Abstract group 85752.b: $C_{2382}:C_{36}$

• Label: 85752.b
• Order: \( 2^{3} \cdot 3^{3} \cdot 397 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 397 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{3} \cdot 11 \cdot 397 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \cdot 11 \)
• Perm deg.: $402$
• Trans deg.: $2382$
• Rank: $2$

Group information

Description:	$C_{2382}:C_{36}$
Order:	\(85752\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 397 \)
Exponent:	\(14292\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 397 \)
Automorphism group:	$C_{1191}.C_{198}.C_2^3$, of order \(1886544\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 11 \cdot 397 \)
Composition factors:	$C_2$ x 3, $C_3$ x 3, $C_{397}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.

Group statistics

Order	1	2	3	4	6	9	12	18	36	397	794	1191	2382
Elements	1	795	2384	1588	8736	7146	12704	21438	28584	396	396	792	792	85752
Conjugacy classes	1	3	8	4	24	18	32	54	72	11	11	22	22	282
Divisions	1	3	4	2	12	3	8	9	6	1	1	1	1	52
Autjugacy classes	1	3	5	2	15	6	10	18	12	1	1	1	1	76

Dimension	1	2	4	6	12	36	396	792
Irr. complex chars.	216	0	0	0	0	66	0	0	282
Irr. rational chars.	4	18	8	12	6	0	2	2	52

Minimal presentations

Permutation degree:	$402$
Transitive degree:	$2382$
Rank:	$2$
Inequivalent generating pairs:	$864$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	36	72	792
Arbitrary	36	38	398

Constructions

Presentation:	$\langle a, b \mid a^{36}=b^{2382}=1, b^{a}=b^{1741} \rangle$
Permutation group:	Degree $402$ $\langle(1,2,4,14,54,45,144,152,232,84,194,353,225,327,157,199,325,155,324,259,220,282,385,308,130,305,290,358,269,191,348,313,167,121,36,30,104,90,247,226,329,160,143,44,140,276,132,277,126,300,379,241,381,394,374,287,388,397,390,279,253,116,175,105,185,271,314,135,141,207,69,206,363,320,147,319,258,115,33,8,32,110,68,17,67,79,193,86,23,85,233,142,134,41,133,159,50,158,230,372,367,376,245,89,243,182,246,256,94,183,169,53,168,260,218,257,292,250,178,252,215,323,154,322,237,362,205,255,286,198,357,285,164,176,55,173,341,334,392,373,340,299,212,74,112,281,248,360,200,359,371,229,82,21,80,153,48,12,18,71,148,262,311,377,389,339,172,76,217,350,188,328,221,365,209,72,22,5,19,73,210,213,352,296,123,179,83,202,347,263,163,56,102,96,27,26,6,24,87,137,42,10,40,43,139,70,146,117,288,356,197,64,196,81,228,337,264,302,127,38,9,35,103,270,387,295,274,204,361,303,128,195,354,370,383,249,184,58,101,29,7,28,97,124,37,122,211,369,366,301,391,284,332,166,125,186,227,355,342,378,283,336,224,145,177,111,278,386,261,330,234,187,349,304,266,99,265,239,306,318,291,219,235,293,203,119,34,118,289,131,309,280,317,315,136,174,344,393,380,238,108,31,107,92,25,91,61,190,114,39,129,170,120,150,47,149,165,52,13,51,161,310,181,57,180,345,321,151,189,60,88,240,326,223,78,222,162,307,244,382,396,395,331,343,368,346,312,351,216,75,214,95,192,62,15,59,109,275,298,316,138,156,49,20,77,113,242,375,231,208,364,335,254,93,251,384,333,294,236,268,100,267,273,106,272,297,338,171,66,201,98,65,16,63,46,11,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 177 \end{array}\right), \left(\begin{array}{rr} 85 & 0 \\ 0 & 383 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{397})$
Direct product:	$C_2$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_{397}:C_{36})$
Semidirect product:	$C_{2382}$ $\,\rtimes\,$ $C_{36}$	$(C_3\times D_{397})$ $\,\rtimes\,$ $C_{36}$	$D_{397}$ $\,\rtimes\,$ $(C_3\times C_{36})$	$(C_{2382}:C_4)$ $\,\rtimes\,$ $C_9$	all 21
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3\times D_{794})$ . $C_{18}$	$D_{794}$ . $(C_3\times C_{18})$	$D_{397}$ . $(C_6\times C_{18})$	$(C_{2382}:C_6)$ . $C_6$	all 24

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

Homology

Abelianization:	$C_{6} \times C_{36} \simeq C_{2} \times C_{4} \times C_{3} \times C_{9}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 30256 subgroups in 160 conjugacy classes, 84 normal (34 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_{397}:C_{36}$
Commutator:	$G' \simeq$ $C_{397}$	$G/G' \simeq$ $C_6\times C_{36}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{2382}:C_{36}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{2382}$	$G/\operatorname{Fit} \simeq$ $C_{36}$
Radical:	$R \simeq$ $C_{2382}:C_{36}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{2382}$	$G/\operatorname{soc} \simeq$ $C_{36}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$
397-Sylow subgroup:	$P_{ 397 } \simeq$ $C_{397}$

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

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Classes of subgroups up to conjugation

Order 85752: $C_{2382}:C_{36}$

Order 42876: $C_{1191}:C_{36}$ x 2, $C_{2382}:C_{18}$

Order 28584: $C_{794}:C_{36}$ x 3, $C_{2382}:C_{12}$

Order 21438: $C_{1191}:C_{18}$ x 2, $C_{1191}:C_{18}$

Order 14292: $C_{397}:C_{36}$ x 6, $C_{794}:C_{18}$ x 3, $C_{1191}:C_{12}$ x 2, $C_{2382}:C_6$

Order 10719: $C_{1191}:C_9$

Order 9528: $C_{794}:C_{12}$ x 3, $C_{2382}:C_4$

Order 7146: $C_{397}:C_{18}$ x 6, $C_{397}:C_{18}$ x 3, $C_{1191}:C_6$ x 2, $C_{1191}:C_6$

Order 4764: $C_{397}:C_{12}$ x 6, $C_{794}:C_6$ x 3, $C_{1191}:C_4$ x 2, $C_3\times D_{794}$

Order 3573: $C_{397}:C_9$ x 3, $C_{1191}:C_3$

Order 3176: $C_{794}:C_4$

Order 2382: $C_{397}:C_6$ x 6, $C_{397}:C_6$ x 3, $C_3\times D_{397}$ x 2, $C_{2382}$

Order 1588: $C_{397}:C_4$ x 2, $D_{794}$

Order 1191: $C_{397}:C_3$ x 3, $C_{1191}$

Order 794: $D_{397}$ x 2, $C_{794}$

Order 397: $C_{397}$

Order 216: $C_6\times C_{36}$

Order 108: $C_3\times C_{36}$ x 2, $C_6\times C_{18}$

Order 72: $C_2\times C_{36}$ x 3, $C_6\times C_{12}$

Order 54: $C_3\times C_{18}$ x 3

Order 36: $C_{36}$ x 6, $C_2\times C_{18}$ x 3, $C_3\times C_{12}$ x 2, $C_6^2$

Order 27: $C_3\times C_9$

Order 24: $C_2\times C_{12}$ x 4

Order 18: $C_{18}$ x 9, $C_3\times C_6$ x 3

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 4

Order 9: $C_9$ x 3, $C_3^2$

Order 8: $C_2\times C_4$

Order 6: $C_6$ x 12

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 85752: $C_{2382}:C_{36}$

Order 42876: $C_{2382}:C_{18}$, $C_{1191}:C_{36}$

Order 28584: $C_{2382}:C_{12}$, $C_{794}:C_{36}$

Order 21438: $C_{1191}:C_{18}$ x 2, $C_{1191}:C_{18}$

Order 14292: $C_{2382}:C_6$, $C_{1191}:C_{12}$, $C_{397}:C_{36}$, $C_{794}:C_{18}$

Order 10719: $C_{1191}:C_9$

Order 9528: $C_{794}:C_{12}$ x 2, $C_{2382}:C_4$

Order 7146: $C_{1191}:C_6$ x 2, $C_{397}:C_{18}$ x 2, $C_{1191}:C_6$, $C_{397}:C_{18}$

Order 4764: $C_{794}:C_6$ x 2, $C_{397}:C_{12}$ x 2, $C_3\times D_{794}$, $C_{1191}:C_4$

Order 3573: $C_{1191}:C_3$, $C_{397}:C_9$

Order 3176: $C_{794}:C_4$

Order 2382: $C_{397}:C_6$ x 4, $C_3\times D_{397}$ x 2, $C_{397}:C_6$ x 2, $C_{2382}$

Order 1588: $D_{794}$, $C_{397}:C_4$

Order 1191: $C_{397}:C_3$ x 2, $C_{1191}$

Order 794: $D_{397}$ x 2, $C_{794}$

Order 397: $C_{397}$

Order 216: $C_6\times C_{36}$

Order 108: $C_6\times C_{18}$, $C_3\times C_{36}$

Order 72: $C_2\times C_{36}$, $C_6\times C_{12}$

Order 54: $C_3\times C_{18}$ x 3

Order 36: $C_3\times C_{12}$, $C_2\times C_{18}$, $C_{36}$, $C_6^2$

Order 27: $C_3\times C_9$

Order 24: $C_2\times C_{12}$ x 3

Order 18: $C_3\times C_6$ x 3, $C_{18}$ x 3

Order 12: $C_2\times C_6$ x 3, $C_{12}$ x 3

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

Order 8: $C_2\times C_4$

Order 6: $C_6$ x 9

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 85752: $C_{2382}:C_{36}$ ($C_1$)

Order 42876: $C_{1191}:C_{36}$ ($C_2$) x 2, $C_{2382}:C_{18}$ ($C_2$)

Order 28584: $C_{794}:C_{36}$ ($C_3$) x 3, $C_{2382}:C_{12}$ ($C_3$)

Order 21438: $C_{1191}:C_{18}$ ($C_2^2$), $C_{1191}:C_{18}$ ($C_4$), $C_{1191}:C_{18}$ ($C_4$)

Order 14292: $C_{397}:C_{36}$ ($C_6$) x 6, $C_{794}:C_{18}$ ($C_6$) x 3, $C_{1191}:C_{12}$ ($C_6$) x 2, $C_{2382}:C_6$ ($C_6$)

Order 10719: $C_{1191}:C_9$ ($C_2\times C_4$)

Order 9528: $C_{794}:C_{12}$ ($C_9$) x 2, $C_{2382}:C_4$ ($C_9$), $C_{794}:C_{12}$ ($C_3^2$)

Order 7146: $C_{397}:C_{18}$ ($C_{12}$) x 3, $C_{397}:C_{18}$ ($C_2\times C_6$) x 3, $C_{397}:C_{18}$ ($C_{12}$) x 3, $C_{1191}:C_6$ ($C_{12}$), $C_{1191}:C_6$ ($C_2\times C_6$), $C_{1191}:C_6$ ($C_{12}$)

Order 4764: $C_{397}:C_{12}$ ($C_{18}$) x 4, $C_{794}:C_6$ ($C_{18}$) x 2, $C_{1191}:C_4$ ($C_{18}$) x 2, $C_{397}:C_{12}$ ($C_3\times C_6$) x 2, $C_3\times D_{794}$ ($C_{18}$), $C_{794}:C_6$ ($C_3\times C_6$)

Order 3573: $C_{397}:C_9$ ($C_2\times C_{12}$) x 3, $C_{1191}:C_3$ ($C_2\times C_{12}$)

Order 3176: $C_{794}:C_4$ ($C_3\times C_9$)

Order 2382: $C_{397}:C_6$ ($C_{36}$) x 2, $C_{397}:C_6$ ($C_2\times C_{18}$) x 2, $C_{397}:C_6$ ($C_{36}$) x 2, $C_{2382}$ ($C_{36}$), $C_3\times D_{397}$ ($C_2\times C_{18}$), $C_3\times D_{397}$ ($C_{36}$), $C_{397}:C_6$ ($C_3\times C_{12}$), $C_{397}:C_6$ ($C_3\times C_{12}$), $C_{397}:C_6$ ($C_6^2$)

Order 1588: $C_{397}:C_4$ ($C_3\times C_{18}$) x 2, $D_{794}$ ($C_3\times C_{18}$)

Order 1191: $C_{397}:C_3$ ($C_2\times C_{36}$) x 2, $C_{1191}$ ($C_2\times C_{36}$), $C_{397}:C_3$ ($C_6\times C_{12}$)

Order 794: $C_{794}$ ($C_3\times C_{36}$), $D_{397}$ ($C_6\times C_{18}$), $D_{397}$ ($C_3\times C_{36}$)

Order 397: $C_{397}$ ($C_6\times C_{36}$)

Order 6: $C_6$ ($C_{397}:C_{36}$)

Order 3: $C_3$ ($C_{794}:C_{36}$)

Order 2: $C_2$ ($C_{1191}:C_{36}$)

Order 1: $C_1$ ($C_{2382}:C_{36}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 85752: $C_{2382}:C_{36}$ ($C_1$)

Order 42876: $C_{2382}:C_{18}$ ($C_2$), $C_{1191}:C_{36}$ ($C_2$)

Order 28584: $C_{2382}:C_{12}$ ($C_3$), $C_{794}:C_{36}$ ($C_3$)

Order 21438: $C_{1191}:C_{18}$ ($C_2^2$), $C_{1191}:C_{18}$ ($C_4$), $C_{1191}:C_{18}$ ($C_4$)

Order 14292: $C_{2382}:C_6$ ($C_6$), $C_{1191}:C_{12}$ ($C_6$), $C_{397}:C_{36}$ ($C_6$), $C_{794}:C_{18}$ ($C_6$)

Order 10719: $C_{1191}:C_9$ ($C_2\times C_4$)

Order 9528: $C_{2382}:C_4$ ($C_9$), $C_{794}:C_{12}$ ($C_3^2$), $C_{794}:C_{12}$ ($C_9$)

Order 7146: $C_{1191}:C_6$ ($C_{12}$), $C_{1191}:C_6$ ($C_2\times C_6$), $C_{1191}:C_6$ ($C_{12}$), $C_{397}:C_{18}$ ($C_{12}$), $C_{397}:C_{18}$ ($C_2\times C_6$), $C_{397}:C_{18}$ ($C_{12}$)

Order 4764: $C_3\times D_{794}$ ($C_{18}$), $C_{794}:C_6$ ($C_3\times C_6$), $C_{794}:C_6$ ($C_{18}$), $C_{1191}:C_4$ ($C_{18}$), $C_{397}:C_{12}$ ($C_3\times C_6$), $C_{397}:C_{12}$ ($C_{18}$)

Order 3573: $C_{1191}:C_3$ ($C_2\times C_{12}$), $C_{397}:C_9$ ($C_2\times C_{12}$)

Order 3176: $C_{794}:C_4$ ($C_3\times C_9$)

Order 2382: $C_{2382}$ ($C_{36}$), $C_3\times D_{397}$ ($C_2\times C_{18}$), $C_3\times D_{397}$ ($C_{36}$), $C_{397}:C_6$ ($C_3\times C_{12}$), $C_{397}:C_6$ ($C_{36}$), $C_{397}:C_6$ ($C_3\times C_{12}$), $C_{397}:C_6$ ($C_2\times C_{18}$), $C_{397}:C_6$ ($C_{36}$), $C_{397}:C_6$ ($C_6^2$)

Order 1588: $D_{794}$ ($C_3\times C_{18}$), $C_{397}:C_4$ ($C_3\times C_{18}$)

Order 1191: $C_{1191}$ ($C_2\times C_{36}$), $C_{397}:C_3$ ($C_2\times C_{36}$), $C_{397}:C_3$ ($C_6\times C_{12}$)

Order 794: $C_{794}$ ($C_3\times C_{36}$), $D_{397}$ ($C_6\times C_{18}$), $D_{397}$ ($C_3\times C_{36}$)

Order 397: $C_{397}$ ($C_6\times C_{36}$)

Order 6: $C_6$ ($C_{397}:C_{36}$)

Order 3: $C_3$ ($C_{794}:C_{36}$)

Order 2: $C_2$ ($C_{1191}:C_{36}$)

Order 1: $C_1$ ($C_{2382}:C_{36}$)

Series

Derived series

$C_{2382}:C_{36}$

$\rhd$

$C_{397}$

$\rhd$

$C_1$

Chief series

$C_{2382}:C_{36}$

$\rhd$

$C_{2382}:C_{18}$

$\rhd$

$C_{1191}:C_{18}$

$\rhd$

$C_{1191}:C_6$

$\rhd$

$C_{2382}$

$\rhd$

$C_{1191}$

$\rhd$

$C_{397}$

$\rhd$

$C_1$

Lower central series

$C_{2382}:C_{36}$

$\rhd$

$C_{397}$

Upper central series

$C_1$

$\lhd$

$C_6$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $282 \times 282$ 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 $52 \times 52$ rational character table.