Abstract group 1296.157: $(C_3\times C_{36}).D_6$

• Label: 1296.157
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $35$
• Trans deg.: $216$
• Rank: $2$

Group information

Description:	$(C_3\times C_{36}).D_6$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$\He_3.C_6^2.C_2^4$, of order \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	8	9	12	18	24	36	72
Elements	1	109	26	110	242	36	54	268	54	72	108	216	1296
Conjugacy classes	1	2	3	2	5	2	4	6	4	4	8	12	53
Divisions	1	2	3	2	4	1	2	4	2	1	2	1	25
Autjugacy classes	1	2	3	2	4	1	2	4	2	1	2	1	25

Dimension	1	2	4	6	8	12	18	36	72
Irr. complex chars.	4	15	5	28	0	1	0	0	0	53
Irr. rational chars.	4	5	6	2	2	2	2	1	1	25

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$216$
Rank:	$2$
Inequivalent generating pairs:	$18$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	72
Arbitrary	6	10	22

Constructions

Presentation:	${\langle a, b, c, d \mid b^{6}=c^{9}=d^{12}=[a,c]=[c,d]=1, a^{2}=d^{3}, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $35$ $\langle(3,9)(4,5)(7,17)(8,14)(10,19)(11,12)(13,20)(15,24)(16,23)(18,22)(21,26) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(\He_3.C_{24})$ $\,\rtimes\,$ $C_2$	$(C_3.\He_3)$ $\,\rtimes\,$ $\SD_{16}$	$((C_3\times C_{36}).S_3)$ $\,\rtimes\,$ $C_2$	$(C_3\times C_9)$ $\,\rtimes\,$ $(C_3:\SD_{16})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3:D_{36})$ . $S_3$	$(C_3\times C_{36})$ . $D_6$	$(\He_3:C_8)$ . $S_3$	$(C_6.\He_3)$ . $D_4$	all 18

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 1532 subgroups in 112 conjugacy classes, 24 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $\He_3.D_{12}$
Commutator:	$G' \simeq$ $C_{12}.\He_3$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_3\times C_{12}$	$G/\Phi \simeq$ $S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}.\He_3$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $(C_3\times C_{36}).D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_3^2:D_{12}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3.\He_3$

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

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 1296: $(C_3\times C_{36}).D_6$

Order 648: $\He_3.C_{24}$, $(C_3\times C_{36}).S_3$, $\He_3.D_{12}$

Order 432: $C_{12}.S_3^2$, $C_6.D_{36}$

Order 324: $\He_3.D_6$, $C_{12}.\He_3$, $(C_3\times C_{18}).S_3$

Order 216: $C_3:D_{36}$, $C_{36}.C_6$, $C_3^2:D_{12}$, $\He_3:Q_8$, $C_{36}.C_6$, $\He_3:C_8$, $C_3:C_{72}$

Order 162: $C_6.\He_3$, $\He_3.S_3$

Order 144: $C_{72}:C_2$, $C_6.D_{12}$, $C_6.D_{12}$

Order 108: $C_9:C_{12}$ x 2, $C_3^2:C_{12}$, $C_9:C_{12}$, $C_3:D_{18}$, $C_3^2:D_6$, $C_9:C_{12}$, $C_4\times \He_3$, $C_3\times C_{36}$

Order 81: $C_3.\He_3$

Order 72: $C_9:Q_8$ x 2, $C_3:C_{24}$ x 2, $D_{36}$, $C_3:D_{12}$, $C_3^2:Q_8$, $C_3\times D_{12}$, $C_3^2:Q_8$, $C_{72}$

Order 54: $C_3\times C_{18}$, $C_3:D_9$, $C_3^2:C_6$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $C_{24}:C_2$, $Q_8:S_3$, $C_3:\SD_{16}$

Order 36: $C_9:C_4$ x 3, $C_3\times C_{12}$ x 2, $C_{36}$ x 2, $C_3^2:C_4$, $C_3:C_{12}$, $D_{18}$, $C_6:S_3$, $C_6\times S_3$

Order 27: $C_9:C_3$, $\He_3$, $C_3\times C_9$

Order 24: $D_{12}$ x 2, $C_3:Q_8$ x 2, $C_3:C_8$ x 2, $C_{24}$, $C_3\times Q_8$, $C_3\times D_4$

Order 18: $C_3\times C_6$ x 2, $C_{18}$ x 2, $C_3:S_3$, $C_3\times S_3$, $D_9$

Order 16: $\SD_{16}$

Order 12: $C_{12}$ x 4, $D_6$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

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

Order 8: $Q_8$, $D_4$, $C_8$

Order 6: $C_6$ x 4, $S_3$ x 2

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $(C_3\times C_{36}).D_6$

Order 648: $\He_3.C_{24}$, $(C_3\times C_{36}).S_3$, $\He_3.D_{12}$

Order 432: $C_{12}.S_3^2$, $C_6.D_{36}$

Order 324: $\He_3.D_6$, $C_{12}.\He_3$, $(C_3\times C_{18}).S_3$

Order 216: $C_3:D_{36}$, $C_{36}.C_6$, $C_3^2:D_{12}$, $\He_3:Q_8$, $C_{36}.C_6$, $\He_3:C_8$, $C_3:C_{72}$

Order 162: $C_6.\He_3$, $\He_3.S_3$

Order 144: $C_{72}:C_2$, $C_6.D_{12}$, $C_6.D_{12}$

Order 108: $C_9:C_{12}$, $C_3^2:C_{12}$, $C_9:C_{12}$, $C_3:D_{18}$, $C_3^2:D_6$, $C_9:C_{12}$, $C_4\times \He_3$, $C_3\times C_{36}$

Order 81: $C_3.\He_3$

Order 72: $C_9:Q_8$ x 2, $C_3:C_{24}$ x 2, $D_{36}$, $C_3:D_{12}$, $C_3^2:Q_8$, $C_3\times D_{12}$, $C_3^2:Q_8$, $C_{72}$

Order 54: $C_3\times C_{18}$, $C_3:D_9$, $C_3^2:C_6$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $C_{24}:C_2$, $Q_8:S_3$, $C_3:\SD_{16}$

Order 36: $C_3\times C_{12}$ x 2, $C_{36}$ x 2, $C_9:C_4$ x 2, $C_3^2:C_4$, $C_3:C_{12}$, $D_{18}$, $C_6:S_3$, $C_6\times S_3$

Order 27: $C_9:C_3$, $\He_3$, $C_3\times C_9$

Order 24: $D_{12}$ x 2, $C_3:Q_8$ x 2, $C_3:C_8$ x 2, $C_{24}$, $C_3\times Q_8$, $C_3\times D_4$

Order 18: $C_3\times C_6$ x 2, $C_{18}$ x 2, $C_3:S_3$, $C_3\times S_3$, $D_9$

Order 16: $\SD_{16}$

Order 12: $C_{12}$ x 4, $D_6$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

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

Order 8: $Q_8$, $D_4$, $C_8$

Order 6: $C_6$ x 4, $S_3$ x 2

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $(C_3\times C_{36}).D_6$ ($C_1$)

Order 648: $\He_3.C_{24}$ ($C_2$), $(C_3\times C_{36}).S_3$ ($C_2$), $\He_3.D_{12}$ ($C_2$)

Order 324: $C_{12}.\He_3$ ($C_2^2$)

Order 216: $C_3:D_{36}$ ($S_3$), $\He_3:C_8$ ($S_3$)

Order 162: $C_6.\He_3$ ($D_4$)

Order 108: $C_4\times \He_3$ ($D_6$), $C_3\times C_{36}$ ($D_6$)

Order 81: $C_3.\He_3$ ($\SD_{16}$)

Order 54: $C_3\times C_{18}$ ($C_3:D_4$), $C_2\times \He_3$ ($D_{12}$)

Order 36: $C_3\times C_{12}$ ($S_3^2$)

Order 27: $\He_3$ ($C_{24}:C_2$), $C_3\times C_9$ ($C_3:\SD_{16}$)

Order 18: $C_3\times C_6$ ($C_3:D_{12}$)

Order 12: $C_{12}$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_6.D_{12}$)

Order 6: $C_6$ ($C_3^2:D_{12}$)

Order 4: $C_4$ ($C_3^2.S_3^2$)

Order 3: $C_3$ ($C_{12}.S_3^2$)

Order 2: $C_2$ ($\He_3.D_{12}$)

Order 1: $C_1$ ($(C_3\times C_{36}).D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $(C_3\times C_{36}).D_6$ ($C_1$)

Order 648: $\He_3.C_{24}$ ($C_2$), $(C_3\times C_{36}).S_3$ ($C_2$), $\He_3.D_{12}$ ($C_2$)

Order 324: $C_{12}.\He_3$ ($C_2^2$)

Order 216: $C_3:D_{36}$ ($S_3$), $\He_3:C_8$ ($S_3$)

Order 162: $C_6.\He_3$ ($D_4$)

Order 108: $C_4\times \He_3$ ($D_6$), $C_3\times C_{36}$ ($D_6$)

Order 81: $C_3.\He_3$ ($\SD_{16}$)

Order 54: $C_3\times C_{18}$ ($C_3:D_4$), $C_2\times \He_3$ ($D_{12}$)

Order 36: $C_3\times C_{12}$ ($S_3^2$)

Order 27: $\He_3$ ($C_{24}:C_2$), $C_3\times C_9$ ($C_3:\SD_{16}$)

Order 18: $C_3\times C_6$ ($C_3:D_{12}$)

Order 12: $C_{12}$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_6.D_{12}$)

Order 6: $C_6$ ($C_3^2:D_{12}$)

Order 4: $C_4$ ($C_3^2.S_3^2$)

Order 3: $C_3$ ($C_{12}.S_3^2$)

Order 2: $C_2$ ($\He_3.D_{12}$)

Order 1: $C_1$ ($(C_3\times C_{36}).D_6$)

Series

Derived series

$(C_3\times C_{36}).D_6$

$\rhd$

$C_{12}.\He_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$(C_3\times C_{36}).D_6$

$\rhd$

$\He_3.C_{24}$

$\rhd$

$C_{12}.\He_3$

$\rhd$

$C_6.\He_3$

$\rhd$

$C_3.\He_3$

$\rhd$

$\He_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_3\times C_{36}).D_6$

$\rhd$

$C_{12}.\He_3$

$\rhd$

$C_6.\He_3$

$\rhd$

$C_3.\He_3$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

Character theory

Complex character table

See the $53 \times 53$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $25 \times 25$ rational character table.