Abstract group 113764.b: $C_{3346}:C_{34}$

• Label: 113764.b
• Order: \( 2^{2} \cdot 7 \cdot 17 \cdot 239 \)
• Exponent: \( 2 \cdot 7 \cdot 17 \cdot 239 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 7 \cdot 17 \)
• $\card{Z(G)}$: \( 2 \cdot 7 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3 \cdot 7 \cdot 17 \cdot 239 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \cdot 7 \)
• Perm deg.: $248$
• Trans deg.: $3346$
• Rank: $2$

Group information

Description:	$C_{3346}:C_{34}$
Order:	\(113764\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \cdot 239 \)
Exponent:	\(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \)
Automorphism group:	$C_{239}:(C_2^2\times C_{714})$, of order \(682584\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 17 \cdot 239 \)
Composition factors:	$C_2$ x 2, $C_7$, $C_{17}$, $C_{239}$
Derived length:	$2$

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

Group statistics

Order	1	2	7	14	17	34	119	238	239	478	1673	3346
Elements	1	479	6	2874	3824	11472	22944	68832	238	238	1428	1428	113764
Conjugacy classes	1	3	6	18	16	48	96	288	7	7	42	42	574
Divisions	1	3	1	3	1	3	1	3	1	1	1	1	20
Autjugacy classes	1	2	1	2	16	32	16	32	1	1	1	1	106

Minimal presentations

Permutation degree:	$248$
Transitive degree:	$3346$
Rank:	$2$
Inequivalent generating pairs:	$6912$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	34	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{238}=b^{478}=1, b^{a}=b^{427} \rangle$
Permutation group:	Degree $248$ $\langle(1,2,4,14,48,115,37,114,206,239,180,179,121,148,52,147,55,15,53,149,170,66,84,82,93,54,134,44,12,43,131,167,218,221,113,74,81,152,150,89,34,109,216,223,199,90,198,196,166,112,36,10,35,78,187,127,41,126,220,169,192,189,162,177,210,209,95,28,94,204,211,96,164,103,47,13,46,137,219,111,119,226,229,207,124,143,50,142,64,18,63,165,168,151,122,92,201,159,171,203,176,105,39,120,227,128,104,32,102,107,33,9,31,67,172,108,215,205,193,141,230,222,237,174,68,173,73,21,5,19,65,71,20,69,62,156,58,16,57,155,153,214,228,232,133,49,139,191,175,130,42,129,231,123,145,51,144,140,154,56,79,188,97,30,8,29,70,178,87,197,236,136,45,135,200,235,161,186,138,181,213,163,61,17,60,160,83,24,6,23,77,185,183,234,132,233,195,182,76,22,75,146,212,99,101,80,190,184,202,91,27,7,26,88,117,72,98,110,106,208,225,118,38,116,224,125,158,59,157,217,238,194,86,25,85,100,40,11,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 26 & 0 \\ 0 & 83 \end{array}\right), \left(\begin{array}{rr} 238 & 0 \\ 0 & 238 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{239})$
Direct product:	$C_2$ $\, \times\, $ $C_7$ $\, \times\, $ $(C_{239}:C_{34})$
Semidirect product:	$D_{478}$ $\,\rtimes\,$ $C_{119}$	$D_{239}$ $\,\rtimes\,$ $C_{238}$ (2)	$C_{3346}$ $\,\rtimes\,$ $C_{34}$	$C_{478}$ $\,\rtimes\,$ $C_{238}$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{2} \times C_{238} \simeq C_{2}^{2} \times C_{7} \times C_{17}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 3848 subgroups in 40 conjugacy classes, 24 normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{14}$	$G/Z \simeq$ $C_{239}:C_{34}$
Commutator:	$G' \simeq$ $C_{239}$	$G/G' \simeq$ $C_2\times C_{238}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{3346}:C_{34}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{3346}$	$G/\operatorname{Fit} \simeq$ $C_{34}$
Radical:	$R \simeq$ $C_{3346}:C_{34}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{3346}$	$G/\operatorname{soc} \simeq$ $C_{34}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$
239-Sylow subgroup:	$P_{ 239 } \simeq$ $C_{239}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 113764: $C_{3346}:C_{34}$

Order 56882: $C_{1673}:C_{34}$ x 2, $C_{1673}:C_{34}$

Order 28441: $C_{1673}:C_{17}$

Order 16252: $C_{478}:C_{34}$

Order 8126: $C_{239}:C_{34}$ x 2, $C_{239}:C_{34}$

Order 6692: $C_7\times D_{478}$

Order 4063: $C_{239}:C_{17}$

Order 3346: $C_7\times D_{239}$ x 2, $C_{3346}$

Order 1673: $C_{1673}$

Order 956: $D_{478}$

Order 478: $D_{239}$ x 2, $C_{478}$

Order 476: $C_2\times C_{238}$

Order 239: $C_{239}$

Order 238: $C_{238}$ x 3

Order 119: $C_{119}$

Order 68: $C_2\times C_{34}$

Order 34: $C_{34}$ x 3

Order 28: $C_2\times C_{14}$

Order 17: $C_{17}$

Order 14: $C_{14}$ x 3

Order 7: $C_7$

Order 4: $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 113764: $C_{3346}:C_{34}$

Order 56882: $C_{1673}:C_{34}$, $C_{1673}:C_{34}$

Order 28441: $C_{1673}:C_{17}$

Order 16252: $C_{478}:C_{34}$

Order 8126: $C_{239}:C_{34}$, $C_{239}:C_{34}$

Order 6692: $C_7\times D_{478}$

Order 4063: $C_{239}:C_{17}$

Order 3346: $C_{3346}$, $C_7\times D_{239}$

Order 1673: $C_{1673}$

Order 956: $D_{478}$

Order 478: $C_{478}$, $D_{239}$

Order 476: $C_2\times C_{238}$

Order 239: $C_{239}$

Order 238: $C_{238}$ x 2

Order 119: $C_{119}$

Order 68: $C_2\times C_{34}$

Order 34: $C_{34}$ x 2

Order 28: $C_2\times C_{14}$

Order 17: $C_{17}$

Order 14: $C_{14}$ x 2

Order 7: $C_7$

Order 4: $C_2^2$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 113764: $C_{3346}:C_{34}$ ($C_1$)

Order 56882: $C_{1673}:C_{34}$ ($C_2$) x 2, $C_{1673}:C_{34}$ ($C_2$)

Order 28441: $C_{1673}:C_{17}$ ($C_2^2$)

Order 16252: $C_{478}:C_{34}$ ($C_7$)

Order 8126: $C_{239}:C_{34}$ ($C_{14}$) x 2, $C_{239}:C_{34}$ ($C_{14}$)

Order 6692: $C_7\times D_{478}$ ($C_{17}$)

Order 4063: $C_{239}:C_{17}$ ($C_2\times C_{14}$)

Order 3346: $C_7\times D_{239}$ ($C_{34}$) x 2, $C_{3346}$ ($C_{34}$)

Order 1673: $C_{1673}$ ($C_2\times C_{34}$)

Order 956: $D_{478}$ ($C_{119}$)

Order 478: $D_{239}$ ($C_{238}$) x 2, $C_{478}$ ($C_{238}$)

Order 239: $C_{239}$ ($C_2\times C_{238}$)

Order 14: $C_{14}$ ($C_{239}:C_{34}$)

Order 7: $C_7$ ($C_{478}:C_{34}$)

Order 2: $C_2$ ($C_{1673}:C_{34}$)

Order 1: $C_1$ ($C_{3346}:C_{34}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 113764: $C_{3346}:C_{34}$ ($C_1$)

Order 56882: $C_{1673}:C_{34}$ ($C_2$), $C_{1673}:C_{34}$ ($C_2$)

Order 28441: $C_{1673}:C_{17}$ ($C_2^2$)

Order 16252: $C_{478}:C_{34}$ ($C_7$)

Order 8126: $C_{239}:C_{34}$ ($C_{14}$), $C_{239}:C_{34}$ ($C_{14}$)

Order 6692: $C_7\times D_{478}$ ($C_{17}$)

Order 4063: $C_{239}:C_{17}$ ($C_2\times C_{14}$)

Order 3346: $C_{3346}$ ($C_{34}$), $C_7\times D_{239}$ ($C_{34}$)

Order 1673: $C_{1673}$ ($C_2\times C_{34}$)

Order 956: $D_{478}$ ($C_{119}$)

Order 478: $C_{478}$ ($C_{238}$), $D_{239}$ ($C_{238}$)

Order 239: $C_{239}$ ($C_2\times C_{238}$)

Order 14: $C_{14}$ ($C_{239}:C_{34}$)

Order 7: $C_7$ ($C_{478}:C_{34}$)

Order 2: $C_2$ ($C_{1673}:C_{34}$)

Order 1: $C_1$ ($C_{3346}:C_{34}$)

Series

Derived series

$C_{3346}:C_{34}$

$\rhd$

$C_{239}$

$\rhd$

$C_1$

Chief series

$C_{3346}:C_{34}$

$\rhd$

$C_{1673}:C_{34}$

$\rhd$

$C_{239}:C_{34}$

$\rhd$

$C_{478}$

$\rhd$

$C_{239}$

$\rhd$

$C_1$

Lower central series

$C_{3346}:C_{34}$

$\rhd$

$C_{239}$

Upper central series

$C_1$

$\lhd$

$C_{14}$

Supergroups

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

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

Character theory

Complex character table

The $574 \times 574$ character table is not available for this group.

Rational character table

The $20 \times 20$ rational character table is not available for this group.