Abstract group 52212.c: $F_{229}$

• Label: 52212.c
• Order: \( 2^{2} \cdot 3 \cdot 19 \cdot 229 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 19 \cdot 229 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \cdot 19 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3 \cdot 19 \cdot 229 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $229$
• Trans deg.: $229$
• Rank: $2$

Group information

Description:	$F_{229}$
Order:	\(52212\)\(\medspace = 2^{2} \cdot 3 \cdot 19 \cdot 229 \)
Exponent:	\(52212\)\(\medspace = 2^{2} \cdot 3 \cdot 19 \cdot 229 \)
Automorphism group:	$F_{229}$, of order \(52212\)\(\medspace = 2^{2} \cdot 3 \cdot 19 \cdot 229 \)
Composition factors:	$C_2$ x 2, $C_3$, $C_{19}$, $C_{229}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12	19	38	57	76	114	228	229
Elements	1	229	458	458	458	916	4122	4122	8244	8244	8244	16488	228	52212
Conjugacy classes	1	1	2	2	2	4	18	18	36	36	36	72	1	229
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	13
Autjugacy classes	1	1	2	2	2	4	18	18	36	36	36	72	1	229

Dimension	1	2	4	18	36	72	228
Irr. complex chars.	228	0	0	0	0	0	1	229
Irr. rational chars.	2	3	1	2	3	1	1	13

Minimal presentations

Permutation degree:	$229$
Transitive degree:	$229$
Rank:	$2$
Inequivalent generating pairs:	$34560$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	228	228	228
Arbitrary	228	228	228

Constructions

Groups of Lie type:	$\AGL(1,229)$
Presentation:	$\langle a, b \mid a^{228}=b^{229}=1, b^{a}=b^{142} \rangle$
Permutation group:	Degree $229$ $\langle(1,2,4,10,28,15,5,13,37,83,76,33,11,31,71,141,137,68,29,67,136,173,162,86,66,134,116,185,168,93,41,91,42,94,89,40,14,26,9,25,60,87,38,85,90,163,205,161,84,107,99,120,189,124,159,179,171,188,217,160,149,209,183,211,150,148,75,146,101,46,18,45,32,73,121,58,24,57,72,143,206,219,165,125,142,177,174,103,47,59,123,104,176,224,214,216,200,225,226,229,198,151,78,130,195,204,154,98,44,97,170,220,187,119,56,30,69,138,201,194,127,61,126,190,221,172,102,122,192,175,222,210,218,227,202,228,223,199,133,81,36,65,132,197,135,164,96,155,212,213,152,79,35,12,34,77,117,54,64,27,63,129,184,115,166,191,208,147,207,182,181,203,139,193,167,157,82,153,80,131,92,112,158,215,169,100,111,51,95,43,17,6,16,20,50,109,128,62,108,49,106,178,145,74,144,105,110,180,196,140,70,55,118,186,156,88,39,22,53,114,113,52,21,7,19,48,23,8,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 43 & 0 \\ 0 & 96 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{229})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{229}$ $\,\rtimes\,$ $C_{228}$	$(C_{229}:C_{76})$ $\,\rtimes\,$ $C_3$	$(C_{229}:C_{57})$ $\,\rtimes\,$ $C_4$	$(C_{229}:C_4)$ $\,\rtimes\,$ $C_{57}$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{229}$ . $C_{114}$	$(C_{229}:C_{38})$ . $C_6$	$(C_{229}:C_6)$ . $C_{38}$	$(C_{229}:C_{114})$ . $C_2$	more information
Aut. group:	$\Aut(D_{229})$	$\Aut(C_{229}:C_3)$	$\Aut(C_{229}:C_4)$	$\Aut(C_{229}:C_6)$	all 13

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

Homology

Abelianization:	$C_{228} \simeq C_{4} \times C_{3} \times C_{19}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 2532 subgroups in 24 conjugacy classes, 13 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $F_{229}$
Commutator:	$G' \simeq$ $C_{229}$	$G/G' \simeq$ $C_{228}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $F_{229}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{229}$	$G/\operatorname{Fit} \simeq$ $C_{228}$
Radical:	$R \simeq$ $F_{229}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{229}$	$G/\operatorname{soc} \simeq$ $C_{228}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $C_{19}$
229-Sylow subgroup:	$P_{ 229 } \simeq$ $C_{229}$

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 52212: $F_{229}$

Order 26106: $C_{229}:C_{114}$

Order 17404: $C_{229}:C_{76}$

Order 13053: $C_{229}:C_{57}$

Order 8702: $C_{229}:C_{38}$

Order 4351: $C_{229}:C_{19}$

Order 2748: $C_{229}:C_{12}$

Order 1374: $C_{229}:C_6$

Order 916: $C_{229}:C_4$

Order 687: $C_{229}:C_3$

Order 458: $D_{229}$

Order 229: $C_{229}$

Order 228: $C_{228}$

Order 114: $C_{114}$

Order 76: $C_{76}$

Order 57: $C_{57}$

Order 38: $C_{38}$

Order 19: $C_{19}$

Order 12: $C_{12}$

Order 6: $C_6$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 52212: $F_{229}$

Order 26106: $C_{229}:C_{114}$

Order 17404: $C_{229}:C_{76}$

Order 13053: $C_{229}:C_{57}$

Order 8702: $C_{229}:C_{38}$

Order 4351: $C_{229}:C_{19}$

Order 2748: $C_{229}:C_{12}$

Order 1374: $C_{229}:C_6$

Order 916: $C_{229}:C_4$

Order 687: $C_{229}:C_3$

Order 458: $D_{229}$

Order 229: $C_{229}$

Order 228: $C_{228}$

Order 114: $C_{114}$

Order 76: $C_{76}$

Order 57: $C_{57}$

Order 38: $C_{38}$

Order 19: $C_{19}$

Order 12: $C_{12}$

Order 6: $C_6$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 52212: $F_{229}$ ($C_1$)

Order 26106: $C_{229}:C_{114}$ ($C_2$)

Order 17404: $C_{229}:C_{76}$ ($C_3$)

Order 13053: $C_{229}:C_{57}$ ($C_4$)

Order 8702: $C_{229}:C_{38}$ ($C_6$)

Order 4351: $C_{229}:C_{19}$ ($C_{12}$)

Order 2748: $C_{229}:C_{12}$ ($C_{19}$)

Order 1374: $C_{229}:C_6$ ($C_{38}$)

Order 916: $C_{229}:C_4$ ($C_{57}$)

Order 687: $C_{229}:C_3$ ($C_{76}$)

Order 458: $D_{229}$ ($C_{114}$)

Order 229: $C_{229}$ ($C_{228}$)

Order 1: $C_1$ ($F_{229}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 52212: $F_{229}$ ($C_1$)

Order 26106: $C_{229}:C_{114}$ ($C_2$)

Order 17404: $C_{229}:C_{76}$ ($C_3$)

Order 13053: $C_{229}:C_{57}$ ($C_4$)

Order 8702: $C_{229}:C_{38}$ ($C_6$)

Order 4351: $C_{229}:C_{19}$ ($C_{12}$)

Order 2748: $C_{229}:C_{12}$ ($C_{19}$)

Order 1374: $C_{229}:C_6$ ($C_{38}$)

Order 916: $C_{229}:C_4$ ($C_{57}$)

Order 687: $C_{229}:C_3$ ($C_{76}$)

Order 458: $D_{229}$ ($C_{114}$)

Order 229: $C_{229}$ ($C_{228}$)

Order 1: $C_1$ ($F_{229}$)

Series

Derived series

$F_{229}$

$\rhd$

$C_{229}$

$\rhd$

$C_1$

Chief series

$F_{229}$

$\rhd$

$C_{229}:C_{114}$

$\rhd$

$C_{229}:C_{57}$

$\rhd$

$C_{229}:C_{19}$

$\rhd$

$C_{229}$

$\rhd$

$C_1$

Lower central series

$F_{229}$

$\rhd$

$C_{229}$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 3 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 $229 \times 229$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	4A	6A	12A	19A	38A	57A	76A	114A	228A	229A
	Size	1	229	458	458	458	916	4122	4122	8244	8244	8244	16488	228
	2 P	1A	1A	3A	2A	3A	6A	19A	19A	57A	38A	57A	114A	229A
	3 P	1A	2A	1A	4A	2A	4A	19A	38A	19A	76A	38A	76A	229A
	19 P	1A	2A	3A	4A	6A	12A	1A	2A	3A	4A	6A	12A	229A
	229 P	1A	2A	3A	4A	6A	12A	19A	38A	57A	76A	114A	228A	1A
52212.c.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
52212.c.1b		$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$
52212.c.1c		$2$	$2$	$-1$	$2$	$-1$	$-1$	$2$	$2$	$-1$	$2$	$-1$	$-1$	$2$
52212.c.1d		$2$	$-2$	$2$	$0$	$-2$	$0$	$2$	$-2$	$2$	$0$	$-2$	$0$	$2$
52212.c.1e		$2$	$2$	$-1$	$-2$	$-1$	$1$	$2$	$2$	$-1$	$-2$	$-1$	$1$	$2$
52212.c.1f		$4$	$-4$	$-2$	$0$	$2$	$0$	$4$	$-4$	$-2$	$0$	$2$	$0$	$4$
52212.c.1g		$18$	$18$	$18$	$18$	$18$	$18$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$18$
52212.c.1h		$18$	$18$	$18$	$-18$	$18$	$-18$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$18$
52212.c.1i		$36$	$36$	$-18$	$36$	$-18$	$-18$	$-2$	$-2$	$1$	$-2$	$1$	$1$	$36$
52212.c.1j		$36$	$-36$	$36$	$0$	$-36$	$0$	$-2$	$2$	$-2$	$0$	$2$	$0$	$36$
52212.c.1k		$36$	$36$	$-18$	$-36$	$-18$	$18$	$-2$	$-2$	$1$	$2$	$1$	$-1$	$36$
52212.c.1l		$72$	$-72$	$-36$	$0$	$36$	$0$	$-4$	$4$	$2$	$0$	$-2$	$0$	$72$
52212.c.228a		$228$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$