Abstract group 168.28: $C_{21}:D_4$

• Label: 168.28
• Order: \( 2^{3} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $84$
• Rank: $2$

Group information

Description:	$C_{21}:D_4$
Order:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^3\times F_7$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_3$, $C_7$
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	42
Elements	1	17	2	14	34	6	28	18	12	36	168
Conjugacy classes	1	3	2	1	6	3	2	9	6	18	51
Divisions	1	3	1	1	3	1	1	2	1	2	16
Autjugacy classes	1	3	1	1	3	1	1	2	1	2	16

Dimension	1	2	4	6	12	24
Irr. complex chars.	12	39	0	0	0	0	51
Irr. rational chars.	4	5	1	2	3	1	16

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$84$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	24
Arbitrary	2	4	10

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{42}=[a,b]=[b,c]=1, c^{a}=bc^{13} \rangle$
Permutation group:	Degree $14$ $\langle(1,2)(3,4)(9,10)(11,12)(13,14), (2,4), (5,6,7), (1,3)(2,4), (8,9,11,13,14,12,10)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 27 & 0 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 28 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{43})$
Direct product:	$C_3$ $\, \times\, $ $(C_7:D_4)$
Semidirect product:	$D_{14}$ $\,\rtimes\,$ $C_6$	$C_{21}$ $\,\rtimes\,$ $D_4$	$(C_2\times C_6)$ $\,\rtimes\,$ $D_7$	$C_7$ $\,\rtimes\,$ $(C_3\times D_4)$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $D_{14}$	$C_{42}$ . $C_2^2$	$C_2$ . $(C_3\times D_{14})$	$C_{14}$ . $(C_2\times C_6)$	more information

Elements of the group are displayed as matrices in $\GL_{2}(\F_{43})$.

Homology

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

Subgroups

There are 100 subgroups in 32 conjugacy classes, 18 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $D_{14}$
Commutator:	$G' \simeq$ $C_{14}$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_3\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{42}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{21}:D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
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 168: $C_{21}:D_4$

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

Order 56: $C_7:D_4$

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

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

Order 24: $C_3\times D_4$

Order 21: $C_{21}$

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

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

Order 8: $D_4$

Order 7: $C_7$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 168: $C_{21}:D_4$

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

Order 56: $C_7:D_4$

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

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

Order 24: $C_3\times D_4$

Order 21: $C_{21}$

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

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

Order 8: $D_4$

Order 7: $C_7$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 168: $C_{21}:D_4$ ($C_1$)

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

Order 56: $C_7:D_4$ ($C_3$)

Order 42: $C_{42}$ ($C_2^2$)

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

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

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

Order 12: $C_2\times C_6$ ($D_7$)

Order 7: $C_7$ ($C_3\times D_4$)

Order 6: $C_6$ ($D_{14}$)

Order 4: $C_2^2$ ($C_3\times D_7$)

Order 3: $C_3$ ($C_7:D_4$)

Order 2: $C_2$ ($C_3\times D_{14}$)

Order 1: $C_1$ ($C_{21}:D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 168: $C_{21}:D_4$ ($C_1$)

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

Order 56: $C_7:D_4$ ($C_3$)

Order 42: $C_{42}$ ($C_2^2$)

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

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

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

Order 12: $C_2\times C_6$ ($D_7$)

Order 7: $C_7$ ($C_3\times D_4$)

Order 6: $C_6$ ($D_{14}$)

Order 4: $C_2^2$ ($C_3\times D_7$)

Order 3: $C_3$ ($C_7:D_4$)

Order 2: $C_2$ ($C_3\times D_{14}$)

Order 1: $C_1$ ($C_{21}:D_4$)

Series

Derived series

$C_{21}:D_4$

$\rhd$

$C_{14}$

$\rhd$

$C_1$

Chief series

$C_{21}:D_4$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{21}:D_4$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_2\times C_6$

Supergroups

This group is a maximal subgroup of 37 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 $51 \times 51$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	3A	4A	6A	6B	6C	7A	12A	14A	14B	21A	42A	42B
	Size	1	1	2	14	2	14	2	4	28	6	28	6	12	12	12	24
	2 P	1A	1A	1A	1A	3A	2A	3A	3A	3A	7A	6A	7A	7A	21A	21A	21A
	3 P	1A	2A	2B	2C	1A	4A	2A	2B	2C	7A	4A	14A	14B	7A	14A	14B
	7 P	1A	2A	2B	2C	3A	4A	6A	6B	6C	1A	12A	2A	2B	3A	6A	6B
168.28.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
168.28.1b		$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$
168.28.1c		$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$
168.28.1d		$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$
168.28.1e		$2$	$2$	$2$	$2$	$-1$	$2$	$-1$	$-1$	$-1$	$2$	$-1$	$2$	$2$	$-1$	$-1$	$-1$
168.28.1f		$2$	$2$	$-2$	$-2$	$-1$	$2$	$-1$	$1$	$1$	$2$	$-1$	$2$	$-2$	$-1$	$-1$	$1$
168.28.1g		$2$	$2$	$-2$	$2$	$-1$	$-2$	$-1$	$1$	$-1$	$2$	$1$	$2$	$-2$	$-1$	$-1$	$1$
168.28.1h		$2$	$2$	$2$	$-2$	$-1$	$-2$	$-1$	$-1$	$1$	$2$	$1$	$2$	$2$	$-1$	$-1$	$-1$
168.28.2a		$2$	$-2$	$0$	$0$	$2$	$0$	$-2$	$0$	$0$	$2$	$0$	$-2$	$0$	$2$	$-2$	$0$
168.28.2b		$4$	$-4$	$0$	$0$	$-2$	$0$	$2$	$0$	$0$	$4$	$0$	$-4$	$0$	$-2$	$2$	$0$
168.28.2c		$6$	$6$	$6$	$0$	$6$	$0$	$6$	$6$	$0$	$-1$	$0$	$-1$	$-1$	$-1$	$-1$	$-1$
168.28.2d		$6$	$6$	$-6$	$0$	$6$	$0$	$6$	$-6$	$0$	$-1$	$0$	$-1$	$1$	$-1$	$-1$	$1$
168.28.2e		$12$	$12$	$12$	$0$	$-6$	$0$	$-6$	$-6$	$0$	$-2$	$0$	$-2$	$-2$	$1$	$1$	$1$
168.28.2f		$12$	$-12$	$0$	$0$	$12$	$0$	$-12$	$0$	$0$	$-2$	$0$	$2$	$0$	$-2$	$2$	$0$
168.28.2g		$12$	$12$	$-12$	$0$	$-6$	$0$	$-6$	$6$	$0$	$-2$	$0$	$-2$	$2$	$1$	$1$	$-1$
168.28.2h		$24$	$-24$	$0$	$0$	$-12$	$0$	$12$	$0$	$0$	$-4$	$0$	$4$	$0$	$2$	$-2$	$0$