Abstract group 672.1205: $C_{84}:C_2^3$

• Label: 672.1205
• Order: \( 2^{5} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 3^{2} \)
• Perm deg.: $18$
• Trans deg.: $336$
• Rank: $4$

Group information

Description:	$C_{84}:C_2^3$
Order:	\(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^4.C_2^4.C_{21}.C_6.C_2^3$, of order \(258048\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 7 \)
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	12	14	21	28	42	84
Elements	1	119	2	8	238	6	16	42	12	48	84	96	672
Conjugacy classes	1	15	2	4	30	3	8	21	6	24	42	48	204
Divisions	1	15	1	4	15	1	4	7	1	4	7	4	64
Autjugacy classes	1	3	1	1	3	1	1	2	1	1	2	1	18

Dimension	1	2	4	6	12	24
Irr. complex chars.	48	156	0	0	0	0	204
Irr. rational chars.	16	20	4	8	12	4	64

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$336$
Rank:	$4$
Inequivalent generating quadruples:	$239400$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{2}=d^{84}=[a,b]=[a,c]=[a,d]=[b,c]=[b,d]=1, d^{c}=d^{55} \rangle$
Permutation group:	Degree $18$ $\langle(2,3)(4,5)(6,7)(13,14)(15,16)(17,18), (15,17)(16,18), (11,12)(13,14)(15,16) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 27 & 7 \\ 14 & 27 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 0 & 15 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 14 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/28\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $D_{28}$
Semidirect product:	$C_{84}$ $\,\rtimes\,$ $C_2^3$	$(C_2\times C_{42})$ $\,\rtimes\,$ $D_4$	$C_{42}$ $\,\rtimes\,$ $(C_2\times D_4)$	$C_{14}$ $\,\rtimes\,$ $(C_6\times D_4)$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{42}$ . $C_2^4$	$(C_2^2\times C_6)$ . $D_{14}$	$C_2^3$ . $(C_3\times D_{14})$	$C_6$ . $(C_2^2\times D_{14})$	all 12

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

Homology

Abelianization:	$C_{2}^{3} \times C_{6} \simeq C_{2}^{4} \times C_{3}$
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 2204 subgroups in 472 conjugacy classes, 210 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_6$	$G/Z \simeq$ $D_{14}$
Commutator:	$G' \simeq$ $C_{14}$	$G/G' \simeq$ $C_2^3\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{42}:C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{84}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{84}:C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
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

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 672: $C_{84}:C_2^3$

Order 336: $C_6\times D_{28}$ x 12, $C_{42}:C_2^3$ x 2, $C_2^2\times C_{84}$

Order 224: $C_2^2\times D_{28}$

Order 168: $C_6\times D_{14}$ x 20, $C_3\times D_{28}$ x 16, $C_2\times C_{84}$ x 6, $C_2^2\times C_{42}$

Order 112: $C_2\times D_{28}$ x 12, $C_2^2\times D_{14}$ x 2, $C_2^2\times C_{28}$

Order 96: $C_{12}:C_2^3$

Order 84: $C_3\times D_{14}$ x 32, $C_2\times C_{42}$ x 7, $C_{84}$ x 4

Order 56: $C_2\times D_{14}$ x 20, $D_{28}$ x 16, $C_2\times C_{28}$ x 6, $C_2^2\times C_{14}$

Order 48: $C_6\times D_4$ x 12, $C_2^3\times C_6$ x 2, $C_2^2\times C_{12}$

Order 42: $C_3\times D_7$ x 8, $C_{42}$ x 7

Order 32: $C_2^2\times D_4$

Order 28: $D_{14}$ x 32, $C_2\times C_{14}$ x 7, $C_{28}$ x 4

Order 24: $C_2^2\times C_6$ x 21, $C_3\times D_4$ x 16, $C_2\times C_{12}$ x 6

Order 21: $C_{21}$

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$

Order 14: $D_7$ x 8, $C_{14}$ x 7

Order 12: $C_2\times C_6$ x 39, $C_{12}$ x 4

Order 8: $C_2^3$ x 21, $D_4$ x 16, $C_2\times C_4$ x 6

Order 7: $C_7$

Order 6: $C_6$ x 15

Order 4: $C_2^2$ x 39, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 672: $C_{84}:C_2^3$

Order 336: $C_{42}:C_2^3$, $C_2^2\times C_{84}$, $C_6\times D_{28}$

Order 224: $C_2^2\times D_{28}$

Order 168: $C_6\times D_{14}$ x 2, $C_2^2\times C_{42}$, $C_2\times C_{84}$, $C_3\times D_{28}$

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

Order 96: $C_{12}:C_2^3$

Order 84: $C_2\times C_{42}$ x 2, $C_3\times D_{14}$ x 2, $C_{84}$

Order 56: $C_2\times D_{14}$ x 2, $C_2\times C_{28}$, $D_{28}$, $C_2^2\times C_{14}$

Order 48: $C_2^3\times C_6$, $C_6\times D_4$, $C_2^2\times C_{12}$

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

Order 32: $C_2^2\times D_4$

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

Order 24: $C_2^2\times C_6$ x 3, $C_2\times C_{12}$, $C_3\times D_4$

Order 21: $C_{21}$

Order 16: $C_2^4$, $C_2\times D_4$, $C_2^2\times C_4$

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

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

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

Order 7: $C_7$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 672: $C_{84}:C_2^3$ ($C_1$)

Order 336: $C_6\times D_{28}$ ($C_2$) x 12, $C_{42}:C_2^3$ ($C_2$) x 2, $C_2^2\times C_{84}$ ($C_2$)

Order 224: $C_2^2\times D_{28}$ ($C_3$)

Order 168: $C_3\times D_{28}$ ($C_2^2$) x 16, $C_6\times D_{14}$ ($C_2^2$) x 12, $C_2\times C_{84}$ ($C_2^2$) x 6, $C_2^2\times C_{42}$ ($C_2^2$)

Order 112: $C_2\times D_{28}$ ($C_6$) x 12, $C_2^2\times D_{14}$ ($C_6$) x 2, $C_2^2\times C_{28}$ ($C_6$)

Order 84: $C_3\times D_{14}$ ($C_2^3$) x 8, $C_{84}$ ($C_2^3$) x 4, $C_2\times C_{42}$ ($D_4$) x 4, $C_2\times C_{42}$ ($C_2^3$) x 3

Order 56: $D_{28}$ ($C_2\times C_6$) x 16, $C_2\times D_{14}$ ($C_2\times C_6$) x 12, $C_2\times C_{28}$ ($C_2\times C_6$) x 6, $C_2^2\times C_{14}$ ($C_2\times C_6$)

Order 48: $C_2^2\times C_{12}$ ($D_7$)

Order 42: $C_{42}$ ($C_2\times D_4$) x 6, $C_{42}$ ($C_2^4$)

Order 28: $D_{14}$ ($C_2^2\times C_6$) x 8, $C_2\times C_{14}$ ($C_3\times D_4$) x 4, $C_{28}$ ($C_2^2\times C_6$) x 4, $C_2\times C_{14}$ ($C_2^2\times C_6$) x 3

Order 24: $C_2\times C_{12}$ ($D_{14}$) x 6, $C_2^2\times C_6$ ($D_{14}$)

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

Order 16: $C_2^2\times C_4$ ($C_3\times D_7$)

Order 14: $C_{14}$ ($C_6\times D_4$) x 6, $C_{14}$ ($C_2^3\times C_6$)

Order 12: $C_2\times C_6$ ($D_{28}$) x 4, $C_{12}$ ($C_2\times D_{14}$) x 4, $C_2\times C_6$ ($C_2\times D_{14}$) x 3

Order 8: $C_2\times C_4$ ($C_3\times D_{14}$) x 6, $C_2^3$ ($C_3\times D_{14}$)

Order 7: $C_7$ ($C_{12}:C_2^3$)

Order 6: $C_6$ ($C_2\times D_{28}$) x 6, $C_6$ ($C_2^2\times D_{14}$)

Order 4: $C_2^2$ ($C_3\times D_{28}$) x 4, $C_4$ ($C_6\times D_{14}$) x 4, $C_2^2$ ($C_6\times D_{14}$) x 3

Order 3: $C_3$ ($C_2^2\times D_{28}$)

Order 2: $C_2$ ($C_6\times D_{28}$) x 6, $C_2$ ($C_{42}:C_2^3$)

Order 1: $C_1$ ($C_{84}:C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 672: $C_{84}:C_2^3$ ($C_1$)

Order 336: $C_{42}:C_2^3$ ($C_2$), $C_2^2\times C_{84}$ ($C_2$), $C_6\times D_{28}$ ($C_2$)

Order 224: $C_2^2\times D_{28}$ ($C_3$)

Order 168: $C_2^2\times C_{42}$ ($C_2^2$), $C_6\times D_{14}$ ($C_2^2$), $C_2\times C_{84}$ ($C_2^2$), $C_3\times D_{28}$ ($C_2^2$)

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

Order 84: $C_{84}$ ($C_2^3$), $C_2\times C_{42}$ ($C_2^3$), $C_2\times C_{42}$ ($D_4$), $C_3\times D_{14}$ ($C_2^3$)

Order 56: $C_2\times C_{28}$ ($C_2\times C_6$), $D_{28}$ ($C_2\times C_6$), $C_2^2\times C_{14}$ ($C_2\times C_6$), $C_2\times D_{14}$ ($C_2\times C_6$)

Order 48: $C_2^2\times C_{12}$ ($D_7$)

Order 42: $C_{42}$ ($C_2^4$), $C_{42}$ ($C_2\times D_4$)

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

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

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

Order 16: $C_2^2\times C_4$ ($C_3\times D_7$)

Order 14: $C_{14}$ ($C_2^3\times C_6$), $C_{14}$ ($C_6\times D_4$)

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

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

Order 7: $C_7$ ($C_{12}:C_2^3$)

Order 6: $C_6$ ($C_2^2\times D_{14}$), $C_6$ ($C_2\times D_{28}$)

Order 4: $C_2^2$ ($C_6\times D_{14}$), $C_2^2$ ($C_3\times D_{28}$), $C_4$ ($C_6\times D_{14}$)

Order 3: $C_3$ ($C_2^2\times D_{28}$)

Order 2: $C_2$ ($C_{42}:C_2^3$), $C_2$ ($C_6\times D_{28}$)

Order 1: $C_1$ ($C_{84}:C_2^3$)

Series

Derived series

$C_{84}:C_2^3$

$\rhd$

$C_{14}$

$\rhd$

$C_1$

Chief series

$C_{84}:C_2^3$

$\rhd$

$C_{42}:C_2^3$

$\rhd$

$C_2^2\times C_{42}$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{84}:C_2^3$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_6$

$\lhd$

$C_2^2\times C_{12}$

Supergroups

This group is a maximal subgroup of 93 larger groups in the database.

This group is a maximal quotient of 34 larger groups in the database.

Character theory

Complex character table

See the $204 \times 204$ 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 $64 \times 64$ rational character table.