Abstract group 52488.dh: $(C_2\times \He_3^2).S_3^2$

• Label: 52488.dh
• Order: \( 2^{3} \cdot 3^{8} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{2} \)
• Perm deg.: $22$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$(C_2\times \He_3^2).S_3^2$
Order:	\(52488\)\(\medspace = 2^{3} \cdot 3^{8} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^6.C_3^2.C_6^2.C_2^4$, of order \(3779136\)\(\medspace = 2^{6} \cdot 3^{10} \)
Composition factors:	$C_2$ x 3, $C_3$ x 8
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	1459	2024	324	13688	4536	14256	4536	11664	52488
Conjugacy classes	1	3	50	4	60	32	28	32	8	218
Divisions	1	3	43	2	49	17	12	17	2	146
Autjugacy classes	1	2	15	1	18	7	4	7	1	56

Dimension	1	2	4	8	12	18	24	36
Irr. complex chars.	24	36	30	0	72	32	0	24	218
Irr. rational chars.	4	14	18	14	24	16	24	32	146

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid b^{6}=c^{3}=d^{3}=e^{3}=f^{3}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $22$ $\langle(1,3,2,4,7,12)(5,9,6,11,16,15)(8,13,17,18,10,14)(19,20,21,22), (1,2)(3,5,8,12,16,17)(4,6,10)(11,15)(14,18)\rangle$
Transitive group:	36T16213				more information
Direct product:	not computed
Semidirect product:	$(C_3^4.C_3^4)$ $\,\rtimes\,$ $(C_2\times C_4)$				more information
Trans. wreath product:	not computed
Possibly split product:	$(C_3^6.C_6)$ . $C_{12}$ (2)	$C_3^6$ . $(C_6:C_{12})$	$(C_3^5:C_6)$ . $S_3^2$ (2)	$C_3^5$ . $(C_6.S_3^2)$ (2)	all 42

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

Homology

Abelianization:	$C_{2} \times C_{12} \simeq C_{2} \times C_{4} \times C_{3}$
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 66 normal subgroups (26 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:	a subgroup isomorphic to $C_3^2\wr C_3$
Frattini:	a subgroup isomorphic to $C_3^3\times C_6$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4.C_3^4$

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 81 (order at least 648) are shown, as well as all normal subgroups of any index.

Order 52488: $(C_2\times \He_3^2).S_3^2$

Order 26244: $C_3^2\wr C_3:C_{12}$ x 2, $C_2\times C_3^6.C_3.C_6$

Order 17496: $\He_3^2:(C_4\times S_3)$ x 2, $(C_3^5\times C_6).D_6$, $C_3^5.(S_3\times C_{12})$

Order 13122: $C_3^6.C_3.C_6$ x 2, $C_2\times C_3^6.C_3^2$

Order 8748: $C_2\times C_3^6.C_6$ x 3, $C_2\times C_3^5.C_3.C_6$ x 3, $C_3^2\wr C_3:C_4$ x 2, $C_3^5.C_3:C_{12}$ x 2, $(C_2\times \He_3^2).S_3$ x 2, $\He_3^2:C_{12}$ x 2, $C_9^2:C_3^2:C_{12}$ x 2, $C_3^6.C_{12}$ x 2

Order 6561: $C_3^4.C_3^4$

Order 5832: $(C_3^4\times C_6).D_6$ x 2, $C_3^5:(C_2\times C_{12})$ x 2, $C_3^5.D_6.C_2$, $\He_3^2:(C_2\times C_4)$

Order 4374: $C_2\times C_3^5.C_3^2$ x 13, $C_3^5.C_3.C_6$ x 6, $C_3^6.C_6$ x 4, $C_3^6.C_6$ x 2, $C_2\times C_3^6.C_3$ x 2, $C_2:(C_3^6.C_3)$

Order 2916: $C_2\times (C_3^5.C_3):C_2$ x 17, $C_2\times (C_3^4.C_3^2):C_2$ x 5, $C_2\times (C_3^2\times C_3\wr C_3):C_2$ x 3, $C_3^6.C_4$ x 2, $(C_3^4\times C_6).S_3$ x 2, $(C_3^4\times C_6).S_3$ x 2, $C_3^4.C_3:C_{12}$ x 2, $C_3^4.C_3:C_{12}$ x 2, $(C_3^4\times C_6).S_3$ x 2, $\He_3^2:C_4$ x 2, $C_3^5:C_{12}$ x 2, $C_3^4.C_3:C_{12}$ x 2, $C_3^5.C_{12}$ x 2, $C_9^2:C_3:C_{12}$ x 2, $C_3^5:C_{12}$ x 2, $C_2\times C_3^6:C_2$, $C_2\times (C_3^3.C_3^3):C_2$

Order 2187: $C_3^4.C_3^3$ x 4, $C_3^5.C_3^2$ x 2, $C_3^2\wr C_3$ x 2, $C_3^4.C_3^3$ x 2, $C_3^4.C_3^3$ x 2, $C_3^5.C_3^2$, $C_3^3\times C_3\wr C_3$, $C_3^5.C_3^2$, $C_3^3.C_3^4$

Order 1944: $C_3^4:(C_4\times S_3)$ x 10, $(C_3^3\times C_6).D_6$ x 2, $C_3^3:(S_3\times C_{12})$ x 2, $C_3^4:(C_4\times S_3)$ x 2, $(C_3^3\times C_6).D_6$, $C_3^4:(C_2\times C_{12})$

Order 1458: $C_3^5:C_6$ x 31, $C_3^5:C_6$ x 30, $C_3^5:C_6$ x 12, $C_3^5:C_6$ x 7, $C_3^5.C_6$ x 6, $C_{18}:C_3\wr C_3$ x 4, $C_2\times C_3^3.C_3^3$ x 4, $C_2\times C_3^3.C_3^3$ x 4, $C_2\times C_3^3.C_3^3$ x 4, $C_2\times C_3^3.C_3^3$ x 4, $C_3^4:(C_3\times C_6)$ x 4, $C_2\times C_3^4:C_3^2$ x 3, $C_2\times C_9^2:C_3^2$ x 2, $(C_9\times \He_3):C_6$ x 2, $(C_3^3\times C_9):C_6$ x 2, $(C_3^2\times C_{18}):C_3^2$ x 2, $C_3^5:S_3$ x 2, $C_3^5:C_6$ x 2, $C_2\times C_3^3.C_3^3$ x 2, $C_3^5:C_6$ x 2, $\He_3^2:C_2$ x 2, $C_2\times C_3^3.C_3^3$ x 2, $C_3^5\times C_6$, $C_{18}:C_3^4$, $C_3^5:C_6$, $C_9:C_6\times \He_3$, $C_2\times \He_3^2$, $C_3\wr C_3\times C_{18}$

Order 972: $C_2\times C_3^4:C_6$ x 51, $C_3^4:D_6$ x 35, $C_3^4:D_6$ x 15, $C_3^4:C_{12}$ x 10, $C_3^4:C_{12}$ x 10, $(C_3\times \He_3):C_{12}$ x 4, $C_3^4:C_{12}$ x 4, $C_3^3:C_6^2$ x 3, $(C_6\times \He_3).S_3$ x 2, $C_3^2:C_9.C_{12}$ x 2, $C_2\times C_3^4:C_6$ x 2, $C_3^3:C_6^2$ x 2, $(C_3\times \He_3):C_{12}$ x 2, $C_3\wr C_3:C_{12}$ x 2, $(C_9\times C_{18}).S_3$ x 2, $(C_3^3\times C_6).S_3$ x 2, $C_3^4:C_{12}$ x 2, $C_3^4:C_{12}$ x 2, $C_9^2:C_{12}$ x 2, $C_3^4.C_{12}$ x 2, $(C_3^2\times C_6).D_9$ x 2, $C_3^3:C_{36}$ x 2, $(C_3\times \He_3).C_{12}$ x 2, $C_3^4:D_6$, $C_3^3:C_6^2$

Order 729: $C_3^5:C_3$ x 31, $C_3^5:C_3$ x 7, $C_3^5.C_3$ x 6, $C_3^3.C_3^3$ x 4, $C_3^3.C_3^3$ x 4, $C_3^3.C_3^3$ x 4, $C_3^3.C_3^3$ x 4, $C_3^3.C_3^3$ x 4, $C_3^4:C_3^2$ x 3, $C_9^2:C_3^2$ x 2, $C_3^3.C_3^3$ x 2, $C_3^3.C_3^3$ x 2, $C_3^3.C_3^3$ x 2, $C_3^3.C_3^3$ x 2, $C_3^3.C_3^3$ x 2, $C_3^6$, $C_9:C_3^4$, $C_3^3\times \He_3$, $C_3^2.C_3^4$, $\He_3^2$, $C_3^3.C_3^3$

Order 648: $C_3^3:(C_4\times S_3)$ x 57, $C_3^3:(C_4\times S_3)$ x 10, $\He_3:(C_4\times S_3)$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^2:(S_3\times C_{12})$ x 2, $C_4\times C_3^3:C_6$ x 2, $C_3\times C_6.S_3^2$

Order 486: $C_3^4\times C_6$ x 2, $C_3^4:C_6$ x 2

Order 243: $C_3^5$ x 2, $C_3^4:C_3$ x 2

Order 162: $C_3^3\times C_6$

Order 81: $C_3^4$

Order 54: $C_3^2\times C_6$ x 2

Order 27: $C_3^3$ x 2

Order 18: $C_3\times C_6$

Order 9: $C_3^2$

Order 6: $C_6$ x 2

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 52488: $(C_2\times \He_3^2).S_3^2$ ($C_1$)

Order 26244: $C_3^2\wr C_3:C_{12}$ ($C_2$) x 2, $C_2\times C_3^6.C_3.C_6$ ($C_2$)

Order 17496: $(C_3^5\times C_6).D_6$ ($C_3$)

Order 13122: $C_3^6.C_3.C_6$ ($C_4$) x 2, $C_2\times C_3^6.C_3^2$ ($C_2^2$)

Order 8748: $C_3^2\wr C_3:C_4$ ($C_6$) x 2, $(C_2\times \He_3^2).S_3$ ($S_3$) x 2, $C_2\times C_3^6.C_6$ ($C_6$), $C_2\times C_3^6.C_6$ ($S_3$)

Order 6561: $C_3^4.C_3^4$ ($C_2\times C_4$)

Order 4374: $C_2\times C_3^5.C_3^2$ ($D_6$) x 2, $C_3^6.C_6$ ($C_{12}$) x 2, $C_3^6.C_6$ ($C_3:C_4$) x 2, $C_2\times C_3^6.C_3$ ($C_2\times C_6$), $C_2\times C_3^6.C_3$ ($D_6$)

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

Order 2187: $C_3^4.C_3^3$ ($C_4\times S_3$) x 2, $C_3^2\wr C_3$ ($C_2\times C_{12}$), $C_3^2\wr C_3$ ($C_6:C_4$)

Order 1458: $C_3^5:C_6$ ($C_6\times S_3$) x 2, $C_3^5:C_6$ ($S_3^2$) x 2, $C_3^5:S_3$ ($C_3:C_{12}$) x 2, $C_3^5\times C_6$ ($C_6\times S_3$), $C_2\times \He_3^2$ ($S_3^2$)

Order 729: $C_3^5:C_3$ ($S_3\times C_{12}$) x 2, $C_3^5:C_3$ ($C_6.D_6$) x 2, $C_3^6$ ($C_6:C_{12}$), $\He_3^2$ ($C_6.D_6$)

Order 486: $C_3^4\times C_6$ ($C_3\times S_3^2$) x 2, $C_3^4:C_6$ ($C_3:S_3^2$), $C_3^4:C_6$ ($C_3\times S_3^2$)

Order 243: $C_3^5$ ($C_6.S_3^2$) x 2, $C_3^4:C_3$ ($C_6.S_3^2$), $C_3^4:C_3$ ($C_6.S_3^2$)

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

Order 81: $C_3^4$ ($C_3\times C_6.S_3^2$)

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

Order 27: $C_3^3$ ($C_3^3:(S_3\times C_{12})$) x 2

Order 18: $C_3\times C_6$ ($C_3^4:S_3^2$)

Order 9: $C_3^2$ ($\He_3^2:(C_2\times C_4)$)

Order 6: $C_6$ ($C_3^5:(C_6\times S_3)$) x 2

Order 3: $C_3$ ($C_3^5.C_6^2.C_2$), $C_3$ ($C_3^4:C_3.(C_{12}\times S_3)$)

Order 2: $C_2$ ($\He_3^2:S_3^2$)

Order 1: $C_1$ ($(C_2\times \He_3^2).S_3^2$)

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 8 larger groups in the database.

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

Character theory

Complex character table

See the $218 \times 218$ 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 $146 \times 146$ rational character table (warning: may be slow to load).