Abstract group 1728.20951: $(C_3^2\times C_{12}).D_8$

• Label: 1728.20951
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $25$
• Trans deg.: $192$
• Rank: $2$

Group information

Description:	$(C_3^2\times C_{12}).D_8$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$(C_3\times C_6).C_2^6.C_2^3$, of order \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	3	26	124	78	192	632	672	1728
Conjugacy classes	1	3	8	7	24	8	44	40	135
Divisions	1	3	5	7	15	2	21	5	59
Autjugacy classes	1	3	5	6	15	2	16	4	52

Dimension	1	2	4	8	16	32
Irr. complex chars.	24	30	75	6	0	0	135
Irr. rational chars.	4	8	15	24	6	2	59

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$192$
Rank:	$2$
Inequivalent generating pairs:	$96$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	6	10	12

Constructions

Presentation:	$\langle a, b, c, d \mid b^{24}=c^{3}=d^{12}=[a,c]=[c,d]=1, a^{2}=b^{12}d^{9}, b^{a}=b^{19}d^{3}, d^{a}=d^{5}, c^{b}=d^{8}, d^{b}=cd^{3} \rangle$
Permutation group:	Degree $25$ $\langle(4,5)(7,8)(9,11)(10,13)(12,14)(15,16,19,20)(17,18,21,22), (1,2)(3,5)(4,6) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 22 & 21 \\ 21 & 43 \end{array}\right), \left(\begin{array}{rr} 43 & 14 \\ 35 & 29 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 22 & 21 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 12 & 16 \\ 1 & 2 \end{array}\right), \left(\begin{array}{rr} 55 & 0 \\ 0 & 55 \end{array}\right), \left(\begin{array}{rr} 1 & 42 \\ 21 & 1 \end{array}\right), \left(\begin{array}{rr} 46 & 36 \\ 18 & 10 \end{array}\right), \left(\begin{array}{rr} 55 & 16 \\ 1 & 53 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/63\Z)$
Direct product:	$C_3$ $\, \times\, $ $((C_3\times C_{12}).D_8)$
Semidirect product:	$C_3^3$ $\,\rtimes\,$ $(C_4.D_8)$	$C_3^2$ $\,\rtimes\,$ $(C_{12}.D_8)$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6^2.D_6)$ . $C_4$ (2)	$(C_6^2.C_6)$ . $D_4$	$C_6$ . $(C_6^2.D_4)$	$C_4$ . $(C_3^3:D_8)$	all 46

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

Homology

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

Subgroups

There are 1556 subgroups in 294 conjugacy classes, 54 normal (50 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $S_3^2:C_4$
Commutator:	$G' \simeq$ $C_6.D_6$	$G/G' \simeq$ $C_2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $S_3^2:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6\times C_{12}$	$G/\operatorname{Fit} \simeq$ $D_4$
Radical:	$R \simeq$ $(C_3^2\times C_{12}).D_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^2:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.D_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

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

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Classes of subgroups up to conjugation

Order 1728: $(C_3^2\times C_{12}).D_8$

Order 864: $C_3\times C_6^2.C_2^3$, $(C_3\times C_{12}):C_{24}$, $C_6.(S_3\times C_{24})$

Order 576: $(C_3\times C_{12}).D_8$

Order 432: $C_6^2.D_6$ x 3, $C_{12}.C_6^2$, $C_6^2.C_{12}$, $C_6^2.D_6$, $C_{12}.C_6^2$

Order 288: $C_6^2.C_2^3$, $C_6^2.C_2^3$, $(C_3\times C_{12}):C_8$, $C_3\times C_{12}.D_4$, $C_6.(S_3\times C_8)$

Order 216: $C_6^2.C_6$ x 2, $C_6.C_6^2$ x 2, $C_3^3:Q_8$ x 2, $C_3^2:C_{24}$, $C_3\times C_6\times C_{12}$, $C_3^2:C_{24}$

Order 192: $C_{12}.D_8$

Order 144: $C_6^2.C_2^2$ x 4, $C_6:C_{24}$ x 3, $C_2^2.S_3^2$ x 3, $C_{12}.D_6$ x 3, $C_{12}:C_{12}$ x 2, $C_3^2:C_4^2$, $C_4.C_6^2$, $C_6^2.C_4$, $C_6\times C_{24}$

Order 108: $C_3^2:C_{12}$ x 3, $C_3^2\times C_{12}$ x 2, $C_3^2:C_{12}$ x 2, $C_3\times C_6^2$

Order 96: $C_4:C_{24}$ x 2, $C_{12}.D_4$, $C_{12}:Q_8$, $C_{12}.D_4$

Order 72: $C_6:C_{12}$ x 10, $C_3^2:Q_8$ x 8, $C_6\times C_{12}$ x 7, $C_3:C_{24}$ x 4, $Q_8\times C_3^2$ x 2, $C_6.D_6$ x 2, $C_3^2:C_8$, $C_3\times C_{24}$

Order 64: $C_4.D_8$

Order 54: $C_3^2\times C_6$ x 3

Order 48: $C_2\times C_{24}$ x 4, $C_6.D_4$ x 4, $C_6\times Q_8$ x 3, $C_4:C_{12}$ x 3, $C_{12}:C_4$ x 2, $C_6:C_8$, $C_6:Q_8$, $C_4\times C_{12}$

Order 36: $C_3:C_{12}$ x 15, $C_3\times C_{12}$ x 12, $C_6^2$ x 5, $C_3^2:C_4$ x 3

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

Order 27: $C_3^3$

Order 24: $C_2\times C_{12}$ x 13, $C_3\times Q_8$ x 8, $C_6:C_4$ x 6, $C_{24}$ x 5, $C_3:Q_8$ x 2, $C_3:C_8$

Order 18: $C_3\times C_6$ x 15

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 21, $C_3:C_4$ x 9, $C_2\times C_6$ x 5

Order 9: $C_3^2$ x 5

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: $(C_3^2\times C_{12}).D_8$

Order 864: $C_3\times C_6^2.C_2^3$, $(C_3\times C_{12}):C_{24}$, $C_6.(S_3\times C_{24})$

Order 576: $(C_3\times C_{12}).D_8$

Order 432: $C_6^2.D_6$ x 2, $C_{12}.C_6^2$, $C_6^2.C_{12}$, $C_6^2.D_6$, $C_{12}.C_6^2$

Order 288: $C_6^2.C_2^3$, $C_6^2.C_2^3$, $(C_3\times C_{12}):C_8$, $C_3\times C_{12}.D_4$, $C_6.(S_3\times C_8)$

Order 216: $C_6^2.C_6$ x 2, $C_3^3:Q_8$ x 2, $C_3^2:C_{24}$, $C_3\times C_6\times C_{12}$, $C_6.C_6^2$, $C_3^2:C_{24}$

Order 192: $C_{12}.D_8$

Order 144: $C_6:C_{24}$ x 3, $C_{12}.D_6$ x 3, $C_6^2.C_2^2$ x 2, $C_{12}:C_{12}$ x 2, $C_2^2.S_3^2$ x 2, $C_3^2:C_4^2$, $C_4.C_6^2$, $C_6^2.C_4$, $C_6\times C_{24}$

Order 108: $C_3^2:C_{12}$ x 3, $C_3^2\times C_{12}$ x 2, $C_3\times C_6^2$, $C_3^2:C_{12}$

Order 96: $C_4:C_{24}$ x 2, $C_{12}.D_4$, $C_{12}:Q_8$, $C_{12}.D_4$

Order 72: $C_6:C_{12}$ x 7, $C_6\times C_{12}$ x 6, $C_3^2:Q_8$ x 6, $C_3:C_{24}$ x 3, $Q_8\times C_3^2$ x 2, $C_6.D_6$ x 2, $C_3^2:C_8$, $C_3\times C_{24}$

Order 64: $C_4.D_8$

Order 54: $C_3^2\times C_6$ x 3

Order 48: $C_2\times C_{24}$ x 4, $C_6\times Q_8$ x 3, $C_4:C_{12}$ x 2, $C_6.D_4$ x 2, $C_{12}:C_4$ x 2, $C_6:C_8$, $C_6:Q_8$, $C_4\times C_{12}$

Order 36: $C_3\times C_{12}$ x 11, $C_3:C_{12}$ x 9, $C_6^2$ x 5, $C_3^2:C_4$ x 3

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

Order 27: $C_3^3$

Order 24: $C_2\times C_{12}$ x 10, $C_3\times Q_8$ x 6, $C_6:C_4$ x 5, $C_{24}$ x 4, $C_3:Q_8$ x 2, $C_3:C_8$

Order 18: $C_3\times C_6$ x 15

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 16, $C_3:C_4$ x 7, $C_2\times C_6$ x 5

Order 9: $C_3^2$ x 5

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

Order 6: $C_6$ x 15

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

Order 3: $C_3$ x 5

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $(C_3^2\times C_{12}).D_8$ ($C_1$)

Order 864: $C_3\times C_6^2.C_2^3$ ($C_2$), $(C_3\times C_{12}):C_{24}$ ($C_2$), $C_6.(S_3\times C_{24})$ ($C_2$)

Order 576: $(C_3\times C_{12}).D_8$ ($C_3$)

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

Order 288: $C_6^2.C_2^3$ ($C_6$), $(C_3\times C_{12}):C_8$ ($C_6$), $C_6.(S_3\times C_8)$ ($C_6$)

Order 216: $C_3\times C_6\times C_{12}$ ($D_4$), $C_6^2.C_6$ ($D_4$), $C_6^2.C_6$ ($C_2\times C_4$)

Order 144: $C_2^2.S_3^2$ ($C_{12}$) x 2, $C_3^2:C_4^2$ ($C_2\times C_6$)

Order 108: $C_3\times C_6^2$ ($C_2^2:C_4$), $C_3^2\times C_{12}$ ($\SD_{16}$), $C_3^2\times C_{12}$ ($D_8$), $C_3^2:C_{12}$ ($Q_{16}$), $C_3^2:C_{12}$ ($\SD_{16}$)

Order 72: $C_6\times C_{12}$ ($C_3\times D_4$), $C_6.D_6$ ($C_2\times C_{12}$), $C_6.D_6$ ($C_3\times D_4$)

Order 54: $C_3^2\times C_6$ ($D_4:C_4$), $C_3^2\times C_6$ ($C_4.D_4$), $C_3^2\times C_6$ ($Q_8:C_4$)

Order 36: $C_3\times C_{12}$ ($C_3\times \SD_{16}$), $C_3\times C_{12}$ ($C_3\times D_8$), $C_3^2:C_4$ ($C_3\times Q_{16}$), $C_3^2:C_4$ ($C_3\times \SD_{16}$), $C_6^2$ ($C_2^2:C_{12}$)

Order 27: $C_3^3$ ($C_4.D_8$)

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

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 12: $C_2\times C_6$ ($S_3^2:C_4$), $C_{12}$ ($C_3^2:\SD_{16}$), $C_{12}$ ($C_3^2:D_8$)

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

Order 8: $C_2\times C_4$ ($S_3^2:C_6$)

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

Order 4: $C_2^2$ ($S_3^2:C_{12}$), $C_4$ ($C_3^3:\SD_{16}$), $C_4$ ($C_3^3:D_8$)

Order 3: $C_3$ ($(C_3\times C_{12}).D_8$)

Order 2: $C_2$ ($C_3\times C_6^2.D_4$), $C_2$ ($C_{12}.\SOPlus(4,2)$), $C_2$ ($C_{12}.\SOPlus(4,2)$)

Order 1: $C_1$ ($(C_3^2\times C_{12}).D_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $(C_3^2\times C_{12}).D_8$ ($C_1$)

Order 864: $C_3\times C_6^2.C_2^3$ ($C_2$), $(C_3\times C_{12}):C_{24}$ ($C_2$), $C_6.(S_3\times C_{24})$ ($C_2$)

Order 576: $(C_3\times C_{12}).D_8$ ($C_3$)

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

Order 288: $C_6^2.C_2^3$ ($C_6$), $(C_3\times C_{12}):C_8$ ($C_6$), $C_6.(S_3\times C_8)$ ($C_6$)

Order 216: $C_3\times C_6\times C_{12}$ ($D_4$), $C_6^2.C_6$ ($D_4$), $C_6^2.C_6$ ($C_2\times C_4$)

Order 144: $C_3^2:C_4^2$ ($C_2\times C_6$), $C_2^2.S_3^2$ ($C_{12}$)

Order 108: $C_3\times C_6^2$ ($C_2^2:C_4$), $C_3^2\times C_{12}$ ($\SD_{16}$), $C_3^2\times C_{12}$ ($D_8$), $C_3^2:C_{12}$ ($Q_{16}$), $C_3^2:C_{12}$ ($\SD_{16}$)

Order 72: $C_6\times C_{12}$ ($C_3\times D_4$), $C_6.D_6$ ($C_2\times C_{12}$), $C_6.D_6$ ($C_3\times D_4$)

Order 54: $C_3^2\times C_6$ ($D_4:C_4$), $C_3^2\times C_6$ ($C_4.D_4$), $C_3^2\times C_6$ ($Q_8:C_4$)

Order 36: $C_3\times C_{12}$ ($C_3\times \SD_{16}$), $C_3\times C_{12}$ ($C_3\times D_8$), $C_3^2:C_4$ ($C_3\times Q_{16}$), $C_3^2:C_4$ ($C_3\times \SD_{16}$), $C_6^2$ ($C_2^2:C_{12}$)

Order 27: $C_3^3$ ($C_4.D_8$)

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

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 12: $C_2\times C_6$ ($S_3^2:C_4$), $C_{12}$ ($C_3^2:\SD_{16}$), $C_{12}$ ($C_3^2:D_8$)

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

Order 8: $C_2\times C_4$ ($S_3^2:C_6$)

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

Order 4: $C_2^2$ ($S_3^2:C_{12}$), $C_4$ ($C_3^3:\SD_{16}$), $C_4$ ($C_3^3:D_8$)

Order 3: $C_3$ ($(C_3\times C_{12}).D_8$)

Order 2: $C_2$ ($C_3\times C_6^2.D_4$), $C_2$ ($C_{12}.\SOPlus(4,2)$), $C_2$ ($C_{12}.\SOPlus(4,2)$)

Order 1: $C_1$ ($(C_3^2\times C_{12}).D_8$)

Series

Derived series

$(C_3^2\times C_{12}).D_8$

$\rhd$

$C_6.D_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$(C_3^2\times C_{12}).D_8$

$\rhd$

$C_6.(S_3\times C_{24})$

$\rhd$

$C_6^2.D_6$

$\rhd$

$C_3^2:C_4^2$

$\rhd$

$C_6.D_6$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$(C_3^2\times C_{12}).D_8$

$\rhd$

$C_6.D_6$

$\rhd$

$C_6^2$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

$\lhd$

$C_2\times C_{12}$

Character theory

Complex character table

See the $135 \times 135$ 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 $59 \times 59$ rational character table.