Abstract group 336.72: $C_{21}:Q_{16}$

• Label: 336.72
• Order: \( 2^{4} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $26$
• Trans deg.: $336$
• Rank: $2$

Group information

Description:	$C_{21}:Q_{16}$
Order:	\(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^2\times D_4\times F_7$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
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	8	12	14	21	24	28	42	84
Elements	1	1	2	34	2	6	28	68	6	12	56	36	12	72	336
Conjugacy classes	1	1	2	3	2	3	2	6	3	6	4	9	6	18	66
Divisions	1	1	1	3	1	1	1	3	1	1	1	2	1	2	20
Autjugacy classes	1	1	1	3	1	1	1	3	1	1	1	2	1	2	20

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	12	45	9	0	0	0	0	66
Irr. rational chars.	4	5	2	2	1	4	2	20

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$336$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	24
Arbitrary	4	6	14

Constructions

Presentation:	$\langle a, b, c \mid b^{8}=c^{21}=[a,c]=1, a^{2}=b^{4}, b^{a}=b^{7}, c^{b}=c^{13} \rangle$
Permutation group:	Degree $26$ $\langle(2,3)(4,5)(6,7)(11,12,15,18)(13,19,21,26)(14,22,24,17)(16,23,25,20), (11,13,15,21) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 48 & 44 \\ 36 & 4 \end{array}\right), \left(\begin{array}{rr} 66 & 13 \\ 52 & 53 \end{array}\right), \left(\begin{array}{rr} 64 & 14 \\ 42 & 22 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right), \left(\begin{array}{rr} 25 & 39 \\ 78 & 38 \end{array}\right), \left(\begin{array}{rr} 27 & 0 \\ 0 & 27 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/91\Z)$
Direct product:	$C_3$ $\, \times\, $ $(C_7:Q_{16})$
Semidirect product:	$C_{21}$ $\,\rtimes\,$ $Q_{16}$	$C_7$ $\,\rtimes\,$ $(C_3\times Q_{16})$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{42}$ . $D_4$	$C_{12}$ . $D_{14}$	$C_{84}$ . $C_2^2$	$(C_3\times Q_8)$ . $D_7$	all 16

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

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 104 subgroups in 36 conjugacy classes, 22 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_7:D_4$
Commutator:	$G' \simeq$ $C_{28}$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_3\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{21}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{21}:Q_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}$
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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Classes of subgroups up to conjugation

Order 336: $C_{21}:Q_{16}$

Order 168: $Q_8\times C_{21}$, $C_7:C_{24}$, $C_{21}:Q_8$

Order 112: $C_7:Q_{16}$

Order 84: $C_{84}$ x 2, $C_7:C_{12}$

Order 56: $C_7:Q_8$, $C_7\times Q_8$, $C_7:C_8$

Order 48: $C_3\times Q_{16}$

Order 42: $C_{42}$

Order 28: $C_{28}$ x 2, $C_7:C_4$

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

Order 21: $C_{21}$

Order 16: $Q_{16}$

Order 14: $C_{14}$

Order 12: $C_{12}$ x 3

Order 8: $Q_8$ x 2, $C_8$

Order 7: $C_7$

Order 6: $C_6$

Order 4: $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 336: $C_{21}:Q_{16}$

Order 168: $Q_8\times C_{21}$, $C_7:C_{24}$, $C_{21}:Q_8$

Order 112: $C_7:Q_{16}$

Order 84: $C_{84}$ x 2, $C_7:C_{12}$

Order 56: $C_7:Q_8$, $C_7\times Q_8$, $C_7:C_8$

Order 48: $C_3\times Q_{16}$

Order 42: $C_{42}$

Order 28: $C_{28}$ x 2, $C_7:C_4$

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

Order 21: $C_{21}$

Order 16: $Q_{16}$

Order 14: $C_{14}$

Order 12: $C_{12}$ x 3

Order 8: $Q_8$ x 2, $C_8$

Order 7: $C_7$

Order 6: $C_6$

Order 4: $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 336: $C_{21}:Q_{16}$ ($C_1$)

Order 168: $Q_8\times C_{21}$ ($C_2$), $C_7:C_{24}$ ($C_2$), $C_{21}:Q_8$ ($C_2$)

Order 112: $C_7:Q_{16}$ ($C_3$)

Order 84: $C_{84}$ ($C_2^2$)

Order 56: $C_7:Q_8$ ($C_6$), $C_7\times Q_8$ ($C_6$), $C_7:C_8$ ($C_6$)

Order 42: $C_{42}$ ($D_4$)

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

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

Order 21: $C_{21}$ ($Q_{16}$)

Order 14: $C_{14}$ ($C_3\times D_4$)

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

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

Order 7: $C_7$ ($C_3\times Q_{16}$)

Order 6: $C_6$ ($C_7:D_4$)

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

Order 3: $C_3$ ($C_7:Q_{16}$)

Order 2: $C_2$ ($C_{21}:D_4$)

Order 1: $C_1$ ($C_{21}:Q_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 336: $C_{21}:Q_{16}$ ($C_1$)

Order 168: $Q_8\times C_{21}$ ($C_2$), $C_7:C_{24}$ ($C_2$), $C_{21}:Q_8$ ($C_2$)

Order 112: $C_7:Q_{16}$ ($C_3$)

Order 84: $C_{84}$ ($C_2^2$)

Order 56: $C_7:Q_8$ ($C_6$), $C_7\times Q_8$ ($C_6$), $C_7:C_8$ ($C_6$)

Order 42: $C_{42}$ ($D_4$)

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

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

Order 21: $C_{21}$ ($Q_{16}$)

Order 14: $C_{14}$ ($C_3\times D_4$)

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

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

Order 7: $C_7$ ($C_3\times Q_{16}$)

Order 6: $C_6$ ($C_7:D_4$)

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

Order 3: $C_3$ ($C_7:Q_{16}$)

Order 2: $C_2$ ($C_{21}:D_4$)

Order 1: $C_1$ ($C_{21}:Q_{16}$)

Series

Derived series

$C_{21}:Q_{16}$

$\rhd$

$C_{28}$

$\rhd$

$C_1$

Chief series

$C_{21}:Q_{16}$

$\rhd$

$Q_8\times C_{21}$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{21}:Q_{16}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_{12}$

$\lhd$

$C_3\times Q_8$

Supergroups

This group is a maximal subgroup of 31 larger groups in the database.

This group is a maximal quotient of 15 larger groups in the database.

Character theory

Complex character table

See the $66 \times 66$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.