Abstract group 139968.hj: $C_3^4.D_6^2:D_6$

• Label: 139968.hj
• Order: \( 2^{6} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $22$
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$C_3^4.D_6^2:D_6$
Order:	\(139968\)\(\medspace = 2^{6} \cdot 3^{7} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^6.C_6.C_2.C_6.C_2^6$, of order \(3359232\)\(\medspace = 2^{9} \cdot 3^{8} \)
Composition factors:	$C_2$ x 6, $C_3$ x 7
Derived length:	$3$

This group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	5295	890	23760	64446	1296	24840	15552	3888	139968
Conjugacy classes	1	21	31	6	220	2	17	10	1	309
Divisions	1	21	31	6	220	2	17	10	1	309
Autjugacy classes	1	12	15	4	66	2	10	6	1	117

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$36$
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j \mid c^{4}=d^{2}=e^{6}=f^{3}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $22$ $\langle(1,4,5)(3,9,13,17,11,14)(6,8,16,7,15,18)(19,22)(20,21), (1,2,5,12)(3,8) \!\cdots\! \rangle$
Transitive group:	36T21097				more information
Direct product:	not computed
Semidirect product:	$(C_3^2\wr C_3)$ $\,\rtimes\,$ $(C_2^3:D_4)$				more information
Trans. wreath product:	not computed
Possibly split product:	$(C_3^4.S_3^3)$ . $D_4$ (8)	$C_3^4$ . $(D_6^2:D_6)$	$C_3^6$ . $(C_2^4:D_6)$	$(C_3^6.D_6:D_4)$ . $C_2$ (2)	all 69

Elements of the group are displayed as permutations of degree 22.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 148 normal subgroups (66 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_3^3\times C_6$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\wr C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 32 (order at least 4374) are shown, as well as all normal subgroups of any index.

Order 139968: $C_3^4.D_6^2:D_6$

Order 69984: $C_3^6.(D_4\times D_6)$ x 2, $C_3^4.S_3^2:D_{12}$ x 2, $C_3^6.D_6:D_4$ x 2, $C_3^4.S_3^3:C_4$ x 2, $C_2^2\times C_3^4.S_3^3$, $C_3^4.D_6^2:S_3$, $C_3^6.D_6:D_4$, $C_6^2.C_3^4:D_{12}$, $C_3^6.D_6.D_4$, $(C_3^2\times C_6^2).S_3^3$, $C_3^4.D_6^2:C_6$

Order 46656: $C_3^6.C_2^5.C_2$

Order 34992: $C_3^5.D_6^2$ x 15, $C_3^6.(S_3\times D_4)$ x 8, $C_3^6.C_6:D_4$ x 4, $C_2^2\times C_3^6.D_6$ x 2, $C_3^6.C_6.D_4$ x 2, $C_3^5:S_3.D_{12}$ x 2, $(C_3^5\times C_6).D_{12}$ x 2, $C_2\times C_3^6.D_{12}$ x 2, $C_3^5:S_3.D_{12}$ x 2, $C_3^6.(C_4\times D_6)$ x 2, $C_3^4.S_3^2:C_{12}$ x 2, $C_3^6.(C_6\times D_4)$ x 2, $C_3^4.\He_3.C_2^4$, $C_2^2\times C_3^6.C_6.C_2$, $C_2^2\times C_3^3.\He_3.D_6$, $(C_3^3\times C_6).S_3^3$, $(C_3^3\times C_6).S_3^3$, $(C_3^3\times C_6).S_3^3$, $(C_3^3\times C_6).S_3^3$, $(C_3^3\times C_6).S_3^3$, $C_3^6.(S_3\times D_4)$, $(C_3^3\times C_6^2).S_3^2$, $C_3^6.C_6.D_4$, $C_3^6.(C_6\times D_4)$, $C_6^2.C_3^4:C_{12}$, $C_3^6.(S_3\times D_4)$

Order 23328: $C_3^4.D_6\wr C_2$ x 8, $C_3^6.C_2^4.C_2$ x 5, $C_2^2\times C_3^6.C_2^3$, $C_2^2\times C_3^3.S_3^3$, $C_2\times C_3^6.C_2^3.C_2$, $(C_3\times C_6^2).S_3^3$

Order 17496: $C_3^4.S_3^3$ x 40, $C_2\times C_3^6:D_6$ x 13, $C_2\times C_3\wr D_6$ x 13, $C_2\times C_3^6.D_6$ x 8, $C_3^2\wr C_3:C_2^3$ x 8, $C_3^2\wr C_3:D_4$ x 8, $C_3^6.D_{12}$ x 4, $C_3^6:(C_4\times S_3)$ x 4, $C_3^2\wr C_3:D_4$ x 4, $C_2\times C_3^3.\He_3.D_6$ x 3, $C_3^5.C_6^2.C_2$ x 2, $C_2\times C_3^6.D_6$ x 2, $C_2\times C_3^5.S_3^2$ x 2, $C_3^5:S_3.D_6$ x 2, $C_2\times C_3^6.C_{12}$ x 2, $C_3^4.C_3^3.C_2^3$, $C_2^2\times C_3^6.S_3$, $C_2^2\times C_3^6.C_6$, $C_2\times C_3^4.C_3^2.D_6$, $C_3^6:(C_4\times S_3)$, $C_3^6:D_{12}$, $C_3^2\wr C_3:D_4$, $C_3^2\wr C_3:D_4$, $(C_3^5\times C_6).D_6$, $C_3^2\wr C_3:D_4$, $C_3^5.(S_3\times C_{12})$, $C_3^6:D_{12}$, $C_3^6:(C_4\times S_3)$, $C_3^2\wr C_3:D_4$, $(C_3^5\times C_6).D_6$, $C_2^2.C_3\wr C_6$

Order 15552: $C_6^2:S_3^2:D_6$

Order 11664: $C_3^4.D_6^2$ x 25, $C_3^6.C_2^3.C_2$ x 21, $C_2\times C_3^6.C_2^3$ x 15, $C_3^6.(C_2\times D_4)$ x 8, $C_2^2\times C_3^5.D_6$ x 6, $C_2\times C_3^6.D_4$ x 4, $C_3^4.D_6:D_6$ x 3, $C_3^6.C_2^4$, $C_3^5.(C_6\times D_4)$, $C_3^4.D_6^2$, $C_3^4.C_6^2.C_2^2$, $C_3^3.C_6^3.C_2$, $C_2^2\times C_3^5.C_6.C_2$, $C_2^2\times C_3^4.(C_6\times S_3)$, $C_2^2\times C_3^3.C_3^3.C_2^2$, $C_2^2\times C_3.C_3^4:C_6.C_2$, $C_2\times C_6\times C_3^5:C_2^2$, $C_2\times C_3^6.C_2^2.C_2$, $C_3^5:(S_3\times D_4)$, $C_3^5:(S_3\times D_4)$, $(C_3^2\times C_6).S_3^3$, $(C_3^2\times C_6).S_3^3$, $C_3^5:(S_3\times D_4)$, $C_3^5:(S_3\times D_4)$, $C_3^5:(C_2\times D_{12})$, $(C_3^3\times C_6^2):D_6$, $(C_3^3\times C_6^2):D_6$, $(C_3^3\times C_6^2):D_6$, $(C_3^3\times C_6^2):D_6$, $C_3^5:(C_6\times D_4)$, $(C_3^3\times C_6^2).D_6$, $(C_3^3\times C_6^2):D_6$

Order 8748: $C_3^6:D_6$ x 18, $C_3\wr D_6$ x 18, $C_3^6.D_6$ x 12, $C_3^2\wr C_3:C_2^2$ x 12, $C_3^6.D_6$ x 10, $C_3^6.D_6$ x 7, $C_3^5.C_6^2$ x 7, $C_3^4:C_3.C_6^2$ x 4, $C_2\times C_3^6.S_3$ x 4, $C_2\times C_3^6.C_6$ x 4, $C_3^2\wr C_3:C_4$ x 2, $C_3^6.C_{12}$ x 2, $C_3^5.C_6^2$, $C_3^2\wr C_3:C_4$, $C_3^6.C_{12}$

Order 7776: $C_3^5.C_2^4.C_2$ x 5, $C_2^2\times C_3^5.C_2^3$ x 5, $C_2^2\times C_3^2.S_3^3$ x 2, $C_3^4:(D_4\times D_6)$ x 2, $C_3^3:D_6\wr C_2$ x 2, $C_3^3:D_6\wr C_2$ x 2, $C_3^2.S_3^3:C_4$ x 2, $C_3^3:D_6\wr C_2$, $C_3^3:D_6\wr C_2$, $(C_3^2\times C_6^2):D_{12}$, $C_3^4:D_6.D_4$, $C_6^2.S_3^3$, $C_6^2.S_3^3$, $C_3^3:D_6\wr C_2$

Order 5832: $C_3^3.S_3^3$ x 68, $C_3^6:C_2^3$ x 40, $C_2\times C_3^5.D_6$ x 37, $C_3^6:D_4$ x 16, $C_3^6.C_2^3$ x 15, $C_2\times C_3^5:D_6$ x 13, $C_2\times C_3^5:D_6$ x 13, $C_2\times C_3^5:D_6$ x 13, $C_2\times C_3^5:D_6$ x 13, $(C_6\times \He_3):S_3^2$ x 13, $C_2\times C_3^5.D_6$ x 13, $C_2\times C_3^5:D_6$ x 13, $C_6\times C_3^5:C_2^2$ x 12, $C_3^4.C_6^2.C_2$ x 9, $C_2^2\times (C_3^5.C_3):C_2$ x 6, $C_3^6.C_2^2.C_2$ x 4, $C_2\times C_3^4:C_3.D_6$ x 4, $C_3^6:(C_2\times C_4)$ x 4, $C_2\times C_3^4.S_3^2$ x 3, $C_3^6.D_4$ x 2, $C_3^5.D_6.C_2$ x 2, $C_3^3.C_3^3.C_2^3$ x 2, $C_2\times C_3^6.C_4$ x 2, $C_2\times C_3^6.C_2^2$ x 2, $(C_3^4\times C_6^2).C_2$ x 2, $C_6^2\times C_3\times C_3^3:C_2$, $C_3^5.(C_3\times D_4)$, $C_2^2\times C_3^6:C_2$, $(C_3^4\times C_6).D_6$, $(C_3^4\times C_6).D_6$, $(C_3^4\times C_6).D_6$, $C_3^5:D_{12}$, $C_3^5.(C_4\times S_3)$, $(C_3^4\times C_6).D_6$, $C_3^5:(C_4\times S_3)$, $(C_3^4\times C_6).D_6$, $C_3^5:D_{12}$, $C_3^5:D_{12}$, $C_3^5:(C_4\times S_3)$, $(C_3^4\times C_6).D_6$, $C_3^5:(C_3\times D_4)$, $C_3^5:(C_2\times C_{12})$, $C_3^5:(C_3\times D_4)$, $C_3^4.(S_3\times C_{12})$, $C_3^5.D_{12}$, $(C_3^4\times C_6).D_6$, $C_3^5:(C_4\times S_3)$, $(C_3^4\times C_6).D_6$, $C_3^5:D_{12}$, $(C_3^3\times C_6^2):S_3$, $(C_3^3\times C_6^2).S_3$, $(C_3^3\times C_6^2):S_3$, $(C_3^3\times C_6^2):C_6$, $(C_3^3\times C_6^2).C_6$, $(C_3^3\times C_6^2):C_6$

Order 5184: $C_3:S_3^3:D_4$ x 4, $C_3^4.C_2^5.C_2$

Order 4374: $C_3\wr S_3$ x 5, $C_3\wr C_6$ x 5, $C_3^6:S_3$ x 3, $C_3^6.C_6$ x 3, $C_3^2\wr S_3$ x 3, $C_2:(C_3^6.C_3)$ x 2

Order 2916: $C_3^5.D_6$ x 4, $C_2\times C_3^6:C_2$ x 2, $C_3^4\times C_6^2$

Order 2187: $C_3^2\wr C_3$

Order 1944: $C_2\times C_3^4:D_6$

Order 1458: $C_3^5:S_3$ x 2, $C_3^5\times C_6$

Order 972: $C_3^4:D_6$ x 2, $C_3^3:C_6^2$

Order 729: $C_3^6$

Order 486: $C_3^4:S_3$ x 2, $C_3^4:C_6$

Order 324: $C_3^2\times C_6^2$

Order 243: $C_3^4:C_3$

Order 162: $C_3^3\times C_6$

Order 81: $C_3^4$

Order 36: $C_6^2$

Order 18: $C_3\times C_6$

Order 9: $C_3^2$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 139968: $C_3^4.D_6^2:D_6$ ($C_1$)

Order 69984: $C_3^6.(D_4\times D_6)$ ($C_2$) x 2, $C_3^4.S_3^2:D_{12}$ ($C_2$) x 2, $C_3^6.D_6:D_4$ ($C_2$) x 2, $C_3^4.S_3^3:C_4$ ($C_2$) x 2, $C_2^2\times C_3^4.S_3^3$ ($C_2$), $C_3^4.D_6^2:S_3$ ($C_2$), $C_3^6.D_6:D_4$ ($C_2$), $C_6^2.C_3^4:D_{12}$ ($C_2$), $C_3^6.D_6.D_4$ ($C_2$), $(C_3^2\times C_6^2).S_3^3$ ($C_2$), $C_3^4.D_6^2:C_6$ ($C_2$)

Order 34992: $C_3^5.D_6^2$ ($C_2^2$) x 5, $C_3^6.C_6:D_4$ ($C_2^2$) x 4, $C_3^6.C_6.D_4$ ($C_2^2$) x 2, $C_3^5:S_3.D_{12}$ ($C_2^2$) x 2, $(C_3^5\times C_6).D_{12}$ ($C_2^2$) x 2, $C_2\times C_3^6.D_{12}$ ($C_2^2$) x 2, $C_3^5:S_3.D_{12}$ ($C_2^2$) x 2, $C_3^6.(C_4\times D_6)$ ($C_2^2$) x 2, $C_3^4.S_3^2:C_{12}$ ($C_2^2$) x 2, $C_3^6.(C_6\times D_4)$ ($C_2^2$) x 2, $C_2^2\times C_3^6.D_6$ ($C_2^2$), $C_2^2\times C_3^6.C_6.C_2$ ($C_2^2$), $C_2^2\times C_3^3.\He_3.D_6$ ($C_2^2$), $(C_3^3\times C_6).S_3^3$ ($C_2^2$), $(C_3^3\times C_6).S_3^3$ ($C_2^2$), $(C_3^3\times C_6).S_3^3$ ($C_2^2$), $(C_3^3\times C_6^2).S_3^2$ ($C_2^2$), $C_3^6.C_6.D_4$ ($C_2^2$), $C_3^6.(C_6\times D_4)$ ($C_2^2$), $C_6^2.C_3^4:C_{12}$ ($C_2^2$)

Order 23328: $C_3^4.D_6\wr C_2$ ($S_3$)

Order 17496: $C_3^4.S_3^3$ ($D_4$) x 8, $C_2\times C_3^6.D_6$ ($C_2^3$) x 3, $C_3^2\wr C_3:C_2^3$ ($C_2^3$) x 3, $C_2\times C_3^6.D_6$ ($D_4$) x 2, $C_3^5:S_3.D_6$ ($C_2^3$) x 2, $C_2\times C_3^6.C_{12}$ ($C_2^3$) x 2, $C_3^4.C_3^3.C_2^3$ ($D_4$), $C_2^2\times C_3^6.S_3$ ($D_4$), $C_2^2\times C_3^6.C_6$ ($C_2^3$), $C_2\times C_3^5.S_3^2$ ($C_2^3$), $C_2\times C_3^3.\He_3.D_6$ ($C_2^3$), $(C_3^5\times C_6).D_6$ ($C_2^3$), $C_3^5.(S_3\times C_{12})$ ($C_2^3$)

Order 11664: $C_3^6.C_2^3.C_2$ ($D_6$) x 4, $C_3^4.D_6:D_6$ ($D_6$), $C_2^2\times C_3^5.D_6$ ($D_6$), $C_2\times C_3^6.D_4$ ($D_6$)

Order 8748: $C_3^6.D_6$ ($C_2\times D_4$) x 4, $C_3^6.D_6$ ($C_2\times D_4$) x 4, $C_3^2\wr C_3:C_2^2$ ($C_2\times D_4$) x 4, $C_3^4:C_3.C_6^2$ ($C_2\times D_4$) x 2, $C_2\times C_3^6.S_3$ ($C_2\times D_4$) x 2, $C_3^5.C_6^2$ ($C_2\times D_4$), $C_2\times C_3^6.C_6$ ($C_2^4$), $C_2\times C_3^6.C_6$ ($C_2\times D_4$)

Order 5832: $C_2\times C_3^5.D_6$ ($C_2\times D_6$) x 3, $C_3^6.C_2^2.C_2$ ($C_2\times D_6$), $C_3^5.D_6.C_2$ ($C_2\times D_6$), $C_2^2\times C_3^6:C_2$ ($C_2\times D_6$), $C_2\times C_3^6.C_4$ ($C_2\times D_6$)

Order 4374: $C_3^6:S_3$ ($C_2^2\wr C_2$) x 2, $C_3^6.C_6$ ($C_2^2\times D_4$) x 2, $C_3^2\wr S_3$ ($C_2^2\wr C_2$) x 2, $C_2:(C_3^6.C_3)$ ($C_2^2\times D_4$)

Order 2916: $C_3^5.D_6$ ($S_3\times D_4$) x 4, $C_3^4\times C_6^2$ ($S_3\times D_4$), $C_2\times C_3^6:C_2$ ($C_2^2\times D_6$), $C_2\times C_3^6:C_2$ ($S_3\times D_4$)

Order 2187: $C_3^2\wr C_3$ ($C_2^3:D_4$)

Order 1944: $C_2\times C_3^4:D_6$ ($\SOPlus(4,2)$)

Order 1458: $C_3^5:S_3$ ($D_4\times D_6$) x 2, $C_3^5\times C_6$ ($D_4\times D_6$)

Order 972: $C_3^4:D_6$ ($S_3^2:C_2^2$) x 2, $C_3^3:C_6^2$ ($S_3^2:C_2^2$)

Order 729: $C_3^6$ ($C_2^4:D_6$)

Order 486: $C_3^4:S_3$ ($D_6\wr C_2$) x 2, $C_3^4:C_6$ ($C_6^2:D_4$)

Order 324: $C_3^2\times C_6^2$ ($S_3^3:C_2$)

Order 243: $C_3^4:C_3$ ($D_6^2:C_2^2$)

Order 162: $C_3^3\times C_6$ ($S_3^3:C_2^2$)

Order 81: $C_3^4$ ($D_6^2:D_6$)

Order 36: $C_6^2$ ($C_3^4:(S_3\times D_4)$)

Order 18: $C_3\times C_6$ ($C_3^4:(D_4\times D_6)$)

Order 9: $C_3^2$ ($C_6^2:S_3^2:D_6$)

Order 4: $C_2^2$ ($C_3^6.(S_3\times D_4)$)

Order 2: $C_2$ ($C_3^6.D_6.C_2^3$)

Order 1: $C_1$ ($C_3^4.D_6^2:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

The $309 \times 309$ rational character table is not available for this group.