Abstract group 1944.3414: $(C_6\times \He_3).D_6$

• Label: 1944.3414
• Order: \( 2^{3} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{8} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 3^{4} \cdot 5 \)
• Perm deg.: $33$
• Trans deg.: $216$
• Rank: $5$

Group information

Description:	$(C_6\times \He_3).D_6$
Order:	\(1944\)\(\medspace = 2^{3} \cdot 3^{5} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^4.C_2^3.C_2.\SU(4,2).C_2$, of order \(67184640\)\(\medspace = 2^{11} \cdot 3^{8} \cdot 5 \)
Composition factors:	$C_2$ x 3, $C_3$ x 5
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	3	242	324	726	648	1944
Conjugacy classes	1	3	42	4	126	8	184
Divisions	1	3	41	2	123	2	172
Autjugacy classes	1	2	2	1	4	1	11

Minimal presentations

Permutation degree:	$33$
Transitive degree:	$216$
Rank:	$5$
Inequivalent generating 5-tuples:	$211526640$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{4}=b^{3}=c^{3}=d^{3}=e^{3}=f^{6}=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $33$ $\langle(2,5)(3,8)(6,13)(7,16)(9,19)(11,14)(12,17)(15,24)(18,20)(21,27)(22,25)(23,26) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $((C_6\times \He_3).S_3)$
Semidirect product:	$C_3^3$ $\,\rtimes\,$ $(C_6.D_6)$	$\He_3$ $\,\rtimes\,$ $(C_6.D_6)$	$(C_6.C_3^4)$ $\,\rtimes\,$ $C_4$	$(C_6\times \He_3)$ $\,\rtimes\,$ $(C_3:C_4)$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_3^3:D_6)$	$(C_6\times \He_3)$ . $D_6$	$C_6$ . $(C_3^4:C_4)$	$(C_3^2:C_6^2)$ . $S_3$	all 18

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

Homology

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

Subgroups

There are 15384 subgroups in 2344 conjugacy classes, 1068 normal (10 characteristic).

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

Special subgroups

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

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1944: $(C_6\times \He_3).D_6$

Order 972: $(C_6\times \He_3).S_3$ x 2, $C_3^3:C_6^2$

Order 648: $(C_6\times \He_3):C_4$ x 40

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

Order 324: $(C_3\times \He_3):C_4$ x 80, $C_3^2:C_6^2$ x 40

Order 243: $C_3^3:C_3^2$

Order 216: $C_6^2.S_3$ x 90, $C_6^2.C_6$ x 40

Order 162: $C_6\times \He_3$ x 120

Order 108: $\He_3:C_4$ x 180, $C_2^2\times \He_3$ x 90, $C_3^2:C_{12}$ x 80, $C_3\times C_6^2$ x 40

Order 81: $C_3\times \He_3$ x 40

Order 72: $C_6.D_6$ x 40, $C_6:C_{12}$ x 40

Order 54: $C_2\times \He_3$ x 270, $C_3^2\times C_6$ x 120

Order 36: $C_3^2:C_4$ x 80, $C_3:C_{12}$ x 80, $C_6^2$ x 80

Order 27: $\He_3$ x 90, $C_3^3$ x 40

Order 24: $C_6:C_4$ x 40, $C_2\times C_{12}$

Order 18: $C_3\times C_6$ x 240

Order 12: $C_3:C_4$ x 80, $C_2\times C_6$ x 41, $C_{12}$ x 2

Order 9: $C_3^2$ x 80

Order 8: $C_2\times C_4$

Order 6: $C_6$ x 123

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

Order 3: $C_3$ x 41

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1944: $(C_6\times \He_3).D_6$

Order 972: $C_3^3:C_6^2$, $(C_6\times \He_3).S_3$

Order 648: $(C_6\times \He_3):C_4$

Order 486: $C_6.C_3^4$ x 2

Order 324: $C_3^2:C_6^2$, $(C_3\times \He_3):C_4$

Order 243: $C_3^3:C_3^2$

Order 216: $C_6^2.S_3$, $C_6^2.C_6$

Order 162: $C_6\times \He_3$ x 2

Order 108: $C_3\times C_6^2$, $C_3^2:C_{12}$, $C_2^2\times \He_3$, $\He_3:C_4$

Order 81: $C_3\times \He_3$

Order 72: $C_6.D_6$, $C_6:C_{12}$

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

Order 36: $C_6^2$ x 2, $C_3^2:C_4$, $C_3:C_{12}$

Order 27: $C_3^3$, $\He_3$

Order 24: $C_2\times C_{12}$, $C_6:C_4$

Order 18: $C_3\times C_6$ x 4

Order 12: $C_2\times C_6$ x 2, $C_{12}$, $C_3:C_4$

Order 9: $C_3^2$ x 2

Order 8: $C_2\times C_4$

Order 6: $C_6$ x 4

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1944: $(C_6\times \He_3).D_6$ ($C_1$)

Order 972: $(C_6\times \He_3).S_3$ ($C_2$) x 2, $C_3^3:C_6^2$ ($C_2$)

Order 486: $C_6.C_3^4$ ($C_4$) x 2, $C_6.C_3^4$ ($C_2^2$)

Order 324: $C_3^2:C_6^2$ ($S_3$) x 40

Order 243: $C_3^3:C_3^2$ ($C_2\times C_4$)

Order 162: $C_6\times \He_3$ ($C_3:C_4$) x 80, $C_6\times \He_3$ ($D_6$) x 40

Order 108: $C_2^2\times \He_3$ ($C_3:S_3$) x 90, $C_3\times C_6^2$ ($C_3:S_3$) x 40

Order 81: $C_3\times \He_3$ ($C_6:C_4$) x 40

Order 54: $C_2\times \He_3$ ($C_3^2:C_4$) x 180, $C_2\times \He_3$ ($C_6:S_3$) x 90, $C_3^2\times C_6$ ($C_3^2:C_4$) x 80, $C_3^2\times C_6$ ($C_6:S_3$) x 40

Order 36: $C_6^2$ ($C_3^2:S_3$) x 40

Order 27: $\He_3$ ($C_6.D_6$) x 90, $C_3^3$ ($C_6.D_6$) x 40

Order 18: $C_3\times C_6$ ($C_3^3:C_4$) x 80, $C_3\times C_6$ ($C_3^2:D_6$) x 40

Order 12: $C_2\times C_6$ ($C_3^3:S_3$)

Order 9: $C_3^2$ ($C_6^2.S_3$) x 40

Order 6: $C_6$ ($C_3^4:C_4$) x 2, $C_6$ ($C_3^3:D_6$)

Order 4: $C_2^2$ ($(C_3\times \He_3):S_3$)

Order 3: $C_3$ ($C_2\times C_3^4:C_4$)

Order 2: $C_2$ ($(C_6\times \He_3).S_3$) x 2, $C_2$ ($(C_6\times \He_3):S_3$)

Order 1: $C_1$ ($(C_6\times \He_3).D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1944: $(C_6\times \He_3).D_6$ ($C_1$)

Order 972: $C_3^3:C_6^2$ ($C_2$), $(C_6\times \He_3).S_3$ ($C_2$)

Order 486: $C_6.C_3^4$ ($C_2^2$), $C_6.C_3^4$ ($C_4$)

Order 324: $C_3^2:C_6^2$ ($S_3$)

Order 243: $C_3^3:C_3^2$ ($C_2\times C_4$)

Order 162: $C_6\times \He_3$ ($D_6$), $C_6\times \He_3$ ($C_3:C_4$)

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

Order 81: $C_3\times \He_3$ ($C_6:C_4$)

Order 54: $C_3^2\times C_6$ ($C_3^2:C_4$), $C_3^2\times C_6$ ($C_6:S_3$), $C_2\times \He_3$ ($C_3^2:C_4$), $C_2\times \He_3$ ($C_6:S_3$)

Order 36: $C_6^2$ ($C_3^2:S_3$)

Order 27: $C_3^3$ ($C_6.D_6$), $\He_3$ ($C_6.D_6$)

Order 18: $C_3\times C_6$ ($C_3^2:D_6$), $C_3\times C_6$ ($C_3^3:C_4$)

Order 12: $C_2\times C_6$ ($C_3^3:S_3$)

Order 9: $C_3^2$ ($C_6^2.S_3$)

Order 6: $C_6$ ($C_3^3:D_6$), $C_6$ ($C_3^4:C_4$)

Order 4: $C_2^2$ ($(C_3\times \He_3):S_3$)

Order 3: $C_3$ ($C_2\times C_3^4:C_4$)

Order 2: $C_2$ ($(C_6\times \He_3):S_3$), $C_2$ ($(C_6\times \He_3).S_3$)

Order 1: $C_1$ ($(C_6\times \He_3).D_6$)

Series

Derived series

$(C_6\times \He_3).D_6$

$\rhd$

$C_3^3:C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$(C_6\times \He_3).D_6$

$\rhd$

$C_3^3:C_6^2$

$\rhd$

$C_6.C_3^4$

$\rhd$

$C_6\times \He_3$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_6\times \He_3).D_6$

$\rhd$

$C_3^3:C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

Character theory

Complex character table

The $184 \times 184$ character table is not available for this group.

Rational character table

The $172 \times 172$ rational character table is not available for this group.