Abstract group 672.200: $D_6:C_{56}$

• Label: 672.200
• Order: \( 2^{5} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 7 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 7 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $22$
• Trans deg.: $336$
• Rank: $2$

Group information

Description:	$D_6:C_{56}$
Order:	\(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_6^2.C_2^5$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
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	15	2	16	6	6	32	8	90	12	16	96	36	192	48	96	672
Conjugacy classes	1	5	1	6	3	6	8	4	30	6	8	36	18	48	24	48	252
Divisions	1	5	1	3	3	1	2	2	5	1	1	3	3	2	2	1	36
Autjugacy classes	1	4	1	3	3	1	2	2	4	1	1	3	3	2	2	1	34

Dimension	1	2	4	6	8	12	24	48
Irr. complex chars.	112	140	0	0	0	0	0	0	252
Irr. rational chars.	4	6	6	4	2	6	6	2	36

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$336$
Rank:	$2$
Inequivalent generating pairs:	$96$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	4	14

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{168}=[a,b]=[b,c]=1, c^{a}=bc^{113} \rangle$
Permutation group:	Degree $22$ $\langle(9,10)(11,12)(13,14), (11,13,14,12)(15,16,17,19,18,20,21,22), (1,2,3,4,5,6,7) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 31 & 0 \\ 0 & 31 \end{array}\right), \left(\begin{array}{rr} 40 & 39 \\ 13 & 27 \end{array}\right), \left(\begin{array}{rr} 9 & 32 \\ 12 & 37 \end{array}\right), \left(\begin{array}{rr} 40 & 13 \\ 13 & 40 \end{array}\right), \left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 31 & 22 \\ 18 & 21 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/52\Z)$
Direct product:	$C_7$ $\, \times\, $ $(D_6:C_8)$
Semidirect product:	$D_6$ $\,\rtimes\,$ $C_{56}$ (2)	$(C_2\times C_{56})$ $\,\rtimes\,$ $S_3$	$(S_3\times C_{14})$ $\,\rtimes\,$ $C_8$ (2)	$(C_6:C_{56})$ $\,\rtimes\,$ $C_2$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{84}$ . $D_4$ (2)	$C_{28}$ . $D_{12}$	$C_{42}$ . $\OD_{16}$	$C_{14}$ . $(S_3\times C_8)$	all 32

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

Homology

Abelianization:	$C_{2} \times C_{56} \simeq C_{2} \times C_{8} \times C_{7}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 228 subgroups in 100 conjugacy classes, 50 normal (46 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{28}$	$G/Z \simeq$ $D_6$
Commutator:	$G' \simeq$ $C_6$	$G/G' \simeq$ $C_2\times C_{56}$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $S_3\times C_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{168}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_6:C_{56}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:C_8$
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

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 672: $D_6:C_{56}$

Order 336: $C_6:C_{56}$, $D_6\times C_{28}$, $C_2\times C_{168}$

Order 224: $C_2^2:C_{56}$

Order 168: $S_3\times C_{28}$ x 2, $C_{168}$, $D_6\times C_{14}$, $C_2\times C_{84}$, $C_6:C_{28}$, $C_3:C_{56}$

Order 112: $C_2\times C_{56}$ x 2, $C_2^2\times C_{28}$

Order 96: $D_6:C_8$

Order 84: $S_3\times C_{14}$ x 4, $C_{84}$ x 2, $C_3:C_{28}$, $C_2\times C_{42}$

Order 56: $C_2\times C_{28}$ x 4, $C_{56}$ x 2, $C_2^2\times C_{14}$

Order 48: $C_6:C_8$, $C_4\times D_6$, $C_2\times C_{24}$

Order 42: $C_{42}$ x 3, $S_3\times C_7$ x 2

Order 32: $C_2^2:C_8$

Order 28: $C_2\times C_{14}$ x 5, $C_{28}$ x 3

Order 24: $C_4\times S_3$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $C_{24}$, $C_2\times D_6$, $C_3:C_8$

Order 21: $C_{21}$

Order 16: $C_2\times C_8$ x 2, $C_2^2\times C_4$

Order 14: $C_{14}$ x 5

Order 12: $D_6$ x 4, $C_{12}$ x 2, $C_2\times C_6$, $C_3:C_4$

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

Order 7: $C_7$

Order 6: $C_6$ x 3, $S_3$ x 2

Order 4: $C_2^2$ x 5, $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 672: $D_6:C_{56}$

Order 336: $C_6:C_{56}$, $D_6\times C_{28}$, $C_2\times C_{168}$

Order 224: $C_2^2:C_{56}$

Order 168: $S_3\times C_{28}$ x 2, $C_{168}$, $D_6\times C_{14}$, $C_2\times C_{84}$, $C_6:C_{28}$, $C_3:C_{56}$

Order 112: $C_2\times C_{56}$ x 2, $C_2^2\times C_{28}$

Order 96: $D_6:C_8$

Order 84: $S_3\times C_{14}$ x 3, $C_{84}$ x 2, $C_3:C_{28}$, $C_2\times C_{42}$

Order 56: $C_2\times C_{28}$ x 4, $C_{56}$ x 2, $C_2^2\times C_{14}$

Order 48: $C_6:C_8$, $C_4\times D_6$, $C_2\times C_{24}$

Order 42: $C_{42}$ x 3, $S_3\times C_7$

Order 32: $C_2^2:C_8$

Order 28: $C_2\times C_{14}$ x 4, $C_{28}$ x 3

Order 24: $C_4\times S_3$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $C_{24}$, $C_2\times D_6$, $C_3:C_8$

Order 21: $C_{21}$

Order 16: $C_2\times C_8$ x 2, $C_2^2\times C_4$

Order 14: $C_{14}$ x 4

Order 12: $D_6$ x 3, $C_{12}$ x 2, $C_2\times C_6$, $C_3:C_4$

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

Order 7: $C_7$

Order 6: $C_6$ x 3, $S_3$

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 672: $D_6:C_{56}$ ($C_1$)

Order 336: $C_6:C_{56}$ ($C_2$), $D_6\times C_{28}$ ($C_2$), $C_2\times C_{168}$ ($C_2$)

Order 168: $D_6\times C_{14}$ ($C_4$), $C_2\times C_{84}$ ($C_2^2$), $C_6:C_{28}$ ($C_4$)

Order 112: $C_2\times C_{56}$ ($S_3$)

Order 96: $D_6:C_8$ ($C_7$)

Order 84: $C_{84}$ ($D_4$) x 2, $S_3\times C_{14}$ ($C_8$) x 2, $C_2\times C_{42}$ ($C_2\times C_4$)

Order 56: $C_2\times C_{28}$ ($D_6$)

Order 48: $C_6:C_8$ ($C_{14}$), $C_4\times D_6$ ($C_{14}$), $C_2\times C_{24}$ ($C_{14}$)

Order 42: $C_{42}$ ($\OD_{16}$), $C_{42}$ ($C_2\times C_8$), $C_{42}$ ($C_2^2:C_4$)

Order 28: $C_2\times C_{14}$ ($C_4\times S_3$), $C_{28}$ ($C_3:D_4$), $C_{28}$ ($D_{12}$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_{14}$), $C_6:C_4$ ($C_{28}$), $C_2\times D_6$ ($C_{28}$)

Order 21: $C_{21}$ ($C_2^2:C_8$)

Order 16: $C_2\times C_8$ ($S_3\times C_7$)

Order 14: $C_{14}$ ($C_{24}:C_2$), $C_{14}$ ($S_3\times C_8$), $C_{14}$ ($D_6:C_4$)

Order 12: $D_6$ ($C_{56}$) x 2, $C_{12}$ ($C_7\times D_4$) x 2, $C_2\times C_6$ ($C_2\times C_{28}$)

Order 8: $C_2\times C_4$ ($S_3\times C_{14}$)

Order 7: $C_7$ ($D_6:C_8$)

Order 6: $C_6$ ($C_7\times \OD_{16}$), $C_6$ ($C_2\times C_{56}$), $C_6$ ($C_2^2:C_{28}$)

Order 4: $C_2^2$ ($S_3\times C_{28}$), $C_4$ ($D_6:C_{14}$), $C_4$ ($C_7\times D_{12}$)

Order 3: $C_3$ ($C_2^2:C_{56}$)

Order 2: $C_2$ ($D_6:C_{28}$), $C_2$ ($C_{168}:C_2$), $C_2$ ($S_3\times C_{56}$)

Order 1: $C_1$ ($D_6:C_{56}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 672: $D_6:C_{56}$ ($C_1$)

Order 336: $C_6:C_{56}$ ($C_2$), $D_6\times C_{28}$ ($C_2$), $C_2\times C_{168}$ ($C_2$)

Order 168: $D_6\times C_{14}$ ($C_4$), $C_2\times C_{84}$ ($C_2^2$), $C_6:C_{28}$ ($C_4$)

Order 112: $C_2\times C_{56}$ ($S_3$)

Order 96: $D_6:C_8$ ($C_7$)

Order 84: $C_{84}$ ($D_4$) x 2, $C_2\times C_{42}$ ($C_2\times C_4$), $S_3\times C_{14}$ ($C_8$)

Order 56: $C_2\times C_{28}$ ($D_6$)

Order 48: $C_6:C_8$ ($C_{14}$), $C_4\times D_6$ ($C_{14}$), $C_2\times C_{24}$ ($C_{14}$)

Order 42: $C_{42}$ ($\OD_{16}$), $C_{42}$ ($C_2\times C_8$), $C_{42}$ ($C_2^2:C_4$)

Order 28: $C_2\times C_{14}$ ($C_4\times S_3$), $C_{28}$ ($C_3:D_4$), $C_{28}$ ($D_{12}$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_{14}$), $C_6:C_4$ ($C_{28}$), $C_2\times D_6$ ($C_{28}$)

Order 21: $C_{21}$ ($C_2^2:C_8$)

Order 16: $C_2\times C_8$ ($S_3\times C_7$)

Order 14: $C_{14}$ ($C_{24}:C_2$), $C_{14}$ ($S_3\times C_8$), $C_{14}$ ($D_6:C_4$)

Order 12: $C_{12}$ ($C_7\times D_4$) x 2, $C_2\times C_6$ ($C_2\times C_{28}$), $D_6$ ($C_{56}$)

Order 8: $C_2\times C_4$ ($S_3\times C_{14}$)

Order 7: $C_7$ ($D_6:C_8$)

Order 6: $C_6$ ($C_7\times \OD_{16}$), $C_6$ ($C_2\times C_{56}$), $C_6$ ($C_2^2:C_{28}$)

Order 4: $C_2^2$ ($S_3\times C_{28}$), $C_4$ ($D_6:C_{14}$), $C_4$ ($C_7\times D_{12}$)

Order 3: $C_3$ ($C_2^2:C_{56}$)

Order 2: $C_2$ ($D_6:C_{28}$), $C_2$ ($C_{168}:C_2$), $C_2$ ($S_3\times C_{56}$)

Order 1: $C_1$ ($D_6:C_{56}$)

Series

Derived series

$D_6:C_{56}$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$D_6:C_{56}$

$\rhd$

$D_6\times C_{28}$

$\rhd$

$C_2\times C_{84}$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$D_6:C_{56}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{28}$

$\lhd$

$C_2\times C_{56}$

Supergroups

This group is a maximal subgroup of 124 larger groups in the database.

This group is a maximal quotient of 12 larger groups in the database.

Character theory

Complex character table

See the $252 \times 252$ 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.