Abstract group 336.181: $D_{28}:C_6$

• Label: 336.181
• Order: \( 2^{4} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{2} \)
• Perm deg.: $18$
• Trans deg.: $168$
• Rank: $3$

Group information

Description:	$D_{28}:C_6$
Order:	\(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^2\times S_4\times F_7$, of order \(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_3$, $C_7$
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	12	14	21	28	42	84
Elements	1	43	2	20	86	6	40	6	12	36	12	72	336
Conjugacy classes	1	4	2	5	8	3	10	3	6	9	6	18	75
Divisions	1	4	1	4	4	1	4	1	1	3	1	3	28
Autjugacy classes	1	2	1	2	2	1	2	1	1	1	1	1	16

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	24	42	9	0	0	0	0	75
Irr. rational chars.	8	8	1	4	1	5	1	28

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$168$
Rank:	$3$
Inequivalent generating triples:	$2912$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	48
Arbitrary	4	6	12

Constructions

Presentation:	$\langle a, b, c \mid b^{2}=c^{84}=[a,b]=1, a^{2}=c^{42}, c^{a}=c^{43}, c^{b}=c^{55} \rangle$
Permutation group:	Degree $18$ $\langle(2,3)(4,5)(6,7)(13,17)(15,18), (11,12,14,16)(13,18,17,15), (11,13,14,17)(12,15,16,18), (8,9,10), (11,14)(12,16)(13,17)(15,18), (1,2,4,6,7,5,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 15 & 21 \\ 14 & 15 \end{array}\right), \left(\begin{array}{rr} 1 & 5 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right), \left(\begin{array}{rr} 22 & 0 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 13 & 5 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/35\Z)$
Direct product:	$C_3$ $\, \times\, $ $(D_{28}:C_2)$
Semidirect product:	$D_{28}$ $\,\rtimes\,$ $C_6$ (3)	$(C_3\times Q_8)$ $\,\rtimes\,$ $D_7$	$Q_8$ $\,\rtimes\,$ $(C_3\times D_7)$	$(C_7\times Q_8)$ $\,\rtimes\,$ $C_6$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{12}$ . $D_{14}$ (3)	$C_{84}$ . $C_2^2$ (3)	$C_{42}$ . $C_2^3$	$D_{14}$ . $(C_2\times C_6)$ (3)	all 12

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

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 296 subgroups in 80 conjugacy classes, 46 normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_2\times D_{14}$
Commutator:	$G' \simeq$ $C_{14}$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_6\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{21}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{28}:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2$
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

Classes of subgroups up to automorphism

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.

Classes of subgroups up to conjugation

Order 336: $D_{28}:C_6$

Order 168: $C_3\times D_{28}$ x 3, $C_{12}\times D_7$ x 3, $Q_8\times C_{21}$

Order 112: $D_{28}:C_2$

Order 84: $C_{84}$ x 3, $C_3\times D_{14}$ x 3, $C_7:C_{12}$

Order 56: $D_{28}$ x 3, $C_4\times D_7$ x 3, $C_7\times Q_8$

Order 48: $D_4:C_6$

Order 42: $C_3\times D_7$ x 3, $C_{42}$

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

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

Order 21: $C_{21}$

Order 16: $D_4:C_2$

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

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

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

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 336: $D_{28}:C_6$

Order 168: $Q_8\times C_{21}$, $C_3\times D_{28}$, $C_{12}\times D_7$

Order 112: $D_{28}:C_2$

Order 84: $C_{84}$, $C_7:C_{12}$, $C_3\times D_{14}$

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

Order 48: $D_4:C_6$

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

Order 28: $D_{14}$, $C_{28}$, $C_7:C_4$

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

Order 21: $C_{21}$

Order 16: $D_4:C_2$

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

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

Order 8: $Q_8$, $D_4$, $C_2\times C_4$

Order 7: $C_7$

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 336: $D_{28}:C_6$ ($C_1$)

Order 168: $C_3\times D_{28}$ ($C_2$) x 3, $C_{12}\times D_7$ ($C_2$) x 3, $Q_8\times C_{21}$ ($C_2$)

Order 112: $D_{28}:C_2$ ($C_3$)

Order 84: $C_{84}$ ($C_2^2$) x 3, $C_3\times D_{14}$ ($C_2^2$) x 3, $C_7:C_{12}$ ($C_2^2$)

Order 56: $D_{28}$ ($C_6$) x 3, $C_4\times D_7$ ($C_6$) x 3, $C_7\times Q_8$ ($C_6$)

Order 42: $C_{42}$ ($C_2^3$)

Order 28: $D_{14}$ ($C_2\times C_6$) x 3, $C_{28}$ ($C_2\times C_6$) x 3, $C_7:C_4$ ($C_2\times C_6$)

Order 24: $C_3\times Q_8$ ($D_7$)

Order 21: $C_{21}$ ($D_4:C_2$)

Order 14: $C_{14}$ ($C_2^2\times C_6$)

Order 12: $C_{12}$ ($D_{14}$) x 3

Order 8: $Q_8$ ($C_3\times D_7$)

Order 7: $C_7$ ($D_4:C_6$)

Order 6: $C_6$ ($C_2\times D_{14}$)

Order 4: $C_4$ ($C_3\times D_{14}$) x 3

Order 3: $C_3$ ($D_{28}:C_2$)

Order 2: $C_2$ ($C_6\times D_{14}$)

Order 1: $C_1$ ($D_{28}:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 336: $D_{28}:C_6$ ($C_1$)

Order 168: $Q_8\times C_{21}$ ($C_2$), $C_3\times D_{28}$ ($C_2$), $C_{12}\times D_7$ ($C_2$)

Order 112: $D_{28}:C_2$ ($C_3$)

Order 84: $C_{84}$ ($C_2^2$), $C_7:C_{12}$ ($C_2^2$), $C_3\times D_{14}$ ($C_2^2$)

Order 56: $D_{28}$ ($C_6$), $C_4\times D_7$ ($C_6$), $C_7\times Q_8$ ($C_6$)

Order 42: $C_{42}$ ($C_2^3$)

Order 28: $D_{14}$ ($C_2\times C_6$), $C_{28}$ ($C_2\times C_6$), $C_7:C_4$ ($C_2\times C_6$)

Order 24: $C_3\times Q_8$ ($D_7$)

Order 21: $C_{21}$ ($D_4:C_2$)

Order 14: $C_{14}$ ($C_2^2\times C_6$)

Order 12: $C_{12}$ ($D_{14}$)

Order 8: $Q_8$ ($C_3\times D_7$)

Order 7: $C_7$ ($D_4:C_6$)

Order 6: $C_6$ ($C_2\times D_{14}$)

Order 4: $C_4$ ($C_3\times D_{14}$)

Order 3: $C_3$ ($D_{28}:C_2$)

Order 2: $C_2$ ($C_6\times D_{14}$)

Order 1: $C_1$ ($D_{28}:C_6$)

Series

Derived series

$D_{28}:C_6$

$\rhd$

$C_{14}$

$\rhd$

$C_1$

Chief series

$D_{28}:C_6$

$\rhd$

$C_{12}\times D_7$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$D_{28}:C_6$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_3\times Q_8$

Supergroups

This group is a maximal subgroup of 33 larger groups in the database.

This group is a maximal quotient of 20 larger groups in the database.

Character theory

Complex character table

See the $75 \times 75$ 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.