Abstract group 1728.7359: $C_2^2.(D_6\times C_{36})$

• Label: 1728.7359
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3 \)
• Perm deg.: $32$
• Trans deg.: $576$
• Rank: $3$

Group information

Description:	$C_2^2.(D_6\times C_{36})$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_6.(C_2^5\times C_6).C_2^5$, of order \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	7	8	120	56	18	384	126	1008	1728
Conjugacy classes	1	7	5	20	35	12	76	84	192	432
Divisions	1	7	3	10	21	2	20	14	16	94
Autjugacy classes	1	5	3	5	15	2	9	10	7	57

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	144	252	36	0	0	0	0	432
Irr. rational chars.	8	16	20	8	18	8	16	94

Minimal presentations

Permutation degree:	$32$
Transitive degree:	$576$
Rank:	$3$
Inequivalent generating triples:	$39312$

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:	$\langle a, b, c \mid a^{4}=b^{36}=c^{12}=1, b^{a}=b^{19}c^{6}, c^{a}=b^{18}c, c^{b}=c^{5} \rangle$
Permutation group:	Degree $32$ $\langle(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,14,16,18)(15,17,19,20)(31,32), (1,3,4,7) \!\cdots\! \rangle$
Direct product:	$C_9$ $\, \times\, $ $(C_2^2.(C_4\times D_6))$
Semidirect product:	$(C_{12}:C_{36})$ $\,\rtimes\,$ $C_4$ (4)	$(C_{12}:C_4)$ $\,\rtimes\,$ $C_{36}$ (4)	$C_3$ $\,\rtimes\,$ $(C_4^2:C_{36})$	$(C_3\times C_9)$ $\,\rtimes\,$ $(C_4^2:C_4)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6^2.D_4)$ . $C_6$ (5)	$C_6^2$ . $(D_4:C_6)$ (6)	$(C_{18}.C_4^2)$ . $S_3$	$C_{18}$ . $(C_4^2:S_3)$	all 79

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

Homology

Abelianization:	$C_{2}^{2} \times C_{36} \simeq C_{2}^{2} \times C_{4} \times C_{9}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 1160 subgroups in 490 conjugacy classes, 189 normal (75 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_{18}$	$G/Z \simeq$ $C_2\times D_6$
Commutator:	$G' \simeq$ $C_2\times C_6$	$G/G' \simeq$ $C_2^2\times C_{36}$
Frattini:	$\Phi \simeq$ $C_2^2\times C_6$	$G/\Phi \simeq$ $C_6\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_6.C_{12}^2$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_2^2.(D_6\times C_{36})$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1728: $C_2^2.(D_6\times C_{36})$

Order 864: $(C_2\times C_{12}):C_{36}$ x 5, $C_6.C_{12}^2$, $C_2\times C_{12}:C_{36}$

Order 576: $C_4^2:C_{36}$, $C_2^2.(C_{12}\times D_6)$

Order 432: $C_6^2.C_{12}$ x 4, $C_{12}:C_{36}$ x 4, $C_2\times C_6\times C_{36}$ x 3

Order 288: $C_{18}.C_4^2$ x 7, $C_6^2.D_4$ x 5, $C_6^2.C_2^3$, $C_2.C_{12}^2$, $C_2\times C_4\times C_{36}$

Order 216: $C_6:C_{36}$ x 14, $C_6\times C_{36}$ x 10, $C_2\times C_6\times C_{18}$

Order 192: $C_4^2:C_{12}$, $C_2^2.(C_4\times D_6)$

Order 144: $C_2^2\times C_{36}$ x 10, $C_{12}:C_{12}$ x 4, $C_4\times C_{36}$ x 4, $C_6^2:C_4$ x 4, $C_2\times C_6\times C_{12}$ x 3

Order 108: $C_6\times C_{18}$ x 7, $C_3:C_{36}$ x 6, $C_3\times C_{36}$ x 4

Order 96: $C_6.C_4^2$ x 8, $C_6.C_4^2$ x 5, $C_2\times C_4\times C_{12}$, $C_6:C_4^2$

Order 72: $C_2\times C_{36}$ x 36, $C_6:C_{12}$ x 14, $C_6\times C_{12}$ x 10, $C_2^2\times C_{18}$ x 2, $C_2\times C_6^2$

Order 64: $C_4^2:C_4$

Order 54: $C_3\times C_{18}$ x 7

Order 48: $C_2^2\times C_{12}$ x 13, $C_6.C_2^3$ x 4, $C_4\times C_{12}$ x 4, $C_{12}:C_4$ x 4

Order 36: $C_{36}$ x 16, $C_2\times C_{18}$ x 14, $C_6^2$ x 7, $C_3:C_{12}$ x 6, $C_3\times C_{12}$ x 4

Order 32: $C_2.C_4^2$ x 6, $C_2\times C_4^2$

Order 27: $C_3\times C_9$

Order 24: $C_2\times C_{12}$ x 46, $C_6:C_4$ x 14, $C_2^2\times C_6$ x 3

Order 18: $C_{18}$ x 14, $C_3\times C_6$ x 7

Order 16: $C_2^2\times C_4$ x 7, $C_4^2$ x 4

Order 12: $C_2\times C_6$ x 21, $C_{12}$ x 20, $C_3:C_4$ x 6

Order 9: $C_9$ x 2, $C_3^2$

Order 8: $C_2\times C_4$ x 24, $C_2^3$

Order 6: $C_6$ x 21

Order 4: $C_4$ x 10, $C_2^2$ x 7

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $C_2^2.(D_6\times C_{36})$

Order 864: $(C_2\times C_{12}):C_{36}$ x 3, $C_6.C_{12}^2$, $C_2\times C_{12}:C_{36}$

Order 576: $C_4^2:C_{36}$, $C_2^2.(C_{12}\times D_6)$

Order 432: $C_6^2.C_{12}$ x 3, $C_2\times C_6\times C_{36}$ x 2, $C_{12}:C_{36}$

Order 288: $C_{18}.C_4^2$ x 5, $C_6^2.D_4$ x 3, $C_6^2.C_2^3$, $C_2.C_{12}^2$, $C_2\times C_4\times C_{36}$

Order 216: $C_6:C_{36}$ x 8, $C_6\times C_{36}$ x 5, $C_2\times C_6\times C_{18}$

Order 192: $C_4^2:C_{12}$, $C_2^2.(C_4\times D_6)$

Order 144: $C_2^2\times C_{36}$ x 7, $C_6^2:C_4$ x 3, $C_2\times C_6\times C_{12}$ x 2, $C_{12}:C_{12}$, $C_4\times C_{36}$

Order 108: $C_6\times C_{18}$ x 5, $C_3:C_{36}$ x 3, $C_3\times C_{36}$ x 2

Order 96: $C_6.C_4^2$ x 6, $C_6.C_4^2$ x 3, $C_2\times C_4\times C_{12}$, $C_6:C_4^2$

Order 72: $C_2\times C_{36}$ x 18, $C_6:C_{12}$ x 8, $C_6\times C_{12}$ x 5, $C_2^2\times C_{18}$ x 2, $C_2\times C_6^2$

Order 64: $C_4^2:C_4$

Order 54: $C_3\times C_{18}$ x 5

Order 48: $C_2^2\times C_{12}$ x 9, $C_6.C_2^3$ x 3, $C_4\times C_{12}$, $C_{12}:C_4$

Order 36: $C_2\times C_{18}$ x 10, $C_{36}$ x 7, $C_6^2$ x 5, $C_3:C_{12}$ x 3, $C_3\times C_{12}$ x 2

Order 32: $C_2.C_4^2$ x 4, $C_2\times C_4^2$

Order 27: $C_3\times C_9$

Order 24: $C_2\times C_{12}$ x 23, $C_6:C_4$ x 8, $C_2^2\times C_6$ x 3

Order 18: $C_{18}$ x 10, $C_3\times C_6$ x 5

Order 16: $C_2^2\times C_4$ x 5, $C_4^2$

Order 12: $C_2\times C_6$ x 15, $C_{12}$ x 9, $C_3:C_4$ x 3

Order 9: $C_9$ x 2, $C_3^2$

Order 8: $C_2\times C_4$ x 13, $C_2^3$

Order 6: $C_6$ x 15

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: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $C_2^2.(D_6\times C_{36})$ ($C_1$)

Order 864: $(C_2\times C_{12}):C_{36}$ ($C_2$) x 5, $C_6.C_{12}^2$ ($C_2$), $C_2\times C_{12}:C_{36}$ ($C_2$)

Order 576: $C_2^2.(C_{12}\times D_6)$ ($C_3$)

Order 432: $C_6^2.C_{12}$ ($C_2^2$) x 4, $C_{12}:C_{36}$ ($C_4$) x 4, $C_2\times C_6\times C_{36}$ ($C_2^2$) x 3

Order 288: $C_6^2.D_4$ ($C_6$) x 5, $C_6^2.C_2^3$ ($C_6$), $C_{18}.C_4^2$ ($S_3$), $C_2.C_{12}^2$ ($C_6$)

Order 216: $C_6:C_{36}$ ($C_2\times C_4$) x 4, $C_6\times C_{36}$ ($C_2\times C_4$) x 2, $C_2\times C_6\times C_{18}$ ($C_2^3$)

Order 192: $C_2^2.(C_4\times D_6)$ ($C_9$)

Order 144: $C_{12}:C_{12}$ ($C_{12}$) x 4, $C_6^2:C_4$ ($C_2\times C_6$) x 4, $C_2^2\times C_{36}$ ($D_6$) x 3, $C_2\times C_6\times C_{12}$ ($C_2\times C_6$) x 3

Order 108: $C_6\times C_{18}$ ($D_4:C_2$) x 6, $C_6\times C_{18}$ ($C_2^2\times C_4$)

Order 96: $C_6.C_4^2$ ($C_{18}$) x 5, $C_6.C_4^2$ ($C_3\times S_3$), $C_6.C_4^2$ ($C_{18}$), $C_6:C_4^2$ ($C_{18}$)

Order 72: $C_6:C_{12}$ ($C_2\times C_{12}$) x 4, $C_2\times C_{36}$ ($C_4\times S_3$) x 2, $C_6\times C_{12}$ ($C_2\times C_{12}$) x 2, $C_2\times C_6^2$ ($C_2^2\times C_6$), $C_2^2\times C_{18}$ ($C_2\times D_6$)

Order 54: $C_3\times C_{18}$ ($C_4^2:C_2$) x 4, $C_3\times C_{18}$ ($C_4^2:C_2$) x 3

Order 48: $C_6.C_2^3$ ($C_2\times C_{18}$) x 4, $C_{12}:C_4$ ($C_{36}$) x 4, $C_2^2\times C_{12}$ ($C_2\times C_{18}$) x 3, $C_2^2\times C_{12}$ ($C_6\times S_3$) x 3

Order 36: $C_6^2$ ($D_4:C_6$) x 6, $C_2\times C_{18}$ ($D_4:S_3$) x 3, $C_2\times C_{18}$ ($D_{12}:C_2$) x 2, $C_2\times C_{18}$ ($D_{12}:C_2$), $C_2\times C_{18}$ ($C_4\times D_6$), $C_6^2$ ($C_2^2\times C_{12}$)

Order 32: $C_2.C_4^2$ ($S_3\times C_9$)

Order 27: $C_3\times C_9$ ($C_4^2:C_4$)

Order 24: $C_6:C_4$ ($C_2\times C_{36}$) x 4, $C_2\times C_{12}$ ($C_2\times C_{36}$) x 2, $C_2\times C_{12}$ ($S_3\times C_{12}$) x 2, $C_2^2\times C_6$ ($C_6\times D_6$), $C_2^2\times C_6$ ($C_2^2\times C_{18}$)

Order 18: $C_3\times C_6$ ($C_4^2:C_6$) x 4, $C_3\times C_6$ ($C_4^2:C_6$) x 3, $C_{18}$ ($C_2^3.D_6$) x 2, $C_{18}$ ($(C_2\times C_4).D_6$) x 2, $C_{18}$ ($(C_4\times S_3):C_4$), $C_{18}$ ($C_2^3.D_6$), $C_{18}$ ($C_4^2:S_3$)

Order 16: $C_2^2\times C_4$ ($S_3\times C_{18}$) x 3

Order 12: $C_2\times C_6$ ($D_4:C_{18}$) x 6, $C_2\times C_6$ ($C_{12}.D_6$) x 3, $C_2\times C_6$ ($D_{12}:C_6$) x 2, $C_2\times C_6$ ($C_2^2\times C_{36}$), $C_2\times C_6$ ($D_{12}:C_6$), $C_2\times C_6$ ($C_{12}\times D_6$)

Order 9: $C_3^2$ ($C_4^2:C_{12}$), $C_9$ ($C_2^2.(C_4\times D_6)$)

Order 8: $C_2\times C_4$ ($S_3\times C_{36}$) x 2, $C_2^3$ ($D_6\times C_{18}$)

Order 6: $C_6$ ($C_4^2:C_{18}$) x 4, $C_6$ ($C_4^2:C_{18}$) x 3, $C_6$ ($C_6^2.C_2^3$) x 2, $C_6$ ($C_6^2.C_2^3$) x 2, $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_{12}^2:C_2$)

Order 4: $C_2^2$ ($C_{36}.D_6$) x 3, $C_2^2$ ($D_{12}:C_{18}$) x 2, $C_2^2$ ($D_{12}:C_{18}$), $C_2^2$ ($D_6\times C_{36}$)

Order 3: $C_3$ ($C_4^2:C_{36}$), $C_3$ ($C_2^2.(C_{12}\times D_6)$)

Order 2: $C_2$ ($(C_2\times C_{36}).D_6$) x 2, $C_2$ ($C_9\times C_2^3.D_6$) x 2, $C_2$ ($(C_4\times S_3):C_{36}$), $C_2$ ($C_9\times C_2^3.D_6$), $C_2$ ($C_{12}^2.C_6$)

Order 1: $C_1$ ($C_2^2.(D_6\times C_{36})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $C_2^2.(D_6\times C_{36})$ ($C_1$)

Order 864: $(C_2\times C_{12}):C_{36}$ ($C_2$) x 3, $C_6.C_{12}^2$ ($C_2$), $C_2\times C_{12}:C_{36}$ ($C_2$)

Order 576: $C_2^2.(C_{12}\times D_6)$ ($C_3$)

Order 432: $C_6^2.C_{12}$ ($C_2^2$) x 3, $C_2\times C_6\times C_{36}$ ($C_2^2$) x 2, $C_{12}:C_{36}$ ($C_4$)

Order 288: $C_6^2.D_4$ ($C_6$) x 3, $C_6^2.C_2^3$ ($C_6$), $C_{18}.C_4^2$ ($S_3$), $C_2.C_{12}^2$ ($C_6$)

Order 216: $C_6:C_{36}$ ($C_2\times C_4$) x 2, $C_6\times C_{36}$ ($C_2\times C_4$), $C_2\times C_6\times C_{18}$ ($C_2^3$)

Order 192: $C_2^2.(C_4\times D_6)$ ($C_9$)

Order 144: $C_6^2:C_4$ ($C_2\times C_6$) x 3, $C_2^2\times C_{36}$ ($D_6$) x 2, $C_2\times C_6\times C_{12}$ ($C_2\times C_6$) x 2, $C_{12}:C_{12}$ ($C_{12}$)

Order 108: $C_6\times C_{18}$ ($D_4:C_2$) x 4, $C_6\times C_{18}$ ($C_2^2\times C_4$)

Order 96: $C_6.C_4^2$ ($C_{18}$) x 3, $C_6.C_4^2$ ($C_3\times S_3$), $C_6.C_4^2$ ($C_{18}$), $C_6:C_4^2$ ($C_{18}$)

Order 72: $C_6:C_{12}$ ($C_2\times C_{12}$) x 2, $C_2\times C_{36}$ ($C_4\times S_3$), $C_2\times C_6^2$ ($C_2^2\times C_6$), $C_6\times C_{12}$ ($C_2\times C_{12}$), $C_2^2\times C_{18}$ ($C_2\times D_6$)

Order 54: $C_3\times C_{18}$ ($C_4^2:C_2$) x 3, $C_3\times C_{18}$ ($C_4^2:C_2$) x 2

Order 48: $C_6.C_2^3$ ($C_2\times C_{18}$) x 3, $C_2^2\times C_{12}$ ($C_2\times C_{18}$) x 2, $C_2^2\times C_{12}$ ($C_6\times S_3$) x 2, $C_{12}:C_4$ ($C_{36}$)

Order 36: $C_6^2$ ($D_4:C_6$) x 4, $C_2\times C_{18}$ ($D_4:S_3$) x 2, $C_2\times C_{18}$ ($D_{12}:C_2$), $C_2\times C_{18}$ ($D_{12}:C_2$), $C_2\times C_{18}$ ($C_4\times D_6$), $C_6^2$ ($C_2^2\times C_{12}$)

Order 32: $C_2.C_4^2$ ($S_3\times C_9$)

Order 27: $C_3\times C_9$ ($C_4^2:C_4$)

Order 24: $C_6:C_4$ ($C_2\times C_{36}$) x 2, $C_2\times C_{12}$ ($C_2\times C_{36}$), $C_2\times C_{12}$ ($S_3\times C_{12}$), $C_2^2\times C_6$ ($C_6\times D_6$), $C_2^2\times C_6$ ($C_2^2\times C_{18}$)

Order 18: $C_3\times C_6$ ($C_4^2:C_6$) x 3, $C_3\times C_6$ ($C_4^2:C_6$) x 2, $C_{18}$ ($(C_4\times S_3):C_4$), $C_{18}$ ($C_2^3.D_6$), $C_{18}$ ($C_2^3.D_6$), $C_{18}$ ($C_4^2:S_3$), $C_{18}$ ($(C_2\times C_4).D_6$)

Order 16: $C_2^2\times C_4$ ($S_3\times C_{18}$) x 2

Order 12: $C_2\times C_6$ ($D_4:C_{18}$) x 4, $C_2\times C_6$ ($C_{12}.D_6$) x 2, $C_2\times C_6$ ($C_2^2\times C_{36}$), $C_2\times C_6$ ($D_{12}:C_6$), $C_2\times C_6$ ($D_{12}:C_6$), $C_2\times C_6$ ($C_{12}\times D_6$)

Order 9: $C_3^2$ ($C_4^2:C_{12}$), $C_9$ ($C_2^2.(C_4\times D_6)$)

Order 8: $C_2^3$ ($D_6\times C_{18}$), $C_2\times C_4$ ($S_3\times C_{36}$)

Order 6: $C_6$ ($C_4^2:C_{18}$) x 3, $C_6$ ($C_4^2:C_{18}$) x 2, $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_{12}^2:C_2$)

Order 4: $C_2^2$ ($C_{36}.D_6$) x 2, $C_2^2$ ($D_{12}:C_{18}$), $C_2^2$ ($D_{12}:C_{18}$), $C_2^2$ ($D_6\times C_{36}$)

Order 3: $C_3$ ($C_4^2:C_{36}$), $C_3$ ($C_2^2.(C_{12}\times D_6)$)

Order 2: $C_2$ ($(C_2\times C_{36}).D_6$), $C_2$ ($(C_4\times S_3):C_{36}$), $C_2$ ($C_9\times C_2^3.D_6$), $C_2$ ($C_9\times C_2^3.D_6$), $C_2$ ($C_{12}^2.C_6$)

Order 1: $C_1$ ($C_2^2.(D_6\times C_{36})$)

Series

Derived series

$C_2^2.(D_6\times C_{36})$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$C_2^2.(D_6\times C_{36})$

$\rhd$

$(C_2\times C_{12}):C_{36}$

$\rhd$

$C_2\times C_6\times C_{36}$

$\rhd$

$C_2\times C_6\times C_{18}$

$\rhd$

$C_2\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2^2.(D_6\times C_{36})$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_{18}$

$\lhd$

$C_{18}.C_4^2$

Character theory

Complex character table

See the $432 \times 432$ 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 $94 \times 94$ rational character table.