 896: $C_{16}.D_{28}$

Order 448: $C_8.D_{28}$, $C_{16}.D_{14}$, $D_{56}:C_4$, $C_{56}.C_8$, $C_4\times C_{112}$

Order 224: $C_2\times C_{112}$ x 2, $D_{56}:C_2$, $C_8.D_{14}$, $C_4\times C_{56}$, $C_{112}:C_2$, $D_7\times C_{16}$, $C_{56}.C_4$, $C_7:\OD_{32}$, $D_{28}:C_4$

Order 128: $C_{16}.D_4$

Order 112: $C_{112}$ x 3, $C_2\times C_{56}$ x 2, $C_7:\OD_{16}$, $C_7:Q_{16}$, $D_{56}$, $C_{56}:C_2$, $C_{56}:C_2$, $D_{28}:C_2$, $C_8\times D_7$, $C_4\times C_{28}$, $C_7:C_{16}$

Order 64: $C_8.C_8$, $\OD_{32}:C_2$, $C_4\times C_{16}$, $\OD_{32}:C_2$, $D_8:C_4$

Order 56: $C_{56}$ x 4, $C_2\times C_{28}$ x 2, $C_7:D_4$, $D_{28}$, $C_4\times D_7$, $C_7:Q_8$, $C_7:C_8$

Order 32: $C_2\times C_{16}$ x 3, $\OD_{32}$ x 2, $D_8:C_2$, $\OD_{16}:C_2$, $C_4\times C_8$, $C_8.C_4$, $C_4\wr C_2$

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

Order 16: $C_{16}$ x 4, $C_2\times C_8$ x 3, $\OD_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$, $C_4^2$, $D_4:C_2$

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

Order 8: $C_8$ x 5, $C_2\times C_4$ x 3, $D_4$ x 2, $Q_8$

Order 7: $C_7$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 896: $C_{16}.D_{28}$ ($C_1$)

Order 448: $C_8.D_{28}$ ($C_2$) x 2, $C_{16}.D_{14}$ ($C_2$) x 2, $D_{56}:C_4$ ($C_2$), $C_{56}.C_8$ ($C_2$), $C_4\times C_{112}$ ($C_2$)

Order 224: $C_8.D_{14}$ ($C_2^2$) x 2, $C_2\times C_{112}$ ($C_2^2$) x 2, $C_7:\OD_{32}$ ($C_2^2$) x 2, $D_{28}:C_4$ ($C_4$) x 2, $D_{56}:C_2$ ($C_4$), $C_4\times C_{56}$ ($C_2^2$), $C_{56}.C_4$ ($C_4$)

Order 112: $C_7:\OD_{16}$ ($C_2\times C_4$) x 2, $C_{56}:C_2$ ($C_8$) x 2, $D_{28}:C_2$ ($C_2\times C_4$) x 2, $C_{112}$ ($D_4$) x 2, $C_7:Q_{16}$ ($C_8$), $D_{56}$ ($C_8$), $C_2\times C_{56}$ ($C_2^3$), $C_2\times C_{56}$ ($C_2\times C_4$), $C_4\times C_{28}$ ($C_2\times C_4$)

Order 64: $C_4\times C_{16}$ ($D_7$)

Order 56: $D_{28}$ ($C_2\times C_8$) x 2, $C_7:Q_8$ ($C_2\times C_8$) x 2, $C_{56}$ ($C_2\times C_8$) x 2, $C_2\times C_{28}$ ($C_2^2\times C_4$), $C_{56}$ ($D_4:C_2$), $C_{56}$ ($C_2\times D_4$)

Order 32: $C_2\times C_{16}$ ($D_{14}$) x 2, $C_4\times C_8$ ($D_{14}$)

Order 28: $C_2\times C_{14}$ ($\OD_{16}:C_2$), $C_{28}$ ($C_2^2\times C_8$), $C_{28}$ ($C_4\times D_4$)

Order 16: $C_{16}$ ($D_{28}$) x 2, $C_2\times C_8$ ($C_4\times D_7$), $C_2\times C_8$ ($C_2\times D_{14}$), $C_4^2$ ($C_4\times D_7$)

Order 14: $C_{14}$ ($C_8\times D_4$)

Order 8: $C_8$ ($C_8\times D_7$) x 2, $C_2\times C_4$ ($C_4\times D_{14}$), $C_8$ ($D_{28}:C_2$), $C_8$ ($C_2\times D_{28}$)

Order 7: $C_7$ ($C_{16}.D_4$)

Order 4: $C_2^2$ ($C_8.D_{14}$), $C_4$ ($C_8\times D_{14}$), $C_4$ ($C_4\times D_{28}$)

Order 2: $C_2$ ($C_8\times D_{28}$)

Order 1: $C_1$ ($C_{16}.D_{28}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 896: $C_{16}.D_{28}$ ($C_1$)

Order 448: $C_8.D_{28}$ ($C_2$), $C_{16}.D_{14}$ ($C_2$), $D_{56}:C_4$ ($C_2$), $C_{56}.C_8$ ($C_2$), $C_4\times C_{112}$ ($C_2$)

Order 224: $C_2\times C_{112}$ ($C_2^2$) x 2, $D_{56}:C_2$ ($C_4$), $C_8.D_{14}$ ($C_2^2$), $C_4\times C_{56}$ ($C_2^2$), $C_{56}.C_4$ ($C_4$), $C_7:\OD_{32}$ ($C_2^2$), $D_{28}:C_4$ ($C_4$)

Order 112: $C_{112}$ ($D_4$) x 2, $C_7:\OD_{16}$ ($C_2\times C_4$), $C_7:Q_{16}$ ($C_8$), $D_{56}$ ($C_8$), $C_{56}:C_2$ ($C_8$), $D_{28}:C_2$ ($C_2\times C_4$), $C_2\times C_{56}$ ($C_2^3$), $C_2\times C_{56}$ ($C_2\times C_4$), $C_4\times C_{28}$ ($C_2\times C_4$)

Order 64: $C_4\times C_{16}$ ($D_7$)

Order 56: $C_{56}$ ($C_2\times C_8$) x 2, $C_2\times C_{28}$ ($C_2^2\times C_4$), $D_{28}$ ($C_2\times C_8$), $C_7:Q_8$ ($C_2\times C_8$), $C_{56}$ ($D_4:C_2$), $C_{56}$ ($C_2\times D_4$)

Order 32: $C_2\times C_{16}$ ($D_{14}$) x 2, $C_4\times C_8$ ($D_{14}$)

Order 28: $C_2\times C_{14}$ ($\OD_{16}:C_2$), $C_{28}$ ($C_2^2\times C_8$), $C_{28}$ ($C_4\times D_4$)

Order 16: $C_{16}$ ($D_{28}$) x 2, $C_2\times C_8$ ($C_4\times D_7$), $C_2\times C_8$ ($C_2\times D_{14}$), $C_4^2$ ($C_4\times D_7$)

Order 14: $C_{14}$ ($C_8\times D_4$)

Order 8: $C_8$ ($C_8\times D_7$) x 2, $C_2\times C_4$ ($C_4\times D_{14}$), $C_8$ ($D_{28}:C_2$), $C_8$ ($C_2\times D_{28}$)

Order 7: $C_7$ ($C_{16}.D_4$)

Order 4: $C_2^2$ ($C_8.D_{14}$), $C_4$ ($C_8\times D_{14}$), $C_4$ ($C_4\times D_{28}$)

Order 2: $C_2$ ($C_8\times D_{28}$)

Order 1: $C_1$ ($C_{16}.D_{28}$)

Series

Derived series

$C_{16}.D_{28}$

$\rhd$

$C_{28}$

$\rhd$

$C_1$

Chief series

$C_{16}.D_{28}$

$\rhd$

$C_4\times C_{112}$

$\rhd$

$C_4\times C_{56}$

$\rhd$

$C_2\times C_{56}$

$\rhd$

$C_2\times C_{28}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{16}.D_{28}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_{16}$

$\lhd$

$C_2\times C_{16}$

$\lhd$

$C_4\times C_{16}$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $248 \times 248$ 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 $36 \times 36$ rational character table.