Abstract group 672.797: $D_{56}:C_6$

• Label: 672.797
• Order: \( 2^{5} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $26$
• Trans deg.: $336$
• Rank: $3$

Group information

Description:	$D_{56}:C_6$
Order:	\(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_{28}.(C_2^5\times C_6)$, of order \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	7	8	12	14	21	24	28	42	56	84	168
Elements	1	59	2	60	118	6	8	120	18	12	16	24	36	48	48	96	672
Conjugacy classes	1	4	2	5	8	3	4	10	9	6	8	12	18	24	24	48	186
Divisions	1	4	1	4	4	1	2	4	2	1	2	2	2	2	2	2	36
Autjugacy classes	1	3	1	3	3	1	2	3	2	1	2	2	2	2	2	2	32

Dimension	1	2	4	6	8	12	16	24	48	96
Irr. complex chars.	24	162	0	0	0	0	0	0	0	0	186
Irr. rational chars.	8	10	2	4	1	6	1	2	1	1	36

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$336$
Rank:	$3$
Inequivalent generating triples:	$17472$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	96
Arbitrary	2	4	16

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{8}=c^{42}=[b,c]=1, b^{a}=b^{3}, c^{a}=b^{4}c^{13} \rangle$
Permutation group:	Degree $26$ $\langle(2,3)(4,5)(6,7)(11,12)(13,17)(14,21)(15,24)(16,19)(18,22)(20,25)(23,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 121 & 0 \\ 0 & 39 \end{array}\right), \left(\begin{array}{rr} 227 & 0 \\ 0 & 296 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{337})$
Direct product:	$C_3$ $\, \times\, $ $(D_{56}:C_2)$
Semidirect product:	$D_{56}$ $\,\rtimes\,$ $C_6$	$C_7$ $\,\rtimes\,$ $(D_8:C_6)$	$(C_7:Q_{16})$ $\,\rtimes\,$ $C_6$	$(D_{28}:C_6)$ $\,\rtimes\,$ $C_2$ (2)	all 14
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{84}$ . $D_4$	$C_{24}$ . $D_{14}$ (2)	$C_{12}$ . $D_{28}$	$C_{84}$ . $C_2^3$	all 28

Elements of the group are displayed as matrices in $\GL_{2}(\F_{337})$.

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 604 subgroups in 124 conjugacy classes, 58 normal (42 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{12}$	$G/Z \simeq$ $D_{28}$
Commutator:	$G' \simeq$ $C_{28}$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_6\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{168}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{56}:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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 672: $D_{56}:C_6$

Order 336: $C_{168}:C_2$ x 2, $D_{28}:C_6$ x 2, $C_{21}:Q_{16}$, $C_3\times D_{56}$, $C_2\times C_{168}$

Order 224: $D_{56}:C_2$

Order 168: $C_{168}$ x 2, $C_{21}:D_4$ x 2, $C_3\times D_{28}$ x 2, $C_{12}\times D_7$ x 2, $C_{21}:Q_8$ x 2, $C_2\times C_{84}$

Order 112: $C_{56}:C_2$ x 2, $D_{28}:C_2$ x 2, $C_7:Q_{16}$, $D_{56}$, $C_2\times C_{56}$

Order 96: $D_8:C_6$

Order 84: $C_{84}$ x 2, $C_7:C_{12}$ x 2, $C_3\times D_{14}$ x 2, $C_2\times C_{42}$

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

Order 48: $D_4:C_6$ x 2, $C_3\times \SD_{16}$ x 2, $C_3\times Q_{16}$, $C_3\times D_8$, $C_2\times C_{24}$

Order 42: $C_{42}$ x 2, $C_3\times D_7$ x 2

Order 32: $D_8:C_2$

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

Order 24: $C_3\times D_4$ x 4, $C_2\times C_{12}$ x 3, $C_{24}$ x 2, $C_3\times Q_8$ x 2

Order 21: $C_{21}$

Order 16: $\SD_{16}$ x 2, $D_4:C_2$ x 2, $Q_{16}$, $D_8$, $C_2\times C_8$

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

Order 12: $C_{12}$ x 4, $C_2\times C_6$ x 3

Order 8: $D_4$ x 4, $C_2\times C_4$ x 3, $Q_8$ x 2, $C_8$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 4

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 672: $D_{56}:C_6$

Order 336: $C_{21}:Q_{16}$, $C_3\times D_{56}$, $C_{168}:C_2$, $D_{28}:C_6$, $C_2\times C_{168}$

Order 224: $D_{56}:C_2$

Order 168: $C_{168}$ x 2, $C_2\times C_{84}$, $C_{21}:D_4$, $C_3\times D_{28}$, $C_{12}\times D_7$, $C_{21}:Q_8$

Order 112: $C_7:Q_{16}$, $D_{56}$, $C_{56}:C_2$, $D_{28}:C_2$, $C_2\times C_{56}$

Order 96: $D_8:C_6$

Order 84: $C_{84}$ x 2, $C_7:C_{12}$, $C_2\times C_{42}$, $C_3\times D_{14}$

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

Order 48: $D_4:C_6$, $C_3\times Q_{16}$, $C_3\times \SD_{16}$, $C_3\times D_8$, $C_2\times C_{24}$

Order 42: $C_{42}$ x 2, $C_3\times D_7$

Order 32: $D_8:C_2$

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

Order 24: $C_2\times C_{12}$ x 2, $C_{24}$ x 2, $C_3\times D_4$ x 2, $C_3\times Q_8$

Order 21: $C_{21}$

Order 16: $Q_{16}$, $\SD_{16}$, $D_8$, $C_2\times C_8$, $D_4:C_2$

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

Order 12: $C_{12}$ x 3, $C_2\times C_6$ x 2

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

Order 7: $C_7$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 672: $D_{56}:C_6$ ($C_1$)

Order 336: $C_{168}:C_2$ ($C_2$) x 2, $D_{28}:C_6$ ($C_2$) x 2, $C_{21}:Q_{16}$ ($C_2$), $C_3\times D_{56}$ ($C_2$), $C_2\times C_{168}$ ($C_2$)

Order 224: $D_{56}:C_2$ ($C_3$)

Order 168: $C_{168}$ ($C_2^2$) x 2, $C_3\times D_{28}$ ($C_2^2$) x 2, $C_{21}:Q_8$ ($C_2^2$) x 2, $C_2\times C_{84}$ ($C_2^2$)

Order 112: $C_{56}:C_2$ ($C_6$) x 2, $D_{28}:C_2$ ($C_6$) x 2, $C_7:Q_{16}$ ($C_6$), $D_{56}$ ($C_6$), $C_2\times C_{56}$ ($C_6$)

Order 84: $C_{84}$ ($C_2^3$), $C_{84}$ ($D_4$), $C_2\times C_{42}$ ($D_4$)

Order 56: $D_{28}$ ($C_2\times C_6$) x 2, $C_7:Q_8$ ($C_2\times C_6$) x 2, $C_{56}$ ($C_2\times C_6$) x 2, $C_2\times C_{28}$ ($C_2\times C_6$)

Order 48: $C_2\times C_{24}$ ($D_7$)

Order 42: $C_{42}$ ($C_2\times D_4$)

Order 28: $C_2\times C_{14}$ ($C_3\times D_4$), $C_{28}$ ($C_2^2\times C_6$), $C_{28}$ ($C_3\times D_4$)

Order 24: $C_{24}$ ($D_{14}$) x 2, $C_2\times C_{12}$ ($D_{14}$)

Order 21: $C_{21}$ ($D_8:C_2$)

Order 16: $C_2\times C_8$ ($C_3\times D_7$)

Order 14: $C_{14}$ ($C_6\times D_4$)

Order 12: $C_2\times C_6$ ($D_{28}$), $C_{12}$ ($D_{28}$), $C_{12}$ ($C_2\times D_{14}$)

Order 8: $C_8$ ($C_3\times D_{14}$) x 2, $C_2\times C_4$ ($C_3\times D_{14}$)

Order 7: $C_7$ ($D_8:C_6$)

Order 6: $C_6$ ($C_2\times D_{28}$)

Order 4: $C_2^2$ ($C_3\times D_{28}$), $C_4$ ($C_6\times D_{14}$), $C_4$ ($C_3\times D_{28}$)

Order 3: $C_3$ ($D_{56}:C_2$)

Order 2: $C_2$ ($C_6\times D_{28}$)

Order 1: $C_1$ ($D_{56}:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 672: $D_{56}:C_6$ ($C_1$)

Order 336: $C_{21}:Q_{16}$ ($C_2$), $C_3\times D_{56}$ ($C_2$), $C_{168}:C_2$ ($C_2$), $D_{28}:C_6$ ($C_2$), $C_2\times C_{168}$ ($C_2$)

Order 224: $D_{56}:C_2$ ($C_3$)

Order 168: $C_{168}$ ($C_2^2$) x 2, $C_2\times C_{84}$ ($C_2^2$), $C_3\times D_{28}$ ($C_2^2$), $C_{21}:Q_8$ ($C_2^2$)

Order 112: $C_7:Q_{16}$ ($C_6$), $D_{56}$ ($C_6$), $C_{56}:C_2$ ($C_6$), $D_{28}:C_2$ ($C_6$), $C_2\times C_{56}$ ($C_6$)

Order 84: $C_{84}$ ($C_2^3$), $C_{84}$ ($D_4$), $C_2\times C_{42}$ ($D_4$)

Order 56: $C_{56}$ ($C_2\times C_6$) x 2, $C_2\times C_{28}$ ($C_2\times C_6$), $D_{28}$ ($C_2\times C_6$), $C_7:Q_8$ ($C_2\times C_6$)

Order 48: $C_2\times C_{24}$ ($D_7$)

Order 42: $C_{42}$ ($C_2\times D_4$)

Order 28: $C_2\times C_{14}$ ($C_3\times D_4$), $C_{28}$ ($C_2^2\times C_6$), $C_{28}$ ($C_3\times D_4$)

Order 24: $C_{24}$ ($D_{14}$) x 2, $C_2\times C_{12}$ ($D_{14}$)

Order 21: $C_{21}$ ($D_8:C_2$)

Order 16: $C_2\times C_8$ ($C_3\times D_7$)

Order 14: $C_{14}$ ($C_6\times D_4$)

Order 12: $C_2\times C_6$ ($D_{28}$), $C_{12}$ ($D_{28}$), $C_{12}$ ($C_2\times D_{14}$)

Order 8: $C_8$ ($C_3\times D_{14}$) x 2, $C_2\times C_4$ ($C_3\times D_{14}$)

Order 7: $C_7$ ($D_8:C_6$)

Order 6: $C_6$ ($C_2\times D_{28}$)

Order 4: $C_2^2$ ($C_3\times D_{28}$), $C_4$ ($C_6\times D_{14}$), $C_4$ ($C_3\times D_{28}$)

Order 3: $C_3$ ($D_{56}:C_2$)

Order 2: $C_2$ ($C_6\times D_{28}$)

Order 1: $C_1$ ($D_{56}:C_6$)

Series

Derived series

$D_{56}:C_6$

$\rhd$

$C_{28}$

$\rhd$

$C_1$

Chief series

$D_{56}:C_6$

$\rhd$

$C_2\times C_{168}$

$\rhd$

$C_2\times C_{84}$

$\rhd$

$C_{84}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$D_{56}:C_6$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_{12}$

$\lhd$

$C_2\times C_{12}$

$\lhd$

$C_2\times C_{24}$

Supergroups

This group is a maximal subgroup of 50 larger groups in the database.

This group is a maximal quotient of 22 larger groups in the database.

Character theory

Complex character table

See the $186 \times 186$ 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.