Abstract group 1728.2739: $C_6.S_3^2:C_8$

• Label: 1728.2739
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $34$
• Trans deg.: $288$
• Rank: $2$

Group information

Description:	$C_6.S_3^2:C_8$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$\He_3.C_2^6.C_2^2$, of order \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Derived length:	$4$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	39	26	184	150	288	464	576	1728
Conjugacy classes	1	5	3	8	11	8	16	16	68
Divisions	1	5	3	5	11	2	8	2	37
Autjugacy classes	1	5	3	4	11	2	8	3	37

Dimension	1	2	4	6	8	12	16	24
Irr. complex chars.	16	4	26	16	2	4	0	0	68
Irr. rational chars.	4	4	11	2	7	3	2	4	37

Minimal presentations

Permutation degree:	$34$
Transitive degree:	$288$
Rank:	$2$
Inequivalent generating pairs:	$144$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	10	12	14

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{8}=b^{2}=c^{12}=d^{3}=e^{3}=[b,e]=[d,e]=1, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $34$ $\langle(1,2)(3,7)(4,12)(5,14)(6,10)(8,15)(9,16)(11,18)(13,17)(19,20,22,24)(21,23,25,26) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6.S_3^2)$ $\,\rtimes\,$ $C_8$ (2)	$\He_3$ $\,\rtimes\,$ $(C_2^2.\OD_{16})$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6^2.S_3)$ . $D_4$	$C_6$ . $(S_3^2:C_8)$	$(C_6^2.D_6)$ . $C_4$ (2)	$(C_6.C_6^2)$ . $D_4$	all 25

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

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 2693 subgroups in 259 conjugacy classes, 31 normal (29 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_6.\SOPlus(4,2)$
Commutator:	$G' \simeq$ $C_3^2:D_6$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_{12}$	$G/\Phi \simeq$ $\SOPlus(4,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_6.C_6^2$	$G/\operatorname{Fit} \simeq$ $D_4$
Radical:	$R \simeq$ $C_6.S_3^2:C_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $S_3^2:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2.\OD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

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.

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

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

Classes of subgroups up to conjugation

Order 1728: $C_6.S_3^2:C_8$

Order 864: $(C_2\times \He_3).\OD_{16}$, $(C_3\times C_{12}).D_{12}$, $(C_6\times C_{12}).D_6$

Order 432: $C_6^2.D_6$ x 2, $C_6^2.D_6$, $C_6^2.D_6$, $C_2\times \He_3:C_8$, $C_6^2.D_6$, $C_6^2.C_{12}$

Order 288: $C_6^2.C_2^3$, $C_{12}.D_{12}$

Order 216: $C_6.S_3^2$ x 4, $C_6^2.C_6$ x 2, $C_6.C_6^2$, $C_6^2.S_3$, $C_4\times C_3^2:S_3$, $\He_3:C_8$, $C_3^2:C_{24}$, $C_6^2:S_3$

Order 192: $(C_2\times C_4).D_{12}$

Order 144: $C_{12}\times D_6$ x 2, $D_6.D_6$ x 2, $C_6^2.C_2^2$, $C_{12}.D_6$, $C_6^2.C_2^2$, $C_6:C_{24}$, $C_2^2.S_3^2$, $C_6.D_{12}$

Order 108: $C_3^2:D_6$ x 4, $C_3^2:C_{12}$ x 3, $C_2^2\times \He_3$, $C_4\times \He_3$, $\He_3:C_4$

Order 96: $D_6.D_4$, $C_2^2:C_{24}$, $C_{12}.D_4$, $D_6:C_8$, $C_2^3.D_6$

Order 72: $C_6.D_6$ x 6, $C_6:C_{12}$ x 4, $S_3\times C_{12}$ x 4, $C_6\times D_6$ x 2, $C_6\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_3^2:C_8$, $C_3:C_{24}$

Order 64: $C_2^2.\OD_{16}$

Order 54: $C_2\times \He_3$ x 3, $C_3^2:S_3$ x 2

Order 48: $C_4\times D_6$ x 4, $C_6.D_4$ x 4, $C_6:C_8$ x 2, $C_6.C_2^3$ x 2, $C_2\times C_{24}$ x 2, $C_2^2\times C_{12}$, $C_4:C_{12}$, $C_{12}:C_4$

Order 36: $C_6\times S_3$ x 10, $C_3:C_{12}$ x 5, $C_3\times C_{12}$ x 3, $C_3^2:C_4$ x 3, $C_6^2$ x 2

Order 32: $C_2^2:C_8$ x 2, $C_2^2.D_4$

Order 27: $\He_3$

Order 24: $C_6:C_4$ x 10, $C_4\times S_3$ x 10, $C_2\times C_{12}$ x 7, $C_{24}$ x 2, $C_2\times D_6$ x 2, $C_3:C_8$ x 2, $C_2^2\times C_6$

Order 18: $C_3\times C_6$ x 6, $C_3\times S_3$ x 6

Order 16: $C_2^2\times C_4$ x 3, $C_2\times C_8$ x 2, $C_4:C_4$ x 2

Order 12: $D_6$ x 10, $C_{12}$ x 8, $C_3:C_4$ x 8, $C_2\times C_6$ x 7

Order 9: $C_3^2$ x 2

Order 8: $C_2\times C_4$ x 9, $C_8$ x 2, $C_2^3$

Order 6: $C_6$ x 11, $S_3$ x 6

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $C_6.S_3^2:C_8$

Order 864: $(C_2\times \He_3).\OD_{16}$, $(C_3\times C_{12}).D_{12}$, $(C_6\times C_{12}).D_6$

Order 432: $C_6^2.D_6$ x 2, $C_6^2.D_6$, $C_6^2.D_6$, $C_2\times \He_3:C_8$, $C_6^2.D_6$, $C_6^2.C_{12}$

Order 288: $C_6^2.C_2^3$, $C_{12}.D_{12}$

Order 216: $C_6.S_3^2$ x 3, $C_6^2.C_6$ x 2, $C_6.C_6^2$, $C_6^2.S_3$, $C_4\times C_3^2:S_3$, $\He_3:C_8$, $C_3^2:C_{24}$, $C_6^2:S_3$

Order 192: $(C_2\times C_4).D_{12}$

Order 144: $C_{12}\times D_6$ x 2, $D_6.D_6$ x 2, $C_6^2.C_2^2$, $C_{12}.D_6$, $C_6^2.C_2^2$, $C_6:C_{24}$, $C_2^2.S_3^2$, $C_6.D_{12}$

Order 108: $C_3^2:D_6$ x 4, $C_3^2:C_{12}$ x 2, $C_2^2\times \He_3$, $C_4\times \He_3$, $\He_3:C_4$

Order 96: $D_6.D_4$, $C_2^2:C_{24}$, $C_{12}.D_4$, $D_6:C_8$, $C_2^3.D_6$

Order 72: $C_6:C_{12}$ x 4, $S_3\times C_{12}$ x 3, $C_6.D_6$ x 3, $C_6\times D_6$ x 2, $C_6\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_3^2:C_8$, $C_3:C_{24}$

Order 64: $C_2^2.\OD_{16}$

Order 54: $C_2\times \He_3$ x 3, $C_3^2:S_3$ x 2

Order 48: $C_4\times D_6$ x 4, $C_6.D_4$ x 4, $C_6:C_8$ x 2, $C_6.C_2^3$ x 2, $C_2\times C_{24}$ x 2, $C_2^2\times C_{12}$, $C_4:C_{12}$, $C_{12}:C_4$

Order 36: $C_6\times S_3$ x 8, $C_3:C_{12}$ x 4, $C_3\times C_{12}$ x 3, $C_3^2:C_4$ x 2, $C_6^2$ x 2

Order 32: $C_2^2:C_8$ x 2, $C_2^2.D_4$

Order 27: $\He_3$

Order 24: $C_6:C_4$ x 9, $C_2\times C_{12}$ x 7, $C_4\times S_3$ x 6, $C_{24}$ x 2, $C_2\times D_6$ x 2, $C_3:C_8$ x 2, $C_2^2\times C_6$

Order 18: $C_3\times C_6$ x 6, $C_3\times S_3$ x 4

Order 16: $C_2^2\times C_4$ x 3, $C_2\times C_8$ x 2, $C_4:C_4$ x 2

Order 12: $D_6$ x 8, $C_2\times C_6$ x 7, $C_{12}$ x 7, $C_3:C_4$ x 6

Order 9: $C_3^2$ x 2

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

Order 6: $C_6$ x 11, $S_3$ x 4

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $C_6.S_3^2:C_8$ ($C_1$)

Order 864: $(C_2\times \He_3).\OD_{16}$ ($C_2$), $(C_3\times C_{12}).D_{12}$ ($C_2$), $(C_6\times C_{12}).D_6$ ($C_2$)

Order 432: $C_6^2.D_6$ ($C_4$) x 2, $C_6^2.D_6$ ($C_2^2$)

Order 216: $C_6.S_3^2$ ($C_8$) x 2, $C_6.C_6^2$ ($D_4$), $C_6^2.S_3$ ($D_4$), $C_6^2:S_3$ ($C_2\times C_4$)

Order 108: $C_2^2\times \He_3$ ($C_2^2:C_4$), $C_3^2:D_6$ ($\OD_{16}$), $C_3^2:D_6$ ($C_2\times C_8$)

Order 54: $C_2\times \He_3$ ($C_4.D_4$), $C_2\times \He_3$ ($C_2^3:C_4$), $C_2\times \He_3$ ($C_2^2:C_8$)

Order 27: $\He_3$ ($C_2^2.\OD_{16}$)

Order 24: $C_2\times C_{12}$ ($\SOPlus(4,2)$)

Order 12: $C_2\times C_6$ ($S_3^2:C_4$)

Order 8: $C_2\times C_4$ ($\He_3:D_4$)

Order 6: $C_6$ ($C_6^2.D_4$), $C_6$ ($C_4.\SOPlus(4,2)$), $C_6$ ($S_3^2:C_8$)

Order 4: $C_2^2$ ($C_6.\SOPlus(4,2)$)

Order 3: $C_3$ ($(C_6\times C_{12}).D_4$)

Order 2: $C_2$ ($(C_2^2\times \He_3).D_4$), $C_2$ ($C_{12}.\SOPlus(4,2)$), $C_2$ ($C_{12}.\SOPlus(4,2)$)

Order 1: $C_1$ ($C_6.S_3^2:C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $C_6.S_3^2:C_8$ ($C_1$)

Order 864: $(C_2\times \He_3).\OD_{16}$ ($C_2$), $(C_3\times C_{12}).D_{12}$ ($C_2$), $(C_6\times C_{12}).D_6$ ($C_2$)

Order 432: $C_6^2.D_6$ ($C_4$) x 2, $C_6^2.D_6$ ($C_2^2$)

Order 216: $C_6.C_6^2$ ($D_4$), $C_6^2.S_3$ ($D_4$), $C_6.S_3^2$ ($C_8$), $C_6^2:S_3$ ($C_2\times C_4$)

Order 108: $C_2^2\times \He_3$ ($C_2^2:C_4$), $C_3^2:D_6$ ($\OD_{16}$), $C_3^2:D_6$ ($C_2\times C_8$)

Order 54: $C_2\times \He_3$ ($C_4.D_4$), $C_2\times \He_3$ ($C_2^3:C_4$), $C_2\times \He_3$ ($C_2^2:C_8$)

Order 27: $\He_3$ ($C_2^2.\OD_{16}$)

Order 24: $C_2\times C_{12}$ ($\SOPlus(4,2)$)

Order 12: $C_2\times C_6$ ($S_3^2:C_4$)

Order 8: $C_2\times C_4$ ($\He_3:D_4$)

Order 6: $C_6$ ($C_6^2.D_4$), $C_6$ ($C_4.\SOPlus(4,2)$), $C_6$ ($S_3^2:C_8$)

Order 4: $C_2^2$ ($C_6.\SOPlus(4,2)$)

Order 3: $C_3$ ($(C_6\times C_{12}).D_4$)

Order 2: $C_2$ ($(C_2^2\times \He_3).D_4$), $C_2$ ($C_{12}.\SOPlus(4,2)$), $C_2$ ($C_{12}.\SOPlus(4,2)$)

Order 1: $C_1$ ($C_6.S_3^2:C_8$)

Series

Derived series

$C_6.S_3^2:C_8$

$\rhd$

$C_3^2:D_6$

$\rhd$

$\He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_6.S_3^2:C_8$

$\rhd$

$(C_3\times C_{12}).D_{12}$

$\rhd$

$C_6^2.D_6$

$\rhd$

$C_6.C_6^2$

$\rhd$

$C_2^2\times \He_3$

$\rhd$

$C_2\times \He_3$

$\rhd$

$\He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6.S_3^2:C_8$

$\rhd$

$C_3^2:D_6$

$\rhd$

$C_2\times \He_3$

$\rhd$

$\He_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

Character theory

Complex character table

See the $68 \times 68$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $37 \times 37$ rational character table.