Abstract group 34992.nb: $C_3^7:(C_2\times D_4)$

• Label: 34992.nb
• Order: \( 2^{4} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3^{2} \)
• Perm deg.: $21$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^7:(C_2\times D_4)$
Order:	\(34992\)\(\medspace = 2^{4} \cdot 3^{7} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^6.Q_8^2.C_3^3.C_2^5$, of order \(40310784\)\(\medspace = 2^{11} \cdot 3^{9} \)
Composition factors:	$C_2$ x 4, $C_3$ 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	12
Elements	1	3015	2186	1944	21042	6804	34992
Conjugacy classes	1	7	189	2	129	5	333
Divisions	1	7	189	2	129	5	333
Autjugacy classes	1	7	15	2	20	2	47

Dimension	1	2	4	8	16
Irr. complex chars.	8	22	53	156	94	333
Irr. rational chars.	8	22	53	156	94	333

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	not computed

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:	${\langle a, b, c, d, e, f, g, h \mid a^{2}=b^{6}=c^{12}=d^{3}=e^{3}=f^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $21$ $\langle(1,3,7,11,4,8)(2,5)(6,9)(10,12)(16,17,18)(19,21), (1,2,4,6)(3,5)(7,10)(8,12,11,9)(13,14)(17,18)(19,20,21), (1,4)(2,6)(5,9)(8,11)(14,15)(17,18)(19,21)\rangle$
Transitive group:	36T14275				more information
Direct product:	not computed
Semidirect product:	$C_3^7$ $\,\rtimes\,$ $(C_2\times D_4)$				more information
Trans. wreath product:	not computed
Possibly split product:	$(C_3^6:D_4)$ . $S_3$	$(C_3^6:D_4)$ . $S_3$ (4)	$C_3^6$ . $(S_3\times D_4)$ (5)	$(C_3^5:D_6)$ . $D_6$	all 46

Elements of the group are displayed as permutations of degree 21.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 135 normal subgroups (39 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 $C_1$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^7$

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 27 (order at least 1296) are shown, as well as all normal subgroups of any index.

Order 34992: $C_3^7:(C_2\times D_4)$

Order 17496: $C_3^7.C_2^3$, $C_3^7:D_4$, $C_3^7:D_4$, $C_3^7:(C_2\times C_4)$, $C_3^7:D_4$, $C_3^7:C_2^3$, $C_3^7:D_4$

Order 11664: $C_3^6:(C_2\times D_4)$ x 4, $C_3^6:(C_2\times D_4)$

Order 8748: $C_3^6.D_6$ x 4, $C_3^7.C_2^2$ x 2, $C_3^7:C_2^2$ x 2, $C_3^7:C_4$, $C_3^7:C_4$, $C_3^7:C_2^2$

Order 5832: $(C_3:S_3)^3$ x 10, $C_3^6:C_2^3$ x 8, $C_3^6:D_4$ x 8, $C_3^5:D_{12}$ x 8, $C_3^6.C_2^3$ x 4, $C_3^6:(C_2\times C_4)$ x 4, $C_3^6:D_4$ x 2, $C_3^6:D_4$, $C_3^6:(C_2\times C_4)$, $C_3^5:D_{12}$

Order 4374: $C_3^7.C_2$ x 6, $C_3^2\times C_3\times C_3:(C_3^3:C_2)$

Order 3888: $S_3\times C_3^4:D_4$ x 5, $C_3\wr C_4:D_6$ x 4

Order 2916: $C_3^5.D_6$ x 132, $C_3^4:C_6^2$ x 24, $C_3^5:D_6$ x 16, $C_3^6.C_2^2$ x 12, $C_3^4:C_6^2$ x 12, $C_3^5:D_6$ x 10, $C_3^5:C_{12}$ x 8, $C_3:S_3\times C_3:(C_3^3:C_2)$, $C_3^5:C_{12}$, $C_3^6:C_4$

Order 2187: $C_3^7$

Order 1944: $C_3^2:S_3^3$ x 71, $C_3^2:S_3^3$ x 26, $C_3^2:S_3^3$ x 10, $C_3^5:D_4$ x 10, $C_3\wr C_4:S_3$ x 7, $C_3^4:D_{12}$ x 5, $C_3^5:D_4$ x 5, $C_3^4:C_4\times S_3$ x 5, $C_3^4:D_{12}$ x 4, $C_3^4:D_{12}$ x 4, $C_3^5:D_4$ x 4, $C_3^5:D_4$ x 4, $C_3^5:C_2^3$

Order 1458: $C_3^5:S_3$ x 189, $C_3^5:C_6$ x 87, $C_3^5:C_6$ x 87, $C_3^3\wr C_2$ x 76, $C_3^5:C_6$ x 4, $S_3\times C_3^5$ x 4

Order 1296: $S_3^2:S_3^2$ x 16, $(C_3\times S_3^2):D_6$ x 4, $C_2\times C_3^4:D_4$

Order 972: $C_3^3:S_3^2$ x 5, $C_3^3:S_3^2$, $C_3^5:C_4$

Order 729: $C_3^6$ x 5

Order 486: $C_3^4:C_6$ x 5, $C_3^4:C_6$ x 4

Order 324: $C_3^2:S_3^2$

Order 243: $C_3^5$ x 9

Order 162: $C_3^3:S_3$

Order 81: $C_3^4$ x 21

Order 54: $C_3^2:S_3$

Order 27: $C_3^3$ x 21

Order 9: $C_3^2$ x 9

Order 3: $C_3$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 34992: $C_3^7:(C_2\times D_4)$ ($C_1$)

Order 17496: $C_3^7.C_2^3$ ($C_2$), $C_3^7:D_4$ ($C_2$), $C_3^7:D_4$ ($C_2$), $C_3^7:(C_2\times C_4)$ ($C_2$), $C_3^7:D_4$ ($C_2$), $C_3^7:C_2^3$ ($C_2$), $C_3^7:D_4$ ($C_2$)

Order 8748: $C_3^6.D_6$ ($C_2^2$) x 2, $C_3^7.C_2^2$ ($C_2^2$), $C_3^7:C_4$ ($C_2^2$), $C_3^7:C_4$ ($C_2^2$), $C_3^7:C_2^2$ ($C_2^2$), $C_3^7:C_2^2$ ($C_2^2$)

Order 5832: $C_3^6:D_4$ ($S_3$) x 4, $C_3^6:D_4$ ($S_3$)

Order 4374: $C_3^7.C_2$ ($D_4$) x 2, $C_3^2\times C_3\times C_3:(C_3^3:C_2)$ ($C_2^3$)

Order 2916: $C_3^4:C_6^2$ ($D_6$) x 5, $C_3^5:C_{12}$ ($D_6$) x 4, $C_3^5:D_6$ ($D_6$) x 4, $C_3^6:C_4$ ($D_6$), $C_3^5:D_6$ ($D_6$)

Order 2187: $C_3^7$ ($C_2\times D_4$)

Order 1944: $C_3^5:D_4$ ($C_3:S_3$)

Order 1458: $C_3^5:C_6$ ($C_2\times D_6$) x 5

Order 972: $C_3^3:S_3^2$ ($S_3^2$) x 4, $C_3^3:S_3^2$ ($C_6:S_3$), $C_3^3:S_3^2$ ($C_6:S_3$), $C_3^5:C_4$ ($C_6:S_3$)

Order 729: $C_3^6$ ($S_3\times D_4$) x 5

Order 486: $C_3^4:C_6$ ($S_3\times D_6$) x 4, $C_3^4:C_6$ ($\SOPlus(4,2)$) x 4, $C_3^4:C_6$ ($C_6:D_6$)

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

Order 243: $C_3^5$ ($S_3^2:C_2^2$) x 4, $C_3^5$ ($D_6:D_6$) x 4, $C_3^5$ ($C_{12}:D_6$)

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

Order 81: $C_3^4$ ($S_3^3:C_2$) x 20, $C_3^4$ ($C_6^2:D_6$)

Order 54: $C_3^2:S_3$ ($C_3^4:D_4$)

Order 27: $C_3^3$ ($S_3^2:S_3^2$) x 16, $C_3^3$ ($(C_3\times S_3^2):D_6$) x 4, $C_3^3$ ($C_2\times C_3^4:D_4$)

Order 9: $C_3^2$ ($S_3\times C_3^4:D_4$) x 5, $C_3^2$ ($C_3\wr C_4:D_6$) x 4

Order 3: $C_3$ ($C_3^6:(C_2\times D_4)$) x 4, $C_3$ ($C_3^6:(C_2\times D_4)$)

Order 1: $C_1$ ($C_3^7:(C_2\times D_4)$)

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 10 larger groups in the database.

This group is a maximal quotient of 1 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

See the $333 \times 333$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.