Abstract group 672.827: $C_7:C_{12}\times D_4$

• Label: 672.827
• Order: \( 2^{5} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $18$
• Trans deg.: $336$
• Rank: $3$

Group information

Description:	$C_7:C_{12}\times D_4$
Order:	\(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_{14}.(C_6\times D_4).C_2^4$, of order \(10752\)\(\medspace = 2^{9} \cdot 3 \cdot 7 \)
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	12	14	21	28	42	84
Elements	1	11	2	116	22	6	232	66	12	24	132	48	672
Conjugacy classes	1	7	2	12	14	3	24	21	6	6	42	12	150
Divisions	1	7	1	7	7	1	7	7	1	2	7	2	50
Autjugacy classes	1	4	1	4	4	1	4	4	1	1	4	1	30

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	48	84	18	0	0	0	0	150
Irr. rational chars.	8	14	7	8	1	10	2	50

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$336$
Rank:	$3$
Inequivalent generating triples:	$8736$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	12

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{4}=c^{84}=[a,b]=1, c^{a}=c^{43}, c^{b}=c^{55} \rangle$
Permutation group:	Degree $18$ $\langle(2,3)(4,5)(6,7)(8,9,10,11)(12,13)(14,15), (12,14)(13,15), (14,15), (16,17,18), (8,10)(9,11), (12,13)(14,15), (1,2,4,6,7,5,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 25 & 26 \\ 26 & 38 \end{array}\right), \left(\begin{array}{rr} 5 & 6 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right), \left(\begin{array}{rr} 38 & 0 \\ 13 & 25 \end{array}\right), \left(\begin{array}{rr} 22 & 12 \\ 6 & 37 \end{array}\right), \left(\begin{array}{rr} 14 & 0 \\ 0 & 14 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/39\Z)$
Direct product:	$C_3$ $\, \times\, $ $D_4$ $\, \times\, $ $(C_7:C_4)$
Semidirect product:	$(C_{42}.D_4)$ $\,\rtimes\,$ $C_2$ (2)	$(D_4\times C_{21})$ $\,\rtimes\,$ $C_4$ (4)	$(C_{14}.D_4)$ $\,\rtimes\,$ $C_6$ (2)	$(C_7\times D_4)$ $\,\rtimes\,$ $C_{12}$ (4)	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6\times D_4)$ . $D_7$	$C_6$ . $(D_4\times D_7)$	$C_6$ . $(D_4:D_7)$	$(D_4\times C_{42})$ . $C_2$	all 30

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

Homology

Abelianization:	$C_{2}^{2} \times C_{12} \simeq C_{2}^{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 572 subgroups in 188 conjugacy classes, 102 normal (38 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $C_2\times D_{14}$
Commutator:	$G' \simeq$ $C_{14}$	$G/G' \simeq$ $C_2^2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_6\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_{42}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_7:C_{12}\times D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times D_4$
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

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 672: $C_7:C_{12}\times D_4$

Order 336: $C_{42}.D_4$ x 2, $C_{42}.C_2^3$ x 2, $C_{28}:C_{12}$, $C_{28}:C_{12}$, $D_4\times C_{42}$

Order 224: $C_2^3.D_{14}$

Order 168: $C_{14}:C_{12}$ x 8, $D_4\times C_{21}$ x 4, $C_2^2\times C_{42}$ x 2, $C_2\times C_{84}$

Order 112: $C_2^2.D_{14}$ x 2, $C_{14}.D_4$ x 2, $D_4\times C_{14}$, $C_{28}:C_4$, $C_{28}:C_4$

Order 96: $D_4\times C_{12}$

Order 84: $C_2\times C_{42}$ x 9, $C_7:C_{12}$ x 5, $C_{84}$ x 2

Order 56: $C_{14}:C_4$ x 8, $C_7\times D_4$ x 4, $C_2^2\times C_{14}$ x 2, $C_2\times C_{28}$

Order 48: $C_2^2\times C_{12}$ x 2, $C_2^2:C_{12}$ x 2, $C_6\times D_4$, $C_4:C_{12}$, $C_4\times C_{12}$

Order 42: $C_{42}$ x 7

Order 32: $C_4\times D_4$

Order 28: $C_2\times C_{14}$ x 9, $C_7:C_4$ x 5, $C_{28}$ x 2

Order 24: $C_2\times C_{12}$ x 9, $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2

Order 21: $C_{21}$

Order 16: $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2\times D_4$

Order 14: $C_{14}$ x 7

Order 12: $C_2\times C_6$ x 9, $C_{12}$ x 7

Order 8: $C_2\times C_4$ x 9, $D_4$ x 4, $C_2^3$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 9, $C_4$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 672: $C_7:C_{12}\times D_4$

Order 336: $C_{42}.D_4$, $C_{28}:C_{12}$, $C_{28}:C_{12}$, $D_4\times C_{42}$, $C_{42}.C_2^3$

Order 224: $C_2^3.D_{14}$

Order 168: $C_{14}:C_{12}$ x 4, $C_2^2\times C_{42}$, $D_4\times C_{21}$, $C_2\times C_{84}$

Order 112: $D_4\times C_{14}$, $C_2^2.D_{14}$, $C_{14}.D_4$, $C_{28}:C_4$, $C_{28}:C_4$

Order 96: $D_4\times C_{12}$

Order 84: $C_2\times C_{42}$ x 4, $C_7:C_{12}$ x 3, $C_{84}$

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

Order 48: $C_6\times D_4$, $C_2^2\times C_{12}$, $C_4:C_{12}$, $C_2^2:C_{12}$, $C_4\times C_{12}$

Order 42: $C_{42}$ x 4

Order 32: $C_4\times D_4$

Order 28: $C_2\times C_{14}$ x 4, $C_7:C_4$ x 3, $C_{28}$

Order 24: $C_2\times C_{12}$ x 5, $C_2^2\times C_6$, $C_3\times D_4$

Order 21: $C_{21}$

Order 16: $C_4:C_4$, $C_2^2:C_4$, $C_4^2$, $C_2\times D_4$, $C_2^2\times C_4$

Order 14: $C_{14}$ x 4

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

Order 8: $C_2\times C_4$ x 5, $C_2^3$, $D_4$

Order 7: $C_7$

Order 6: $C_6$ x 4

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 672: $C_7:C_{12}\times D_4$ ($C_1$)

Order 336: $C_{42}.D_4$ ($C_2$) x 2, $C_{42}.C_2^3$ ($C_2$) x 2, $C_{28}:C_{12}$ ($C_2$), $C_{28}:C_{12}$ ($C_2$), $D_4\times C_{42}$ ($C_2$)

Order 224: $C_2^3.D_{14}$ ($C_3$)

Order 168: $D_4\times C_{21}$ ($C_4$) x 4, $C_{14}:C_{12}$ ($C_2^2$) x 4, $C_2^2\times C_{42}$ ($C_2^2$) x 2, $C_2\times C_{84}$ ($C_2^2$)

Order 112: $C_2^2.D_{14}$ ($C_6$) x 2, $C_{14}.D_4$ ($C_6$) x 2, $D_4\times C_{14}$ ($C_6$), $C_{28}:C_4$ ($C_6$), $C_{28}:C_4$ ($C_6$)

Order 84: $C_2\times C_{42}$ ($C_2\times C_4$) x 4, $C_{84}$ ($C_2\times C_4$) x 2, $C_7:C_{12}$ ($D_4$) x 2, $C_2\times C_{42}$ ($C_2^3$)

Order 56: $C_7\times D_4$ ($C_{12}$) x 4, $C_{14}:C_4$ ($C_2\times C_6$) x 4, $C_2^2\times C_{14}$ ($C_2\times C_6$) x 2, $C_2\times C_{28}$ ($C_2\times C_6$)

Order 48: $C_6\times D_4$ ($D_7$)

Order 42: $C_{42}$ ($D_4:C_2$), $C_{42}$ ($C_2\times D_4$), $C_{42}$ ($C_2^2\times C_4$)

Order 28: $C_2\times C_{14}$ ($C_2\times C_{12}$) x 4, $C_{28}$ ($C_2\times C_{12}$) x 2, $C_7:C_4$ ($C_3\times D_4$) x 2, $C_2\times C_{14}$ ($C_2^2\times C_6$)

Order 24: $C_3\times D_4$ ($C_7:C_4$) x 4, $C_2^2\times C_6$ ($D_{14}$) x 2, $C_2\times C_{12}$ ($D_{14}$)

Order 21: $C_{21}$ ($C_4\times D_4$)

Order 16: $C_2\times D_4$ ($C_3\times D_7$)

Order 14: $C_{14}$ ($D_4:C_6$), $C_{14}$ ($C_6\times D_4$), $C_{14}$ ($C_2^2\times C_{12}$)

Order 12: $C_2\times C_6$ ($C_{14}:C_4$) x 4, $C_{12}$ ($C_{14}:C_4$) x 2, $C_2\times C_6$ ($C_2\times D_{14}$)

Order 8: $D_4$ ($C_7:C_{12}$) x 4, $C_2^3$ ($C_3\times D_{14}$) x 2, $C_2\times C_4$ ($C_3\times D_{14}$)

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

Order 6: $C_6$ ($C_2^2.D_{14}$), $C_6$ ($D_4:D_7$), $C_6$ ($D_4\times D_7$)

Order 4: $C_2^2$ ($C_{14}:C_{12}$) x 4, $C_4$ ($C_{14}:C_{12}$) x 2, $C_2^2$ ($C_6\times D_{14}$)

Order 3: $C_3$ ($C_2^3.D_{14}$)

Order 2: $C_2$ ($C_{42}.C_2^3$), $C_2$ ($C_{12}.D_{14}$), $C_2$ ($C_{12}:D_{14}$)

Order 1: $C_1$ ($C_7:C_{12}\times D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 672: $C_7:C_{12}\times D_4$ ($C_1$)

Order 336: $C_{42}.D_4$ ($C_2$), $C_{28}:C_{12}$ ($C_2$), $C_{28}:C_{12}$ ($C_2$), $D_4\times C_{42}$ ($C_2$), $C_{42}.C_2^3$ ($C_2$)

Order 224: $C_2^3.D_{14}$ ($C_3$)

Order 168: $C_{14}:C_{12}$ ($C_2^2$) x 3, $C_2^2\times C_{42}$ ($C_2^2$), $D_4\times C_{21}$ ($C_4$), $C_2\times C_{84}$ ($C_2^2$)

Order 112: $D_4\times C_{14}$ ($C_6$), $C_2^2.D_{14}$ ($C_6$), $C_{14}.D_4$ ($C_6$), $C_{28}:C_4$ ($C_6$), $C_{28}:C_4$ ($C_6$)

Order 84: $C_{84}$ ($C_2\times C_4$), $C_7:C_{12}$ ($D_4$), $C_2\times C_{42}$ ($C_2^3$), $C_2\times C_{42}$ ($C_2\times C_4$)

Order 56: $C_{14}:C_4$ ($C_2\times C_6$) x 3, $C_7\times D_4$ ($C_{12}$), $C_2\times C_{28}$ ($C_2\times C_6$), $C_2^2\times C_{14}$ ($C_2\times C_6$)

Order 48: $C_6\times D_4$ ($D_7$)

Order 42: $C_{42}$ ($D_4:C_2$), $C_{42}$ ($C_2\times D_4$), $C_{42}$ ($C_2^2\times C_4$)

Order 28: $C_2\times C_{14}$ ($C_2\times C_{12}$), $C_2\times C_{14}$ ($C_2^2\times C_6$), $C_{28}$ ($C_2\times C_{12}$), $C_7:C_4$ ($C_3\times D_4$)

Order 24: $C_2\times C_{12}$ ($D_{14}$), $C_2^2\times C_6$ ($D_{14}$), $C_3\times D_4$ ($C_7:C_4$)

Order 21: $C_{21}$ ($C_4\times D_4$)

Order 16: $C_2\times D_4$ ($C_3\times D_7$)

Order 14: $C_{14}$ ($D_4:C_6$), $C_{14}$ ($C_6\times D_4$), $C_{14}$ ($C_2^2\times C_{12}$)

Order 12: $C_2\times C_6$ ($C_{14}:C_4$), $C_2\times C_6$ ($C_2\times D_{14}$), $C_{12}$ ($C_{14}:C_4$)

Order 8: $C_2^3$ ($C_3\times D_{14}$), $D_4$ ($C_7:C_{12}$), $C_2\times C_4$ ($C_3\times D_{14}$)

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

Order 6: $C_6$ ($C_2^2.D_{14}$), $C_6$ ($D_4:D_7$), $C_6$ ($D_4\times D_7$)

Order 4: $C_2^2$ ($C_6\times D_{14}$), $C_2^2$ ($C_{14}:C_{12}$), $C_4$ ($C_{14}:C_{12}$)

Order 3: $C_3$ ($C_2^3.D_{14}$)

Order 2: $C_2$ ($C_{42}.C_2^3$), $C_2$ ($C_{12}.D_{14}$), $C_2$ ($C_{12}:D_{14}$)

Order 1: $C_1$ ($C_7:C_{12}\times D_4$)

Series

Derived series

$C_7:C_{12}\times D_4$

$\rhd$

$C_{14}$

$\rhd$

$C_1$

Chief series

$C_7:C_{12}\times D_4$

$\rhd$

$C_{42}.C_2^3$

$\rhd$

$C_2^2\times C_{42}$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_7:C_{12}\times D_4$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

$\lhd$

$C_6\times D_4$

Supergroups

This group is a maximal subgroup of 124 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 $150 \times 150$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $50 \times 50$ rational character table.