Abstract group 1185408.a: $D_7^3.C_6^2:D_6$

• Label: 1185408.a
• Order: \( 2^{7} \cdot 3^{3} \cdot 7^{3} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{4} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $27$
• Trans deg.: $42$
• Rank: $3$

Group information

Description:	$D_7^3.C_6^2:D_6$
Order:	\(1185408\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 7^{3} \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_7^3.C_2^4:\He_3.C_6.C_2^4$, of order \(14224896\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 7^{3} \)
Composition factors:	$C_2$ x 7, $C_3$ x 3, $C_7$ x 3
Derived length:	$4$

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

Group statistics

Order	1	2	3	4	6	7	12	14	21	28	42
Elements	1	8127	15092	60480	428652	342	296352	62874	86436	87696	139356	1185408
Conjugacy classes	1	23	6	12	118	5	24	51	4	16	34	294
Divisions	1	23	5	12	71	5	12	51	4	16	22	222
Autjugacy classes	1	13	4	4	57	3	8	25	2	5	16	138

Dimension	1	2	3	6	12	18	36	72	108	144	216
Irr. complex chars.	8	16	88	72	0	16	44	12	16	6	16	294
Irr. rational chars.	8	16	24	48	28	16	20	24	16	6	16	222

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$42$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	18	18
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{6}=c^{2}=d^{6}=e^{42}=f^{14}=g^{14}= \!\cdots\! \rangle}$
Permutation group:	Degree $27$ $\langle(1,3,6,14,20,21,4,10,15,11,17,13,19,9,8,16,18,12,7,2,5)(22,24,27,25,23,26) \!\cdots\! \rangle$
Transitive group:	42T1945				more information
Direct product:	not computed
Semidirect product:	$C_7^3$ $\,\rtimes\,$ $(C_2^2\times C_6^2:S_4)$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$D_7^3$ . $(C_6^2:D_6)$ (4)	$(D_7^3.C_3^2:S_4)$ . $C_2$ (4)	$(D_7^3:C_3)$ . $(C_6:S_4)$ (4)	$C_2$ . $(D_7^3.C_3^2:S_4)$	all 49

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$2$

Subgroups

There are 110 normal subgroups (37 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 $C_7^3:(C_6^2:A_4)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^3$

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 37044) are shown, as well as all normal subgroups of any index.

Order 1185408: $D_7^3.C_6^2:D_6$

Order 592704: $D_7^3.C_3^2:S_4$ x 4, $C_2\times C_7^3:(C_6^2:S_4)$, $C_7^3:(C_2\times C_6^2:S_4)$, $D_7^3.C_6^2:C_6$

Order 395136: $D_7^3.(C_6\times S_4)$ x 3, $C_7^3.C_6^2.C_2^5$

Order 296352: $C_7^3:C_6^2:S_4$ x 2, $C_7^3:(C_6^2:S_4)$ x 2, $C_7^3:(C_6^2:S_4)$ x 2, $D_7^3:(C_3^2:A_4)$ x 2, $C_7^3:(C_2^5:\He_3)$, $C_7^3:C_6^2:S_4$

Order 197568: $C_7^3.C_6^2.C_2^4$ x 24, $C_7^3.C_2^4.(C_6\times S_3)$ x 12, $C_2\times C_7^3.C_6^2.C_2^3$ x 3, $C_7^3.C_2^4.(C_6\times S_3)$ x 3, $C_7^3:(C_2^4:C_6^2)$ x 3, $C_7^3.C_2^4.(C_6\times S_3)$ x 3, $C_7^3.C_6.C_6.C_2^4$, $C_2^3\times C_7^3.C_6^2.C_2$, $C_2^2\times C_7^3.C_6.C_6.C_2^2$, $C_2\times C_7^3.C_6.C_6.C_2^3$

Order 148176: $D_7^3:C_3^2:S_3$ x 8, $C_7^3:(C_6^2:D_6)$ x 4, $C_2\times D_7^3:\He_3$ x 3, $C_7^3:(C_6^2:D_6)$ x 2, $C_7^3:(C_6^2:D_6)$ x 2, $C_7^3:(C_6^2:A_4)$, $C_7^3:(C_6^2:D_6)$, $C_7^3:(C_6^2:D_6)$, $C_7^3:C_6^2:A_4$

Order 131712: $D_{14}\wr S_3$ x 3, $C_7^3.C_6.C_2.C_2^5$, $C_7^3.C_2.C_6.C_2^5$

Order 98784: $C_7^3.C_6^2.C_2^3$ x 90, $C_2\times C_7^3.C_6.C_6.C_2^2$ x 33, $C_2^2\times C_7^3.C_6^2.C_2$ x 16, $C_7^3.C_6.C_6.C_2^3$ x 13, $C_2\times C_7^3.C_6^2.C_2^2$ x 8, $C_2^2\times C_7^3.C_6.C_6.C_2$ x 6, $D_7^3:(C_6\times S_3)$ x 6, $(C_7\times C_{14}^2):(C_3\times S_4)$ x 6, $D_7^3:(C_3\times A_4)$ x 6, $(C_7\times C_{14}^2):(C_3\times S_4)$ x 6, $C_7^2.D_6:F_7.C_2^2$ x 3, $C_2^3\times C_7^3:C_6^2$ x 3, $C_{14}^2:(D_6\times F_7)$ x 3, $C_7^3:(C_2^5:C_3^2)$ x 3, $C_2^2\times C_7^2.D_{42}:C_6$ x 2, $C_2^2\times C_7^2.D_6:F_7$ x 2, $C_2\times C_7^3.C_6.C_{12}.C_2$ x 2, $C_2\times C_7^2.D_{42}:C_6.C_2$ x 2, $C_2\times C_7^2.D_6:F_7.C_2$ x 2, $C_2^2\times C_7^3.C_3^2.C_2^3$

Order 74088: $D_7^3:\He_3$ x 6, $C_7^3:C_3^2:S_4$ x 4, $C_7^3:C_3^2:S_4$ x 4, $C_7^3:(C_6^2:C_6)$ x 3, $C_7^3:C_3^2:S_4$ x 2, $C_7^3:C_3^2:S_4$ x 2, $C_7^3:(C_6^2:C_6)$ x 2, $C_7^3:C_6^2:S_3$, $C_7^3:(C_6^2:C_6)$

Order 65856: $C_7^3.C_2.C_6.C_2^4$ x 21, $D_7^3:S_4$ x 12, $C_7^3.C_6.D_4.C_2^2$ x 8, $C_7^3.C_6.C_2.C_2^4$ x 8, $C_2\times C_7^3.C_2.C_6.C_2^3$ x 6, $C_2^2\times C_7^3.C_6.C_2^3$ x 4, $C_2\times C_7^3.C_2^4:S_3$ x 4, $C_7^3.C_2^4:S_3.C_2$ x 3, $C_2^3:(C_7^3:S_4)$ x 3, $C_2^3:(C_7^3:S_4)$ x 3, $D_{14}\wr C_3$ x 3, $C_2^3\times C_7^3:(C_2^2\times C_6)$ x 2, $C_2\times C_7^3.C_6.C_2^3.C_2$ x 2, $C_7^3.D_6.C_2^4$, $C_7^3.C_6.C_2^5$, $C_7^3.C_6.C_2^4.C_2$, $C_7:D_7:C_3.C_{14}.C_2^4$

Order 56448: $C_7^2.C_6^2.C_2^5$

Order 49392: $C_7^3.C_6.C_6.C_2^2$ x 96, $C_2\times C_7^3.C_6^2.C_2$ x 80, $C_7^3.C_6^2.C_2^2$ x 55, $C_2\times C_7^3.C_6.C_6.C_2$ x 45, $C_2^2\times C_7^3:C_6^2$ x 36, $C_7^2.D_{42}:C_6.C_2$ x 25, $C_7^3:(C_6\times S_4)$ x 24, $C_7^2.D_6:F_7.C_2$ x 17, $C_7^3.C_6.C_{12}.C_2$ x 14, $C_2\times C_7^2.D_{42}:C_6$ x 14, $C_2\times C_7^2.D_6:F_7$ x 13, $C_2\times C_7^3.C_3^2.C_2^3$ x 12, $C_7^3:(C_6\times S_4)$ x 12, $C_2\times D_7^3:C_3^2$ x 9, $C_7^3:(C_6^2:C_2^2)$ x 8, $C_7^3:(C_6\times S_4)$ x 6, $C_7^3:(C_6\times S_4)$ x 6, $C_2^2\times C_7^3:(C_6\times S_3)$ x 4, $C_2^2\times C_7^3:(C_3\times C_3:C_4)$ x 4, $C_2^3\times C_7^3:(C_3\times C_6)$ x 3, $C_7^3:(C_2^4:C_3^2)$ x 3, $C_{14}^2:(C_6\times F_7)$ x 3, $C_7^3:(C_6\times S_4)$ x 3, $C_7^3:(C_6\times S_4)$ x 3, $C_7^3.C_3^2.C_2^4$ x 2

Order 43904: $C_7^3.C_2^2.C_2^5$

Order 37044: $C_7^3:C_3^2:D_6$ x 4, $C_7^3:C_3^2:A_4$ x 3, $C_7^3:C_3^2:D_6$, $C_7^3:C_3^2:D_6$, $C_7^3:(C_2^2\times \He_3)$, $(C_7\times C_{14}^2):\He_3$

Order 32928: $C_2^2\times C_7^3:(C_2^2\times C_6)$ x 2, $C_2^3\times C_7^3:(C_2\times C_6)$

Order 24696: $C_7^3.C_6^2.C_2$ x 5, $C_2^2\times C_7^3:(C_3\times C_6)$ x 2, $C_2^3\times C_7^3:C_3^2$, $C_2\times C_7^3:C_6^2$

Order 21952: $C_2^3\times C_7^3:C_2^3$

Order 16464: $C_7^3.C_6.C_2^3$, $C_2^3\times C_7^3:C_6$, $C_2^2\times C_7^3:(C_2\times C_6)$, $C_2\times C_7^3:(C_2^2\times C_6)$

Order 12348: $C_7^3:C_6^2$ x 2, $C_2^2\times C_7^3:C_3^2$, $C_7^3:C_6^2$

Order 10976: $C_2^2\times C_7^3:C_2^3$ x 2, $C_2^3\times C_7^3:C_2^2$

Order 8232: $D_7^3:C_3$ x 4, $C_2^2\times C_7^3:C_6$ x 2, $C_7^3:(C_2^2\times C_6)$, $C_2^3\times C_7^3:C_3$, $C_2\times C_7^3:(C_2\times C_6)$

Order 6174: $C_7^3:(C_3\times C_6)$ x 2, $C_2\times C_7^3:C_3^2$

Order 5488: $C_7^3:C_2^4$, $C_2^3\times C_7^3:C_2$, $C_2^2\times C_7^3:C_2^2$, $C_2\times C_7^3:C_2^3$

Order 4116: $C_7^3:(C_2\times C_6)$ x 2, $C_2^2\times C_7^3:C_3$, $C_2\times C_7^3:C_6$

Order 3087: $C_7^3:C_3^2$

Order 2744: $D_7^3$ x 4, $C_2^2\times C_7^3:C_2$ x 2, $C_{14}^3$, $C_7^3:C_2^3$, $C_2\times C_7^3:C_2^2$

Order 2058: $C_7^3:C_6$ x 2, $C_7^3:C_6$

Order 1372: $C_7:D_7^2$ x 2, $C_7\times C_{14}^2$, $C_7^2:D_{14}$

Order 1029: $C_7^3:C_3$

Order 686: $C_7^3:C_2$ x 2, $C_7^2\times C_{14}$

Order 343: $C_7^3$

Order 8: $C_2^3$

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 1185408: $D_7^3.C_6^2:D_6$ ($C_1$)

Order 592704: $D_7^3.C_3^2:S_4$ ($C_2$) x 4, $C_2\times C_7^3:(C_6^2:S_4)$ ($C_2$), $C_7^3:(C_2\times C_6^2:S_4)$ ($C_2$), $D_7^3.C_6^2:C_6$ ($C_2$)

Order 296352: $C_7^3:(C_6^2:S_4)$ ($C_2^2$) x 2, $C_7^3:(C_6^2:S_4)$ ($C_2^2$) x 2, $D_7^3:(C_3^2:A_4)$ ($C_2^2$) x 2, $C_7^3:(C_2^5:\He_3)$ ($C_2^2$)

Order 197568: $C_7^3:(C_2^4:C_6^2)$ ($S_3$) x 3, $C_2^3\times C_7^3.C_6^2.C_2$ ($S_3$)

Order 148176: $C_7^3:(C_6^2:A_4)$ ($C_2^3$)

Order 98784: $D_7^3:(C_3\times A_4)$ ($D_6$) x 6, $C_7^3:(C_2^5:C_3^2)$ ($D_6$) x 3, $C_2^2\times C_7^3.C_6^2.C_2$ ($D_6$) x 2, $C_2^3\times C_7^3:C_6^2$ ($D_6$)

Order 65856: $C_2^3\times C_7^3:(C_2^2\times C_6)$ ($C_3:S_3$)

Order 49392: $C_7^3:(C_2^4:C_3^2)$ ($C_2\times D_6$) x 3, $C_7^3.C_6^2.C_2^2$ ($S_4$), $C_2^3\times C_7^3:(C_3\times C_6)$ ($S_4$), $C_2^2\times C_7^3:C_6^2$ ($C_2\times D_6$), $C_2\times C_7^3.C_6^2.C_2$ ($S_4$)

Order 32928: $C_2^2\times C_7^3:(C_2^2\times C_6)$ ($C_6:S_3$) x 2, $C_2^3\times C_7^3:(C_2\times C_6)$ ($C_6:S_3$)

Order 24696: $C_7^3.C_6^2.C_2$ ($C_2\times S_4$) x 5, $C_2^2\times C_7^3:(C_3\times C_6)$ ($C_2\times S_4$) x 2, $C_2^3\times C_7^3:C_3^2$ ($C_2\times S_4$), $C_2\times C_7^3:C_6^2$ ($C_2\times S_4$)

Order 21952: $C_2^3\times C_7^3:C_2^3$ ($C_3^2:S_3$)

Order 16464: $C_7^3.C_6.C_2^3$ ($C_3:S_4$), $C_2^3\times C_7^3:C_6$ ($C_3:S_4$), $C_2^2\times C_7^3:(C_2\times C_6)$ ($C_6:D_6$), $C_2\times C_7^3:(C_2^2\times C_6)$ ($C_3:S_4$)

Order 12348: $C_7^3:C_6^2$ ($C_2^2\times S_4$) x 2, $C_2^2\times C_7^3:C_3^2$ ($C_2^2\times S_4$), $C_7^3:C_6^2$ ($C_2^2:S_4$)

Order 10976: $C_2^2\times C_7^3:C_2^3$ ($C_3^2:D_6$) x 2, $C_2^3\times C_7^3:C_2^2$ ($C_3^2:D_6$)

Order 8232: $D_7^3:C_3$ ($C_6:S_4$) x 4, $C_2^2\times C_7^3:C_6$ ($C_6:S_4$) x 2, $C_7^3:(C_2^2\times C_6)$ ($C_6:S_4$), $C_2^3\times C_7^3:C_3$ ($C_6:S_4$), $C_2\times C_7^3:(C_2\times C_6)$ ($C_6:S_4$)

Order 6174: $C_7^3:(C_3\times C_6)$ ($C_2^3:S_4$) x 2, $C_2\times C_7^3:C_3^2$ ($C_2^3:S_4$)

Order 5488: $C_7^3:C_2^4$ ($C_3^2:S_4$), $C_2^3\times C_7^3:C_2$ ($C_3^2:S_4$), $C_2^2\times C_7^3:C_2^2$ ($C_6^2:S_3$), $C_2\times C_7^3:C_2^3$ ($C_3^2:S_4$)

Order 4116: $C_7^3:(C_2\times C_6)$ ($C_2\times C_6:S_4$) x 2, $C_2^2\times C_7^3:C_3$ ($C_2\times C_6:S_4$), $C_2\times C_7^3:C_6$ ($(C_2\times C_6):S_4$)

Order 3087: $C_7^3:C_3^2$ ($C_2^2\wr S_3$)

Order 2744: $D_7^3$ ($C_6^2:D_6$) x 4, $C_2^2\times C_7^3:C_2$ ($C_6^2:D_6$) x 2, $C_{14}^3$ ($C_6^2:D_6$), $C_7^3:C_2^3$ ($C_6^2:D_6$), $C_2\times C_7^3:C_2^2$ ($C_6^2:D_6$)

Order 2058: $C_7^3:C_6$ ($(C_2^2\times C_6):S_4$) x 2, $C_7^3:C_6$ ($(C_2^2\times C_6):S_4$)

Order 1372: $C_7:D_7^2$ ($C_6^2:S_4$) x 2, $C_7\times C_{14}^2$ ($C_6^2:S_4$), $C_7^2:D_{14}$ ($C_6^2:S_4$)

Order 1029: $C_7^3:C_3$ ($(C_2^3\times C_6):S_4$)

Order 686: $C_7^3:C_2$ ($C_2\times C_6^2:S_4$) x 2, $C_7^2\times C_{14}$ ($C_2\times C_6^2:S_4$)

Order 343: $C_7^3$ ($C_2^2\times C_6^2:S_4$)

Order 8: $C_2^3$ ($D_7^3:C_3^2:S_3$)

Order 4: $C_2^2$ ($C_7^3:C_6^2:S_4$)

Order 2: $C_2$ ($D_7^3.C_3^2:S_4$)

Order 1: $C_1$ ($D_7^3.C_6^2:D_6$)

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 2 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $294 \times 294$ 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 $222 \times 222$ rational character table (warning: may be slow to load).