Abstract group 672.459: $C_{12}.D_{28}$

• Label: 672.459
• Order: \( 2^{5} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $26$
• Trans deg.: $336$
• Rank: $3$

Group information

Description:	$C_{12}.D_{28}$
Order:	\(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_{42}.(C_2^4\times C_6).C_2^2$, of order \(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \)
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	87	2	144	6	6	24	120	18	12	24	36	144	48	672
Conjugacy classes	1	3	1	5	3	3	2	6	6	3	9	9	12	12	75
Divisions	1	3	1	5	2	1	2	4	2	1	2	2	2	2	30
Autjugacy classes	1	3	1	4	2	1	1	3	2	1	2	2	1	2	26

Dimension	1	2	4	6	8	12	24	48
Irr. complex chars.	8	34	33	0	0	0	0	0	75
Irr. rational chars.	8	6	3	4	1	6	1	1	30

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$336$
Rank:	$3$
Inequivalent generating triples:	$5376$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	48
Arbitrary	4	8	14

Constructions

Presentation:	$\langle a, b, c \mid c^{84}=1, a^{2}=c^{42}, b^{4}=c^{42}, b^{a}=b^{3}, c^{a}=c^{55}, c^{b}=c^{71} \rangle$
Permutation group:	Degree $26$ $\langle(2,3)(4,5)(6,7)(11,12,15,18)(13,19,20,26)(14,22,23,17)(16,24,25,21), (9,10) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{14}:Q_8)$ $\,\rtimes\,$ $S_3$	$(C_{42}:Q_8)$ $\,\rtimes\,$ $C_2$	$C_7$ $\,\rtimes\,$ $(C_{12}.D_4)$	$C_3$ $\,\rtimes\,$ $(C_4.D_{28})$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{84}$ . $D_4$	$C_{12}$ . $D_{28}$	$D_{84}$ . $C_2^2$	$C_{84}$ . $C_2^3$	all 27

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 872 subgroups in 120 conjugacy classes, 44 normal (32 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_6:D_{28}$
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:	$\operatorname{Fit} \simeq$ $C_2\times C_{84}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{12}.D_{28}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}:C_2$
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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Classes of subgroups up to conjugation

Order 672: $C_{12}.D_{28}$

Order 336: $C_{21}:Q_{16}$ x 2, $C_{21}:\SD_{16}$ x 2, $C_{12}.C_{28}$, $D_{84}:C_2$, $C_{42}:Q_8$

Order 224: $C_4.D_{28}$

Order 168: $C_{21}:Q_8$ x 3, $C_3:C_{56}$ x 2, $C_2\times C_{84}$, $C_{21}:D_4$, $D_{84}$, $C_4\times D_{21}$, $C_{21}:Q_8$, $C_{14}:C_{12}$

Order 112: $C_7:Q_{16}$ x 2, $C_{56}:C_2$ x 2, $D_{28}:C_2$, $C_{14}:Q_8$, $C_7\times \OD_{16}$

Order 96: $C_{12}.D_4$

Order 84: $C_{84}$ x 2, $C_7:C_{12}$ x 2, $C_{21}:C_4$, $C_2\times C_{42}$, $D_{42}$

Order 56: $C_7:Q_8$ x 4, $C_{56}$ x 2, $C_2\times C_{28}$, $C_7:D_4$, $C_{14}:C_4$, $D_{28}$, $C_4\times D_7$

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

Order 42: $C_{42}$ x 2, $D_{21}$

Order 32: $Q_{16}:C_2$

Order 28: $C_7:C_4$ x 3, $C_{28}$ x 2, $C_2\times C_{14}$, $D_{14}$

Order 24: $C_3\times Q_8$ x 3, $C_2\times C_{12}$ x 2, $C_3:C_8$ x 2, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$

Order 21: $C_{21}$

Order 16: $Q_{16}$ x 2, $\SD_{16}$ x 2, $\OD_{16}$, $D_4:C_2$, $C_2\times Q_8$

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

Order 12: $C_{12}$ x 4, $C_2\times C_6$, $D_6$, $C_3:C_4$

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 3, $D_4$ x 2, $C_8$ x 2

Order 7: $C_7$

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

Order 4: $C_4$ x 5, $C_2^2$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 672: $C_{12}.D_{28}$

Order 336: $C_{12}.C_{28}$, $C_{21}:Q_{16}$, $C_{21}:\SD_{16}$, $D_{84}:C_2$, $C_{42}:Q_8$

Order 224: $C_4.D_{28}$

Order 168: $C_{21}:Q_8$ x 2, $C_2\times C_{84}$, $C_{21}:D_4$, $D_{84}$, $C_4\times D_{21}$, $C_{21}:Q_8$, $C_3:C_{56}$, $C_{14}:C_{12}$

Order 112: $C_7:Q_{16}$, $C_{56}:C_2$, $D_{28}:C_2$, $C_{14}:Q_8$, $C_7\times \OD_{16}$

Order 96: $C_{12}.D_4$

Order 84: $C_{84}$ x 2, $C_{21}:C_4$, $C_7:C_{12}$, $C_2\times C_{42}$, $D_{42}$

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

Order 48: $C_6\times Q_8$, $D_{12}:C_2$, $C_3:Q_{16}$, $Q_8:S_3$, $C_3:\OD_{16}$

Order 42: $C_{42}$ x 2, $D_{21}$

Order 32: $Q_{16}:C_2$

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

Order 24: $C_2\times C_{12}$ x 2, $C_3\times Q_8$ x 2, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $C_3:C_8$

Order 21: $C_{21}$

Order 16: $Q_{16}$, $\SD_{16}$, $\OD_{16}$, $D_4:C_2$, $C_2\times Q_8$

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

Order 12: $C_{12}$ x 3, $C_2\times C_6$, $D_6$, $C_3:C_4$

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

Order 7: $C_7$

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

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 672: $C_{12}.D_{28}$ ($C_1$)

Order 336: $C_{21}:Q_{16}$ ($C_2$) x 2, $C_{21}:\SD_{16}$ ($C_2$) x 2, $C_{12}.C_{28}$ ($C_2$), $D_{84}:C_2$ ($C_2$), $C_{42}:Q_8$ ($C_2$)

Order 168: $C_3:C_{56}$ ($C_2^2$) x 2, $C_{21}:Q_8$ ($C_2^2$) x 2, $C_2\times C_{84}$ ($C_2^2$), $D_{84}$ ($C_2^2$), $C_{21}:Q_8$ ($C_2^2$)

Order 112: $C_{14}:Q_8$ ($S_3$)

Order 84: $C_{84}$ ($C_2^3$), $C_{84}$ ($D_4$), $C_2\times C_{42}$ ($D_4$)

Order 56: $C_7:Q_8$ ($D_6$) x 2, $C_2\times C_{28}$ ($D_6$)

Order 48: $C_3:\OD_{16}$ ($D_7$)

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

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

Order 24: $C_3:C_8$ ($D_{14}$) x 2, $C_2\times C_{12}$ ($D_{14}$)

Order 21: $C_{21}$ ($Q_{16}:C_2$)

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

Order 12: $C_2\times C_6$ ($D_{28}$), $C_{12}$ ($D_{28}$), $C_{12}$ ($C_2\times D_{14}$)

Order 8: $C_2\times C_4$ ($S_3\times D_7$)

Order 7: $C_7$ ($C_{12}.D_4$)

Order 6: $C_6$ ($C_2\times D_{28}$)

Order 4: $C_2^2$ ($C_3:D_{28}$), $C_4$ ($S_3\times D_{14}$), $C_4$ ($C_3:D_{28}$)

Order 3: $C_3$ ($C_4.D_{28}$)

Order 2: $C_2$ ($C_6:D_{28}$)

Order 1: $C_1$ ($C_{12}.D_{28}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 672: $C_{12}.D_{28}$ ($C_1$)

Order 336: $C_{12}.C_{28}$ ($C_2$), $C_{21}:Q_{16}$ ($C_2$), $C_{21}:\SD_{16}$ ($C_2$), $D_{84}:C_2$ ($C_2$), $C_{42}:Q_8$ ($C_2$)

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

Order 112: $C_{14}:Q_8$ ($S_3$)

Order 84: $C_{84}$ ($C_2^3$), $C_{84}$ ($D_4$), $C_2\times C_{42}$ ($D_4$)

Order 56: $C_2\times C_{28}$ ($D_6$), $C_7:Q_8$ ($D_6$)

Order 48: $C_3:\OD_{16}$ ($D_7$)

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

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

Order 24: $C_2\times C_{12}$ ($D_{14}$), $C_3:C_8$ ($D_{14}$)

Order 21: $C_{21}$ ($Q_{16}:C_2$)

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

Order 12: $C_2\times C_6$ ($D_{28}$), $C_{12}$ ($D_{28}$), $C_{12}$ ($C_2\times D_{14}$)

Order 8: $C_2\times C_4$ ($S_3\times D_7$)

Order 7: $C_7$ ($C_{12}.D_4$)

Order 6: $C_6$ ($C_2\times D_{28}$)

Order 4: $C_2^2$ ($C_3:D_{28}$), $C_4$ ($S_3\times D_{14}$), $C_4$ ($C_3:D_{28}$)

Order 3: $C_3$ ($C_4.D_{28}$)

Order 2: $C_2$ ($C_6:D_{28}$)

Order 1: $C_1$ ($C_{12}.D_{28}$)

Series

Derived series

$C_{12}.D_{28}$

$\rhd$

$C_{84}$

$\rhd$

$C_1$

Chief series

$C_{12}.D_{28}$

$\rhd$

$C_{21}:\SD_{16}$

$\rhd$

$C_{21}:Q_8$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{12}.D_{28}$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times C_4$

Supergroups

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

This group is a maximal quotient of 24 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 $30 \times 30$ rational character table.