Abstract group 168.16: $C_3:D_{28}$

• Label: 168.16
• Order: \( 2^{3} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $84$
• Rank: $2$

Group information

Description:	$C_3:D_{28}$
Order:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2\times D_6\times F_7$, of order \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \)
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	14	21	28	42
Elements	1	57	2	6	30	6	6	12	36	12	168
Conjugacy classes	1	3	1	1	3	3	3	3	6	3	27
Divisions	1	3	1	1	2	1	1	1	1	1	13
Autjugacy classes	1	3	1	1	2	1	1	1	1	1	13

Dimension	1	2	4	6	12
Irr. complex chars.	4	17	6	0	0	27
Irr. rational chars.	4	3	1	2	3	13

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$84$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	4	12
Arbitrary	4	4	10

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{28}=c^{3}=[a,c]=1, b^{a}=b^{27}, c^{b}=c^{2} \rangle$
Permutation group:	Degree $14$ $\langle(1,2)(3,4)(9,10)(11,12)(13,14), (1,3,4,2)(6,7), (1,4)(2,3), (5,6,7), (8,9,11,13,14,12,10)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 22 & 23 \\ 7 & 6 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 0 & 15 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 22 & 7 \\ 7 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 14 \\ 7 & 27 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/28\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{14}$ $\,\rtimes\,$ $S_3$	$D_{42}$ $\,\rtimes\,$ $C_2$	$C_{21}$ $\,\rtimes\,$ $D_4$	$C_3$ $\,\rtimes\,$ $D_{28}$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{14}$ . $D_6$	$C_6$ . $D_{14}$	$C_{42}$ . $C_2^2$	$C_2$ . $(S_3\times D_7)$	more information

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 192 subgroups in 32 conjugacy classes, 14 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3\times D_7$
Commutator:	$G' \simeq$ $C_{42}$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $S_3\times D_7$
Fitting:	$\operatorname{Fit} \simeq$ $C_{42}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3:D_{28}$	$G/R \simeq$ $C_1$
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: $C_3:D_{28}$

Order 84: $C_3:C_{28}$, $D_{42}$, $C_3\times D_{14}$

Order 56: $D_{28}$

Order 42: $C_{42}$, $D_{21}$, $C_3\times D_7$

Order 28: $D_{14}$ x 2, $C_{28}$

Order 24: $C_3:D_4$

Order 21: $C_{21}$

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

Order 12: $C_2\times C_6$, $D_6$, $C_3:C_4$

Order 8: $D_4$

Order 7: $C_7$

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

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: $C_3:D_{28}$

Order 84: $C_3:C_{28}$, $D_{42}$, $C_3\times D_{14}$

Order 56: $D_{28}$

Order 42: $C_{42}$, $D_{21}$, $C_3\times D_7$

Order 28: $D_{14}$ x 2, $C_{28}$

Order 24: $C_3:D_4$

Order 21: $C_{21}$

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

Order 12: $C_2\times C_6$, $D_6$, $C_3:C_4$

Order 8: $D_4$

Order 7: $C_7$

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

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: $C_3:D_{28}$ ($C_1$)

Order 84: $C_3:C_{28}$ ($C_2$), $D_{42}$ ($C_2$), $C_3\times D_{14}$ ($C_2$)

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

Order 28: $D_{14}$ ($S_3$)

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

Order 14: $C_{14}$ ($D_6$)

Order 12: $C_3:C_4$ ($D_7$)

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

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

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

Order 2: $C_2$ ($S_3\times D_7$)

Order 1: $C_1$ ($C_3:D_{28}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 168: $C_3:D_{28}$ ($C_1$)

Order 84: $C_3:C_{28}$ ($C_2$), $D_{42}$ ($C_2$), $C_3\times D_{14}$ ($C_2$)

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

Order 28: $D_{14}$ ($S_3$)

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

Order 14: $C_{14}$ ($D_6$)

Order 12: $C_3:C_4$ ($D_7$)

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

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

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

Order 2: $C_2$ ($S_3\times D_7$)

Order 1: $C_1$ ($C_3:D_{28}$)

Series

Derived series

$C_3:D_{28}$

$\rhd$

$C_{42}$

$\rhd$

$C_1$

Chief series

$C_3:D_{28}$

$\rhd$

$C_3\times D_{14}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3:D_{28}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

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

This group is a maximal quotient of 28 larger groups in the database.

Character theory

Complex character table

See the $27 \times 27$ 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	7A	14A	21A	28A	42A
	Size	1	1	14	42	2	6	2	28	6	6	12	36	12
	2 P	1A	1A	1A	1A	3A	2A	3A	3A	7A	7A	21A	14A	21A
	3 P	1A	2A	2B	2C	1A	4A	2A	2B	7A	14A	7A	28A	14A
	7 P	1A	2A	2B	2C	3A	4A	6A	6B	1A	2A	3A	4A	6A
168.16.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
168.16.1b		$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$
168.16.1c		$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$
168.16.1d		$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$
168.16.2a		$2$	$2$	$2$	$0$	$-1$	$0$	$-1$	$-1$	$2$	$2$	$-1$	$0$	$-1$
168.16.2b		$2$	$-2$	$0$	$0$	$2$	$0$	$-2$	$0$	$2$	$-2$	$2$	$0$	$-2$
168.16.2c		$2$	$2$	$-2$	$0$	$-1$	$0$	$-1$	$1$	$2$	$2$	$-1$	$0$	$-1$
168.16.2d		$4$	$-4$	$0$	$0$	$-2$	$0$	$2$	$0$	$4$	$-4$	$-2$	$0$	$2$
168.16.2e		$6$	$6$	$0$	$0$	$6$	$6$	$6$	$0$	$-1$	$-1$	$-1$	$-1$	$-1$
168.16.2f		$6$	$6$	$0$	$0$	$6$	$-6$	$6$	$0$	$-1$	$-1$	$-1$	$1$	$-1$
168.16.2g		$12$	$-12$	$0$	$0$	$12$	$0$	$-12$	$0$	$-2$	$2$	$-2$	$0$	$2$
168.16.4a		$12$	$12$	$0$	$0$	$-6$	$0$	$-6$	$0$	$-2$	$-2$	$1$	$0$	$1$
168.16.4b		$12$	$-12$	$0$	$0$	$-6$	$0$	$6$	$0$	$-2$	$2$	$1$	$0$	$-1$