Abstract group 1728.2061: $C_4.D_8\times \He_3$

• Label: 1728.2061
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3 \)
• Perm deg.: $25$
• Trans deg.: $576$
• Rank: $2$

Group information

Description:	$C_4.D_8\times \He_3$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^2\times D_4^2\times \AGL(2,3)$, of order \(110592\)\(\medspace = 2^{12} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Nilpotency class:	$3$
Derived length:	$2$

This group is nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	3	26	28	78	32	728	832	1728
Conjugacy classes	1	3	10	7	30	8	70	80	209
Divisions	1	3	5	7	15	2	35	10	78
Autjugacy classes	1	3	2	6	6	2	12	4	36

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	72	90	16	9	20	0	2	0	209
Irr. rational chars.	4	20	0	21	4	20	4	5	78

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$576$
Rank:	$2$
Inequivalent generating pairs:	$6$

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:	$\langle a, b, c \mid b^{12}=c^{12}=1, a^{12}=b^{6}c^{6}, b^{a}=b^{7}c^{9}, c^{a}=b^{2}c, c^{b}=c^{7} \rangle$
Permutation group:	Degree $25$ $\langle(2,5)(3,7)(6,8)(9,10,11,13,12,14,16,15), (1,2,4,6)(3,8,7,5)(9,10)(11,15) \!\cdots\! \rangle$
Direct product:	$\He_3$ $\, \times\, $ $(C_4.D_8)$
Semidirect product:	$C_3^2$ $\,\rtimes\,$ $(C_{12}.D_8)$	$(C_4^2.C_6^2)$ $\,\rtimes\,$ $C_3$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6.C_6^2)$ . $D_4$ (2)	$C_6$ . $(C_6^2.D_4)$	$C_6$ . $(C_6^2.D_4)$	$(C_4\times \He_3)$ . $D_8$	all 54

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

Homology

Abelianization:	$C_{6} \times C_{12} \simeq C_{2} \times C_{4} \times C_{3}^{2}$
Schur multiplier:	$C_{3} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 1083 subgroups in 352 conjugacy classes, 133 normal (51 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $C_6^2:C_4$
Commutator:	$G' \simeq$ $C_2\times C_{12}$	$G/G' \simeq$ $C_6\times C_{12}$
Frattini:	$\Phi \simeq$ $C_4\times C_{12}$	$G/\Phi \simeq$ $C_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_4.D_8\times \He_3$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_4.D_8\times \He_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_6^2:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.D_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3$

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

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

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.

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: $C_4.D_8\times \He_3$

Order 864: $C_{12}^2.C_6$ x 2, $C_{12}^2.C_6$

Order 576: $C_4^2.C_6^2$ x 4

Order 432: $C_4:C_4\times \He_3$ x 3, $C_{12}.C_6^2$ x 2, $C_{12}.C_6^2$, $C_4^2\times \He_3$

Order 288: $C_{12}:C_{24}$ x 8, $C_{12}^2.C_2$ x 4

Order 216: $C_6.C_6^2$ x 5, $Q_8\times \He_3$ x 2, $C_8\times \He_3$ x 2

Order 192: $C_{12}.D_8$ x 5

Order 144: $C_{12}:C_{12}$ x 12, $C_6\times C_{24}$ x 8, $C_4.C_6^2$ x 4, $C_{12}^2$ x 4

Order 108: $C_4\times \He_3$ x 7, $C_2^2\times \He_3$

Order 96: $C_4:C_{24}$ x 10, $C_{12}:Q_8$ x 5

Order 72: $C_6\times C_{12}$ x 20, $Q_8\times C_3^2$ x 8, $C_3\times C_{24}$ x 8

Order 64: $C_4.D_8$

Order 54: $C_2\times \He_3$ x 3

Order 48: $C_4:C_{12}$ x 15, $C_2\times C_{24}$ x 10, $C_6\times Q_8$ x 5, $C_4\times C_{12}$ x 5

Order 36: $C_3\times C_{12}$ x 28, $C_6^2$ x 4

Order 32: $C_4:C_8$ x 2, $C_4:Q_8$

Order 27: $\He_3$

Order 24: $C_2\times C_{12}$ x 25, $C_{24}$ x 10, $C_3\times Q_8$ x 10

Order 18: $C_3\times C_6$ x 12

Order 16: $C_4:C_4$ x 3, $C_2\times C_8$ x 2, $C_4^2$, $C_2\times Q_8$

Order 12: $C_{12}$ x 35, $C_2\times C_6$ x 5

Order 9: $C_3^2$ x 4

Order 8: $C_2\times C_4$ x 5, $Q_8$ x 2, $C_8$ x 2

Order 6: $C_6$ x 15

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

Order 3: $C_3$ x 5

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $C_4.D_8\times \He_3$

Order 864: $C_{12}^2.C_6$ x 2, $C_{12}^2.C_6$

Order 576: $C_4^2.C_6^2$

Order 432: $C_{12}.C_6^2$ x 2, $C_4:C_4\times \He_3$ x 2, $C_{12}.C_6^2$, $C_4^2\times \He_3$

Order 288: $C_{12}:C_{24}$ x 2, $C_{12}^2.C_2$

Order 216: $C_6.C_6^2$ x 4, $Q_8\times \He_3$ x 2, $C_8\times \He_3$ x 2

Order 192: $C_{12}.D_8$ x 2

Order 144: $C_6\times C_{24}$ x 2, $C_{12}:C_{12}$ x 2, $C_4.C_6^2$, $C_{12}^2$

Order 108: $C_4\times \He_3$ x 6, $C_2^2\times \He_3$

Order 96: $C_4:C_{24}$ x 4, $C_{12}:Q_8$ x 2

Order 72: $C_6\times C_{12}$ x 4, $Q_8\times C_3^2$ x 2, $C_3\times C_{24}$ x 2

Order 64: $C_4.D_8$

Order 54: $C_2\times \He_3$ x 3

Order 48: $C_2\times C_{24}$ x 4, $C_4:C_{12}$ x 4, $C_6\times Q_8$ x 2, $C_4\times C_{12}$ x 2

Order 36: $C_3\times C_{12}$ x 6, $C_6^2$

Order 32: $C_4:C_8$ x 2, $C_4:Q_8$

Order 27: $\He_3$

Order 24: $C_2\times C_{12}$ x 8, $C_{24}$ x 4, $C_3\times Q_8$ x 4

Order 18: $C_3\times C_6$ x 3

Order 16: $C_2\times C_8$ x 2, $C_4:C_4$ x 2, $C_4^2$, $C_2\times Q_8$

Order 12: $C_{12}$ x 12, $C_2\times C_6$ x 2

Order 9: $C_3^2$

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

Order 6: $C_6$ x 6

Order 4: $C_4$ x 6, $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $C_4.D_8\times \He_3$ ($C_1$)

Order 864: $C_{12}^2.C_6$ ($C_2$) x 2, $C_{12}^2.C_6$ ($C_2$)

Order 576: $C_4^2.C_6^2$ ($C_3$) x 4

Order 432: $C_4:C_4\times \He_3$ ($C_4$) x 2, $C_4^2\times \He_3$ ($C_2^2$)

Order 288: $C_{12}:C_{24}$ ($C_6$) x 8, $C_{12}^2.C_2$ ($C_6$) x 4

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

Order 192: $C_{12}.D_8$ ($C_3^2$)

Order 144: $C_{12}:C_{12}$ ($C_{12}$) x 8, $C_{12}^2$ ($C_2\times C_6$) x 4

Order 108: $C_4\times \He_3$ ($\SD_{16}$) x 2, $C_2^2\times \He_3$ ($C_2^2:C_4$), $C_4\times \He_3$ ($Q_{16}$), $C_4\times \He_3$ ($D_8$)

Order 96: $C_4:C_{24}$ ($C_3\times C_6$) x 2, $C_{12}:Q_8$ ($C_3\times C_6$)

Order 72: $C_6\times C_{12}$ ($C_3\times D_4$) x 8, $C_6\times C_{12}$ ($C_2\times C_{12}$) x 4

Order 64: $C_4.D_8$ ($\He_3$)

Order 54: $C_2\times \He_3$ ($D_4:C_4$), $C_2\times \He_3$ ($C_4.D_4$), $C_2\times \He_3$ ($Q_8:C_4$)

Order 48: $C_4:C_{12}$ ($C_3\times C_{12}$) x 2, $C_4\times C_{12}$ ($C_6^2$)

Order 36: $C_3\times C_{12}$ ($C_3\times \SD_{16}$) x 8, $C_3\times C_{12}$ ($C_3\times Q_{16}$) x 4, $C_3\times C_{12}$ ($C_3\times D_8$) x 4, $C_6^2$ ($C_2^2:C_{12}$) x 4

Order 32: $C_4:C_8$ ($C_2\times \He_3$) x 2, $C_4:Q_8$ ($C_2\times \He_3$)

Order 27: $\He_3$ ($C_4.D_8$)

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

Order 18: $C_3\times C_6$ ($Q_8:C_{12}$) x 4, $C_3\times C_6$ ($D_4:C_{12}$) x 4, $C_3\times C_6$ ($C_{12}.D_4$) x 4

Order 16: $C_4:C_4$ ($C_4\times \He_3$) x 2, $C_4^2$ ($C_2^2\times \He_3$)

Order 12: $C_{12}$ ($C_3^2\times \SD_{16}$) x 2, $C_2\times C_6$ ($C_6^2:C_4$), $C_{12}$ ($C_3^2\times Q_{16}$), $C_{12}$ ($C_3^2\times D_8$)

Order 9: $C_3^2$ ($C_{12}.D_8$) x 4

Order 8: $C_2\times C_4$ ($D_4\times \He_3$) x 2, $C_2\times C_4$ ($C_6.C_6^2$)

Order 6: $C_6$ ($C_6^2.D_4$), $C_6$ ($C_6^2.D_4$), $C_6$ ($C_3\times C_{12}.D_4$)

Order 4: $C_4$ ($\SD_{16}\times \He_3$) x 2, $C_2^2$ ($C_6^2:C_{12}$), $C_4$ ($Q_{16}\times \He_3$), $C_4$ ($D_8\times \He_3$)

Order 3: $C_3$ ($C_4^2.C_6^2$)

Order 2: $C_2$ ($Q_8:C_4\times \He_3$), $C_2$ ($(C_6\times C_{24}):C_6$), $C_2$ ($(C_6\times C_{12}).C_{12}$)

Order 1: $C_1$ ($C_4.D_8\times \He_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $C_4.D_8\times \He_3$ ($C_1$)

Order 864: $C_{12}^2.C_6$ ($C_2$) x 2, $C_{12}^2.C_6$ ($C_2$)

Order 576: $C_4^2.C_6^2$ ($C_3$)

Order 432: $C_4:C_4\times \He_3$ ($C_4$), $C_4^2\times \He_3$ ($C_2^2$)

Order 288: $C_{12}:C_{24}$ ($C_6$) x 2, $C_{12}^2.C_2$ ($C_6$)

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

Order 192: $C_{12}.D_8$ ($C_3^2$)

Order 144: $C_{12}:C_{12}$ ($C_{12}$), $C_{12}^2$ ($C_2\times C_6$)

Order 108: $C_4\times \He_3$ ($\SD_{16}$) x 2, $C_2^2\times \He_3$ ($C_2^2:C_4$), $C_4\times \He_3$ ($Q_{16}$), $C_4\times \He_3$ ($D_8$)

Order 96: $C_4:C_{24}$ ($C_3\times C_6$) x 2, $C_{12}:Q_8$ ($C_3\times C_6$)

Order 72: $C_6\times C_{12}$ ($C_3\times D_4$) x 2, $C_6\times C_{12}$ ($C_2\times C_{12}$)

Order 64: $C_4.D_8$ ($\He_3$)

Order 54: $C_2\times \He_3$ ($D_4:C_4$), $C_2\times \He_3$ ($C_4.D_4$), $C_2\times \He_3$ ($Q_8:C_4$)

Order 48: $C_4:C_{12}$ ($C_3\times C_{12}$), $C_4\times C_{12}$ ($C_6^2$)

Order 36: $C_3\times C_{12}$ ($C_3\times \SD_{16}$) x 2, $C_3\times C_{12}$ ($C_3\times Q_{16}$), $C_3\times C_{12}$ ($C_3\times D_8$), $C_6^2$ ($C_2^2:C_{12}$)

Order 32: $C_4:C_8$ ($C_2\times \He_3$) x 2, $C_4:Q_8$ ($C_2\times \He_3$)

Order 27: $\He_3$ ($C_4.D_8$)

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

Order 18: $C_3\times C_6$ ($Q_8:C_{12}$), $C_3\times C_6$ ($D_4:C_{12}$), $C_3\times C_6$ ($C_{12}.D_4$)

Order 16: $C_4:C_4$ ($C_4\times \He_3$), $C_4^2$ ($C_2^2\times \He_3$)

Order 12: $C_{12}$ ($C_3^2\times \SD_{16}$) x 2, $C_2\times C_6$ ($C_6^2:C_4$), $C_{12}$ ($C_3^2\times Q_{16}$), $C_{12}$ ($C_3^2\times D_8$)

Order 9: $C_3^2$ ($C_{12}.D_8$)

Order 8: $C_2\times C_4$ ($D_4\times \He_3$) x 2, $C_2\times C_4$ ($C_6.C_6^2$)

Order 6: $C_6$ ($C_6^2.D_4$), $C_6$ ($C_6^2.D_4$), $C_6$ ($C_3\times C_{12}.D_4$)

Order 4: $C_4$ ($\SD_{16}\times \He_3$) x 2, $C_2^2$ ($C_6^2:C_{12}$), $C_4$ ($Q_{16}\times \He_3$), $C_4$ ($D_8\times \He_3$)

Order 3: $C_3$ ($C_4^2.C_6^2$)

Order 2: $C_2$ ($Q_8:C_4\times \He_3$), $C_2$ ($(C_6\times C_{24}):C_6$), $C_2$ ($(C_6\times C_{12}).C_{12}$)

Order 1: $C_1$ ($C_4.D_8\times \He_3$)

Series

Derived series

$C_4.D_8\times \He_3$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_1$

Chief series

$C_4.D_8\times \He_3$

$\rhd$

$C_{12}^2.C_6$

$\rhd$

$C_4^2\times \He_3$

$\rhd$

$C_{12}^2$

$\rhd$

$C_4\times C_{12}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4.D_8\times \He_3$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

$\lhd$

$C_4^2\times \He_3$

$\lhd$

$C_4.D_8\times \He_3$

Character theory

Complex character table

See the $209 \times 209$ 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 $78 \times 78$ rational character table.