Abstract group 1944.3755: $C_2\times C_3^3:C_6^2$

• Label: 1944.3755
• Order: \( 2^{3} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{4} \)
• Perm deg.: $19$
• Trans deg.: $216$
• Rank: $3$

Group information

Description:	$C_2\times C_3^3:C_6^2$
Order:	\(1944\)\(\medspace = 2^{3} \cdot 3^{5} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$C_3^3.C_6^2.C_3^3.C_2^5$, of order \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \)
Composition factors:	$C_2$ x 3, $C_3$ x 5
Derived length:	$2$

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

Group statistics

Order	1	2	3	6
Elements	1	111	242	1590	1944
Conjugacy classes	1	7	53	191	252
Divisions	1	7	30	106	144
Autjugacy classes	1	2	10	12	25

Dimension	1	2	4	6	12
Irr. complex chars.	72	144	0	36	0	252
Irr. rational chars.	8	48	64	12	12	144

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$216$
Rank:	$3$
Inequivalent generating triples:	$1456$

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:	${\langle a, b, c, d, e \mid a^{6}=b^{3}=c^{3}=d^{6}=e^{6}=[a,b]=[b,c]=[b,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $19$ $\langle(1,2)(3,4)(9,10)(13,17)(15,16)(18,19), (1,2)(3,4), (1,3)(2,4), (5,6,7), (11,12,14) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 27 \\ 27 & 34 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 24 & 13 \end{array}\right), \left(\begin{array}{rr} 35 & 0 \\ 0 & 35 \end{array}\right), \left(\begin{array}{rr} 26 & 31 \\ 27 & 10 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_3^3:C_6)$
Semidirect product:	$(C_3^3:D_6)$ $\,\rtimes\,$ $C_6$	$(C_3^3\times C_6)$ $\,\rtimes\,$ $D_6$	$C_3^4$ $\,\rtimes\,$ $(C_2\times D_6)$	$C_3^3$ $\,\rtimes\,$ $(C_6:D_6)$	all 45
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^3$ . $(C_6\times D_6)$	$C_6^2$ . $(S_3\times C_3^2)$	$C_3^2$ . $(S_3\times C_6^2)$	$C_6^2$ . $(C_3^2:C_6)$	all 12

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{36}\Z)$.

Homology

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

Subgroups

There are 11802 subgroups in 2097 conjugacy classes, 286 normal (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $C_3^3:C_6$
Commutator:	$G' \simeq$ $C_3^3$	$G/G' \simeq$ $C_2\times C_6^2$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3^4:C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $\He_3\times C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_2\times C_3^3:C_6^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_3\times S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\times \He_3$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1944: $C_2\times C_3^3:C_6^2$

Order 972: $C_3^3:C_6^2$ x 6, $\He_3\times C_6^2$

Order 648: $C_2\times C_3^3:D_6$ x 3, $C_3\times C_6^2:C_6$ x 3, $C_3^4:C_2^3$, $C_3^4:C_2^3$

Order 486: $C_3^4:C_6$ x 4, $C_3^4:C_6$ x 3

Order 324: $C_3^2:C_6^2$ x 21, $C_3^3:D_6$ x 18, $C_3^2:C_6^2$ x 18, $C_3^3:D_6$ x 6, $C_3^2:C_6^2$ x 6, $C_3^2\times C_6^2$ x 3

Order 243: $C_3^2\times \He_3$

Order 216: $C_6^2:C_6$ x 10, $C_6^2:C_6$ x 9, $S_3\times C_6^2$ x 4, $C_6^2:S_3$

Order 162: $C_6\times \He_3$ x 63, $C_3^3:C_6$ x 12, $C_3^3:C_6$ x 12, $C_3^3\times C_6$ x 9, $C_3^3:C_6$ x 4, $C_3^2\wr C_2$ x 4

Order 108: $C_3^2:D_6$ x 60, $C_3^2:D_6$ x 54, $C_3\times C_6^2$ x 50, $C_2^2\times \He_3$ x 45, $C_3^2\times D_6$ x 24, $C_3^2:D_6$ x 6

Order 81: $C_3\times \He_3$ x 21, $C_3^4$ x 3

Order 72: $C_6\times D_6$ x 19, $C_6:D_6$ x 7, $C_2\times C_6^2$

Order 54: $C_3^2\times C_6$ x 150, $C_2\times \He_3$ x 135, $C_3^2:C_6$ x 40, $C_3^2:C_6$ x 36, $S_3\times C_3^2$ x 16, $C_3^2:S_3$ x 4

Order 36: $C_6^2$ x 116, $C_6\times S_3$ x 114, $C_6:S_3$ x 42

Order 27: $C_3^3$ x 50, $\He_3$ x 45

Order 24: $C_2\times D_6$ x 7, $C_2^2\times C_6$ x 4

Order 18: $C_3\times C_6$ x 334, $C_3\times S_3$ x 76, $C_3:S_3$ x 28

Order 12: $C_2\times C_6$ x 54, $D_6$ x 42

Order 9: $C_3^2$ x 110

Order 8: $C_2^3$

Order 6: $C_6$ x 106, $S_3$ x 28

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 30

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1944: $C_2\times C_3^3:C_6^2$

Order 972: $\He_3\times C_6^2$, $C_3^3:C_6^2$

Order 648: $C_3^4:C_2^3$, $C_3^4:C_2^3$, $C_2\times C_3^3:D_6$, $C_3\times C_6^2:C_6$

Order 486: $C_3^4:C_6$, $C_3^4:C_6$

Order 324: $C_3^2:C_6^2$ x 5, $C_3^2\times C_6^2$ x 3, $C_3^3:D_6$, $C_3^2:C_6^2$, $C_3^3:D_6$, $C_3^2:C_6^2$

Order 243: $C_3^2\times \He_3$

Order 216: $C_6^2:C_6$ x 4, $S_3\times C_6^2$ x 2, $C_6^2:S_3$, $C_6^2:C_6$

Order 162: $C_6\times \He_3$ x 5, $C_3^3\times C_6$ x 3, $C_3^3:C_6$, $C_3^2\wr C_2$, $C_3^3:C_6$, $C_3^3:C_6$

Order 108: $C_3\times C_6^2$ x 18, $C_3^2:D_6$ x 4, $C_2^2\times \He_3$ x 4, $C_3^2\times D_6$ x 2, $C_3^2:D_6$, $C_3^2:D_6$

Order 81: $C_3\times \He_3$ x 5, $C_3^4$ x 3

Order 72: $C_6\times D_6$ x 5, $C_6:D_6$ x 3, $C_2\times C_6^2$

Order 54: $C_3^2\times C_6$ x 18, $C_3^2:C_6$ x 4, $C_2\times \He_3$ x 4, $S_3\times C_3^2$ x 2, $C_3^2:C_6$, $C_3^2:S_3$

Order 36: $C_6^2$ x 26, $C_6\times S_3$ x 5, $C_6:S_3$ x 3

Order 27: $C_3^3$ x 18, $\He_3$ x 4

Order 24: $C_2\times D_6$ x 3, $C_2^2\times C_6$ x 2

Order 18: $C_3\times C_6$ x 26, $C_3\times S_3$ x 5, $C_3:S_3$ x 3

Order 12: $C_2\times C_6$ x 12, $D_6$ x 3

Order 9: $C_3^2$ x 25

Order 8: $C_2^3$

Order 6: $C_6$ x 12, $S_3$ x 3

Order 4: $C_2^2$ x 2

Order 3: $C_3$ x 10

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1944: $C_2\times C_3^3:C_6^2$ ($C_1$)

Order 972: $C_3^3:C_6^2$ ($C_2$) x 6, $\He_3\times C_6^2$ ($C_2$)

Order 648: $C_2\times C_3^3:D_6$ ($C_3$) x 3, $C_3^4:C_2^3$ ($C_3$)

Order 486: $C_3^4:C_6$ ($C_2^2$) x 4, $C_3^4:C_6$ ($C_2^2$) x 3

Order 324: $C_3^3:D_6$ ($C_6$) x 18, $C_3^3:D_6$ ($C_6$) x 6, $C_3^2:C_6^2$ ($C_6$) x 3, $C_3^2:C_6^2$ ($S_3$) x 3, $C_3^2\times C_6^2$ ($C_6$), $C_3^2\times C_6^2$ ($S_3$)

Order 243: $C_3^2\times \He_3$ ($C_2^3$)

Order 216: $C_6^2:S_3$ ($C_3^2$)

Order 162: $C_3^3:C_6$ ($C_2\times C_6$) x 12, $C_6\times \He_3$ ($C_2\times C_6$) x 9, $C_6\times \He_3$ ($D_6$) x 9, $C_3^3:C_6$ ($C_2\times C_6$) x 4, $C_3^3\times C_6$ ($C_2\times C_6$) x 3, $C_3^3\times C_6$ ($D_6$) x 3

Order 108: $C_2^2\times \He_3$ ($C_3\times S_3$) x 9, $C_3\times C_6^2$ ($C_3\times S_3$) x 7, $C_3^2:D_6$ ($C_3\times C_6$) x 6, $C_3\times C_6^2$ ($C_3\times C_6$), $C_3\times C_6^2$ ($C_3:S_3$)

Order 81: $C_3\times \He_3$ ($C_2^2\times C_6$) x 3, $C_3\times \He_3$ ($C_2\times D_6$) x 3, $C_3^4$ ($C_2^2\times C_6$), $C_3^4$ ($C_2\times D_6$)

Order 54: $C_2\times \He_3$ ($C_6\times S_3$) x 27, $C_3^2\times C_6$ ($C_6\times S_3$) x 21, $C_3^2:S_3$ ($C_6^2$) x 4, $C_3^2\times C_6$ ($C_6^2$) x 3, $C_3^2\times C_6$ ($C_6:S_3$) x 3

Order 36: $C_6^2$ ($C_3^2:C_6$) x 4, $C_6^2$ ($S_3\times C_3^2$) x 4, $C_6^2$ ($C_3^2:C_6$) x 3

Order 27: $\He_3$ ($C_6\times D_6$) x 9, $C_3^3$ ($C_6\times D_6$) x 7, $C_3^3$ ($C_2\times C_6^2$), $C_3^3$ ($C_6:D_6$)

Order 18: $C_3\times C_6$ ($C_3^2:D_6$) x 12, $C_3\times C_6$ ($C_3^2\times D_6$) x 12, $C_3\times C_6$ ($C_3^2:D_6$) x 9

Order 12: $C_2\times C_6$ ($C_3^3:C_6$) x 3, $C_2\times C_6$ ($C_3^2\wr C_2$), $C_2\times C_6$ ($C_3^3:C_6$)

Order 9: $C_3^2$ ($C_6^2:C_6$) x 4, $C_3^2$ ($S_3\times C_6^2$) x 4, $C_3^2$ ($C_6^2:C_6$) x 3

Order 6: $C_6$ ($C_3^2:C_6^2$) x 9, $C_6$ ($C_3^2:C_6^2$) x 3, $C_6$ ($C_3^3:D_6$) x 3

Order 4: $C_2^2$ ($C_3^4:C_6$)

Order 3: $C_3$ ($C_3\times C_6^2:C_6$) x 3, $C_3$ ($C_3^4:C_2^3$), $C_3$ ($C_2\times C_3^3:D_6$)

Order 2: $C_2$ ($C_3^3:C_6^2$) x 3

Order 1: $C_1$ ($C_2\times C_3^3:C_6^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1944: $C_2\times C_3^3:C_6^2$ ($C_1$)

Order 972: $\He_3\times C_6^2$ ($C_2$), $C_3^3:C_6^2$ ($C_2$)

Order 648: $C_3^4:C_2^3$ ($C_3$), $C_2\times C_3^3:D_6$ ($C_3$)

Order 486: $C_3^4:C_6$ ($C_2^2$), $C_3^4:C_6$ ($C_2^2$)

Order 324: $C_3^2\times C_6^2$ ($C_6$), $C_3^2\times C_6^2$ ($S_3$), $C_3^3:D_6$ ($C_6$), $C_3^2:C_6^2$ ($C_6$), $C_3^2:C_6^2$ ($S_3$), $C_3^3:D_6$ ($C_6$)

Order 243: $C_3^2\times \He_3$ ($C_2^3$)

Order 216: $C_6^2:S_3$ ($C_3^2$)

Order 162: $C_3^3\times C_6$ ($C_2\times C_6$), $C_3^3\times C_6$ ($D_6$), $C_3^3:C_6$ ($C_2\times C_6$), $C_6\times \He_3$ ($C_2\times C_6$), $C_6\times \He_3$ ($D_6$), $C_3^3:C_6$ ($C_2\times C_6$)

Order 108: $C_3\times C_6^2$ ($C_3\times S_3$) x 3, $C_3\times C_6^2$ ($C_3\times C_6$), $C_3\times C_6^2$ ($C_3:S_3$), $C_3^2:D_6$ ($C_3\times C_6$), $C_2^2\times \He_3$ ($C_3\times S_3$)

Order 81: $C_3^4$ ($C_2^2\times C_6$), $C_3^4$ ($C_2\times D_6$), $C_3\times \He_3$ ($C_2^2\times C_6$), $C_3\times \He_3$ ($C_2\times D_6$)

Order 54: $C_3^2\times C_6$ ($C_6\times S_3$) x 3, $C_3^2\times C_6$ ($C_6^2$), $C_3^2\times C_6$ ($C_6:S_3$), $C_3^2:S_3$ ($C_6^2$), $C_2\times \He_3$ ($C_6\times S_3$)

Order 36: $C_6^2$ ($C_3^2:C_6$) x 2, $C_6^2$ ($S_3\times C_3^2$) x 2, $C_6^2$ ($C_3^2:C_6$)

Order 27: $C_3^3$ ($C_6\times D_6$) x 3, $C_3^3$ ($C_2\times C_6^2$), $C_3^3$ ($C_6:D_6$), $\He_3$ ($C_6\times D_6$)

Order 18: $C_3\times C_6$ ($C_3^2:D_6$) x 2, $C_3\times C_6$ ($C_3^2\times D_6$) x 2, $C_3\times C_6$ ($C_3^2:D_6$)

Order 12: $C_2\times C_6$ ($C_3^2\wr C_2$), $C_2\times C_6$ ($C_3^3:C_6$), $C_2\times C_6$ ($C_3^3:C_6$)

Order 9: $C_3^2$ ($C_6^2:C_6$) x 2, $C_3^2$ ($S_3\times C_6^2$) x 2, $C_3^2$ ($C_6^2:C_6$)

Order 6: $C_6$ ($C_3^2:C_6^2$), $C_6$ ($C_3^3:D_6$), $C_6$ ($C_3^2:C_6^2$)

Order 4: $C_2^2$ ($C_3^4:C_6$)

Order 3: $C_3$ ($C_3^4:C_2^3$), $C_3$ ($C_2\times C_3^3:D_6$), $C_3$ ($C_3\times C_6^2:C_6$)

Order 2: $C_2$ ($C_3^3:C_6^2$)

Order 1: $C_1$ ($C_2\times C_3^3:C_6^2$)

Series

Derived series

$C_2\times C_3^3:C_6^2$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$C_2\times C_3^3:C_6^2$

$\rhd$

$\He_3\times C_6^2$

$\rhd$

$C_3^2\times C_6^2$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2\times C_3^3:C_6^2$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

Character theory

Complex character table

See the $252 \times 252$ 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 $144 \times 144$ rational character table (warning: may be slow to load).