Abstract group 672.646: $D_{28}:D_6$

• Label: 672.646
• Order: \( 2^{5} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $18$
• Trans deg.: $168$
• Rank: $3$

Group information

Description:	$D_{28}:D_6$
Order:	\(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_{42}.(C_2^5\times C_6)$, of order \(8064\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \)
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	28	42	56	84
Elements	1	139	2	20	86	6	96	40	78	12	36	12	72	72	672
Conjugacy classes	1	5	1	3	4	3	2	5	6	3	6	3	6	9	57
Divisions	1	5	1	3	3	1	2	3	2	1	2	1	1	2	28
Autjugacy classes	1	5	1	3	3	1	2	3	2	1	2	1	1	2	28

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	8	22	24	0	3	0	0	57
Irr. rational chars.	8	6	3	4	1	3	3	28

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$168$
Rank:	$3$
Inequivalent generating triples:	$10752$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	48
Arbitrary	6	6	12

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=d^{84}=[a,b]=[a,c]=1, c^{2}=d^{42}, d^{a}=d^{55}, c^{b}=cd^{42}, d^{b}=cd^{29}, d^{c}=d^{43} \rangle$
Permutation group:	Degree $18$ $\langle(2,3)(4,5)(6,7)(11,12)(14,16), (9,10)(13,17)(14,16)(15,18), (11,13,12,15) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{28}$ $\,\rtimes\,$ $D_6$	$(Q_8:S_3)$ $\,\rtimes\,$ $D_7$	$Q_8$ $\,\rtimes\,$ $(S_3\times D_7)$	$(C_7\times Q_8)$ $\,\rtimes\,$ $D_6$	all 21
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{12}$ . $D_{14}$	$D_{84}$ . $C_2^2$	$C_{84}$ . $C_2^3$	$(C_4\times D_7)$ . $D_6$	all 17

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 1200 subgroups in 136 conjugacy classes, 40 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_6:D_{14}$
Commutator:	$G' \simeq$ $C_{84}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $S_3\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{21}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $D_{28}:D_6$	$G/R \simeq$ $C_1$
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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Subgroup information Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

Order 672: $D_{28}:D_6$

Order 336: $C_{21}:\SD_{16}$, $C_3:D_{56}$, $C_{21}:D_8$, $C_{21}:\OD_{16}$, $D_{28}:C_6$, $D_7\times D_{12}$, $Q_8:D_{21}$

Order 224: $C_8:D_{14}$

Order 168: $C_3\times D_{28}$ x 2, $C_{12}\times D_7$ x 2, $S_3\times D_{14}$, $C_{21}:C_8$, $Q_8\times C_{21}$, $D_{84}$, $C_7\times D_{12}$, $C_3:C_{56}$, $C_7:D_{12}$

Order 112: $D_{56}$, $C_{56}:C_2$, $D_{28}:C_2$, $D_4\times D_7$, $C_7\times \SD_{16}$, $Q_8:D_7$, $C_7:D_8$

Order 96: $D_4:D_6$

Order 84: $S_3\times D_7$ x 2, $C_{84}$ x 2, $C_3\times D_{14}$ x 2, $C_7:C_{12}$, $D_{42}$, $S_3\times C_{14}$

Order 56: $D_{28}$ x 3, $C_4\times D_7$ x 2, $C_7\times D_4$, $C_7:D_4$, $C_{56}$, $C_2\times D_{14}$, $C_7\times Q_8$, $C_7:C_8$

Order 48: $Q_8:S_3$ x 2, $C_3:D_8$ x 2, $D_4:C_6$, $C_2\times D_{12}$, $C_3:\OD_{16}$

Order 42: $C_3\times D_7$ x 2, $C_{42}$, $D_{21}$, $S_3\times C_7$

Order 32: $D_8:C_2$

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

Order 24: $D_{12}$ x 3, $C_2\times C_{12}$ x 2, $C_3\times D_4$ x 2, $C_3:C_8$ x 2, $C_2\times D_6$, $C_3\times Q_8$

Order 21: $C_{21}$

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

Order 14: $D_7$ x 3, $C_{14}$ x 2

Order 12: $D_6$ x 4, $C_{12}$ x 3, $C_2\times C_6$ x 2

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

Order 7: $C_7$

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

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

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 672: $D_{28}:D_6$

Order 336: $C_{21}:\SD_{16}$, $C_3:D_{56}$, $C_{21}:D_8$, $C_{21}:\OD_{16}$, $D_{28}:C_6$, $D_7\times D_{12}$, $Q_8:D_{21}$

Order 224: $C_8:D_{14}$

Order 168: $C_3\times D_{28}$ x 2, $C_{12}\times D_7$ x 2, $S_3\times D_{14}$, $C_{21}:C_8$, $Q_8\times C_{21}$, $D_{84}$, $C_7\times D_{12}$, $C_3:C_{56}$, $C_7:D_{12}$

Order 112: $D_{56}$, $C_{56}:C_2$, $D_{28}:C_2$, $D_4\times D_7$, $C_7\times \SD_{16}$, $Q_8:D_7$, $C_7:D_8$

Order 96: $D_4:D_6$

Order 84: $C_{84}$ x 2, $C_3\times D_{14}$ x 2, $S_3\times D_7$, $C_7:C_{12}$, $D_{42}$, $S_3\times C_{14}$

Order 56: $D_{28}$ x 3, $C_4\times D_7$ x 2, $C_7\times D_4$, $C_7:D_4$, $C_{56}$, $C_2\times D_{14}$, $C_7\times Q_8$, $C_7:C_8$

Order 48: $Q_8:S_3$ x 2, $C_3:D_8$ x 2, $D_4:C_6$, $C_2\times D_{12}$, $C_3:\OD_{16}$

Order 42: $C_3\times D_7$ x 2, $C_{42}$, $D_{21}$, $S_3\times C_7$

Order 32: $D_8:C_2$

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

Order 24: $D_{12}$ x 3, $C_2\times C_{12}$ x 2, $C_3\times D_4$ x 2, $C_3:C_8$ x 2, $C_2\times D_6$, $C_3\times Q_8$

Order 21: $C_{21}$

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

Order 14: $D_7$ x 3, $C_{14}$ x 2

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

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

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: $C_1$

Normal subgroups (quotient in parentheses)

Order 672: $D_{28}:D_6$ ($C_1$)

Order 336: $C_{21}:\SD_{16}$ ($C_2$), $C_3:D_{56}$ ($C_2$), $C_{21}:D_8$ ($C_2$), $C_{21}:\OD_{16}$ ($C_2$), $D_{28}:C_6$ ($C_2$), $D_7\times D_{12}$ ($C_2$), $Q_8:D_{21}$ ($C_2$)

Order 168: $C_{21}:C_8$ ($C_2^2$), $Q_8\times C_{21}$ ($C_2^2$), $D_{84}$ ($C_2^2$), $C_7\times D_{12}$ ($C_2^2$), $C_3:C_{56}$ ($C_2^2$), $C_3\times D_{28}$ ($C_2^2$), $C_{12}\times D_7$ ($C_2^2$)

Order 112: $D_{28}:C_2$ ($S_3$)

Order 84: $C_{84}$ ($C_2^3$), $C_7:C_{12}$ ($D_4$), $C_3\times D_{14}$ ($D_4$)

Order 56: $D_{28}$ ($D_6$), $C_4\times D_7$ ($D_6$), $C_7\times Q_8$ ($D_6$)

Order 48: $Q_8:S_3$ ($D_7$)

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

Order 28: $D_{14}$ ($C_3:D_4$), $C_{28}$ ($C_2\times D_6$), $C_7:C_4$ ($C_3:D_4$)

Order 24: $D_{12}$ ($D_{14}$), $C_3\times Q_8$ ($D_{14}$), $C_3:C_8$ ($D_{14}$)

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

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

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

Order 8: $Q_8$ ($S_3\times D_7$)

Order 7: $C_7$ ($D_4:D_6$)

Order 6: $C_6$ ($D_4\times D_7$)

Order 4: $C_4$ ($S_3\times D_{14}$)

Order 3: $C_3$ ($C_8:D_{14}$)

Order 2: $C_2$ ($D_6:D_{14}$)

Order 1: $C_1$ ($D_{28}:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 672: $D_{28}:D_6$ ($C_1$)

Order 336: $C_{21}:\SD_{16}$ ($C_2$), $C_3:D_{56}$ ($C_2$), $C_{21}:D_8$ ($C_2$), $C_{21}:\OD_{16}$ ($C_2$), $D_{28}:C_6$ ($C_2$), $D_7\times D_{12}$ ($C_2$), $Q_8:D_{21}$ ($C_2$)

Order 168: $C_{21}:C_8$ ($C_2^2$), $Q_8\times C_{21}$ ($C_2^2$), $D_{84}$ ($C_2^2$), $C_7\times D_{12}$ ($C_2^2$), $C_3:C_{56}$ ($C_2^2$), $C_3\times D_{28}$ ($C_2^2$), $C_{12}\times D_7$ ($C_2^2$)

Order 112: $D_{28}:C_2$ ($S_3$)

Order 84: $C_{84}$ ($C_2^3$), $C_7:C_{12}$ ($D_4$), $C_3\times D_{14}$ ($D_4$)

Order 56: $D_{28}$ ($D_6$), $C_4\times D_7$ ($D_6$), $C_7\times Q_8$ ($D_6$)

Order 48: $Q_8:S_3$ ($D_7$)

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

Order 28: $D_{14}$ ($C_3:D_4$), $C_{28}$ ($C_2\times D_6$), $C_7:C_4$ ($C_3:D_4$)

Order 24: $D_{12}$ ($D_{14}$), $C_3\times Q_8$ ($D_{14}$), $C_3:C_8$ ($D_{14}$)

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

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

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

Order 8: $Q_8$ ($S_3\times D_7$)

Order 7: $C_7$ ($D_4:D_6$)

Order 6: $C_6$ ($D_4\times D_7$)

Order 4: $C_4$ ($S_3\times D_{14}$)

Order 3: $C_3$ ($C_8:D_{14}$)

Order 2: $C_2$ ($D_6:D_{14}$)

Order 1: $C_1$ ($D_{28}:D_6$)

Series

Derived series

$D_{28}:D_6$

$\rhd$

$C_{84}$

$\rhd$

$C_1$

Chief series

$D_{28}:D_6$

$\rhd$

$D_7\times D_{12}$

$\rhd$

$C_{12}\times D_7$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$D_{28}:D_6$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$Q_8$

Supergroups

This group is a maximal subgroup of 16 larger groups in the database.

This group is a maximal quotient of 32 larger groups in the database.

Character theory

Complex character table

See the $57 \times 57$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $28 \times 28$ rational character table.