Abstract group 74112.a: $C_2\times F_{193}$

• Label: 74112.a
• Order: \( 2^{7} \cdot 3 \cdot 193 \)
• Exponent: \( 2^{6} \cdot 3 \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $195$
• Trans deg.: $386$
• Rank: $2$

Group information

Description:	$C_2\times F_{193}$
Order:	\(74112\)\(\medspace = 2^{7} \cdot 3 \cdot 193 \)
Exponent:	\(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Automorphism group:	$C_2\times F_{193}$, of order \(74112\)\(\medspace = 2^{7} \cdot 3 \cdot 193 \)
Composition factors:	$C_2$ x 7, $C_3$, $C_{193}$
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	8	12	16	24	32	48	64	96	192	193	386
Elements	1	387	386	772	1158	1544	1544	3088	3088	6176	6176	12352	12352	24704	192	192	74112
Conjugacy classes	1	3	2	4	6	8	8	16	16	32	32	64	64	128	1	1	386
Divisions	1	3	1	2	3	2	2	2	2	2	2	2	2	2	1	1	30
Autjugacy classes	1	3	2	4	6	8	8	16	16	32	32	32	64	64	1	1	290

Dimension	1	2	4	8	16	32	64	192
Irr. complex chars.	384	0	0	0	0	0	0	2	386
Irr. rational chars.	4	6	4	4	4	4	2	2	30

Minimal presentations

Permutation degree:	$195$
Transitive degree:	$386$
Rank:	$2$
Inequivalent generating pairs:	$24576$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	192	192	192
Arbitrary	192	192	192

Constructions

Presentation:	$\langle a, b \mid a^{192}=b^{386}=1, b^{a}=b^{231} \rangle$
Permutation group:	Degree $195$ $\langle(2,4,18,82,119,187,154,147,161,175,181,171,169,113,35,6,20,83,117,30,106,49,56,101,66,153,193,139,92,126,33,91,99,29,5,25,10,24,87,166,138,191,148,78,176,69,162,190,110,27,105,37,93,100,151,120,178,124,36,67,13,55,158,149,112,165,80,70,22,97,31,107,51,103,102,61,41,136,46,38,72,16,57,98,26,108,155,133,116,188,157,137,125,32,71,14,3,11,52,109,156,189,144,90,89,134,111,132,123,167,62,12,54,118,150,50,143,43,19,88,39,95,184,172,152,164,60,146,159,75,15,73,17,58,121,128,174,192,140,47,135,42,96,185,180,74,168,64,94,160,183,127,141,145,59,40,7,21,84,122,177,130,48,44,53,114,77,179,81,163,129,34,65,115,79,68,45,9,23,86,76,182,142,104,131,186,85,173,170,63,28,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 76 & 0 \\ 0 & 28 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 192 & 0 \\ 0 & 192 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_2$ $\, \times\, $ $F_{193}$
Semidirect product:	$D_{193}$ $\,\rtimes\,$ $C_{192}$	$C_{386}$ $\,\rtimes\,$ $C_{192}$	$(C_{386}:C_{64})$ $\,\rtimes\,$ $C_3$	$(C_{193}:C_{64})$ $\,\rtimes\,$ $C_6$ (2)	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{386}$ . $C_{96}$	$D_{193}$ . $(C_2\times C_{96})$	$(C_{386}:C_{96})$ . $C_2$	$(C_{386}:C_{48})$ . $C_4$	all 28
Aut. group:	$\Aut(C_{193}:C_4)$	$\Aut(D_{386})$	$\Aut(C_3\times D_{193})$	$\Aut(C_{193}:C_8)$	all 28

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

Homology

Abelianization:	$C_{2} \times C_{192} \simeq C_{2} \times C_{64} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 7376 subgroups in 80 conjugacy classes, 42 normal (38 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $F_{193}$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_2\times C_{192}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2\times F_{193}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{386}$	$G/\operatorname{Fit} \simeq$ $C_{192}$
Radical:	$R \simeq$ $C_2\times F_{193}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{386}$	$G/\operatorname{soc} \simeq$ $C_{192}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_{64}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
193-Sylow subgroup:	$P_{ 193 } \simeq$ $C_{193}$

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 74112: $C_2\times F_{193}$

Order 37056: $F_{193}$ x 2, $C_{386}:C_{96}$

Order 24704: $C_{386}:C_{64}$

Order 18528: $C_{193}:C_{96}$ x 2, $C_{386}:C_{48}$

Order 12352: $C_{193}:C_{64}$ x 2, $C_{386}:C_{32}$

Order 9264: $C_{193}:C_{48}$ x 2, $C_{386}:C_{24}$

Order 6176: $C_{193}:C_{32}$ x 2, $C_{386}:C_{16}$

Order 4632: $C_{193}:C_{24}$ x 2, $C_{386}:C_{12}$

Order 3088: $C_{193}:C_{16}$ x 2, $C_{386}:C_8$

Order 2316: $C_{193}:C_{12}$ x 2, $C_{386}:C_6$

Order 1544: $C_{193}:C_8$ x 2, $C_{386}:C_4$

Order 1158: $C_{193}:C_6$ x 2, $C_{193}:C_6$

Order 772: $C_{193}:C_4$ x 2, $D_{386}$

Order 579: $C_{193}:C_3$

Order 386: $D_{193}$ x 2, $C_{386}$

Order 384: $C_2\times C_{192}$

Order 193: $C_{193}$

Order 192: $C_{192}$ x 2, $C_2\times C_{96}$

Order 128: $C_2\times C_{64}$

Order 96: $C_{96}$ x 2, $C_2\times C_{48}$

Order 64: $C_{64}$ x 2, $C_2\times C_{32}$

Order 48: $C_{48}$ x 2, $C_2\times C_{24}$

Order 32: $C_{32}$ x 2, $C_2\times C_{16}$

Order 24: $C_{24}$ x 2, $C_2\times C_{12}$

Order 16: $C_{16}$ x 2, $C_2\times C_8$

Order 12: $C_{12}$ x 2, $C_2\times C_6$

Order 8: $C_8$ x 2, $C_2\times C_4$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 74112: $C_2\times F_{193}$

Order 37056: $C_{386}:C_{96}$, $F_{193}$

Order 24704: $C_{386}:C_{64}$

Order 18528: $C_{193}:C_{96}$ x 2, $C_{386}:C_{48}$

Order 12352: $C_{193}:C_{64}$, $C_{386}:C_{32}$

Order 9264: $C_{193}:C_{48}$ x 2, $C_{386}:C_{24}$

Order 6176: $C_{193}:C_{32}$ x 2, $C_{386}:C_{16}$

Order 4632: $C_{193}:C_{24}$ x 2, $C_{386}:C_{12}$

Order 3088: $C_{193}:C_{16}$ x 2, $C_{386}:C_8$

Order 2316: $C_{193}:C_{12}$ x 2, $C_{386}:C_6$

Order 1544: $C_{193}:C_8$ x 2, $C_{386}:C_4$

Order 1158: $C_{193}:C_6$ x 2, $C_{193}:C_6$

Order 772: $C_{193}:C_4$ x 2, $D_{386}$

Order 579: $C_{193}:C_3$

Order 386: $D_{193}$ x 2, $C_{386}$

Order 384: $C_2\times C_{192}$

Order 193: $C_{193}$

Order 192: $C_{192}$, $C_2\times C_{96}$

Order 128: $C_2\times C_{64}$

Order 96: $C_{96}$ x 2, $C_2\times C_{48}$

Order 64: $C_2\times C_{32}$, $C_{64}$

Order 48: $C_{48}$ x 2, $C_2\times C_{24}$

Order 32: $C_{32}$ x 2, $C_2\times C_{16}$

Order 24: $C_{24}$ x 2, $C_2\times C_{12}$

Order 16: $C_{16}$ x 2, $C_2\times C_8$

Order 12: $C_{12}$ x 2, $C_2\times C_6$

Order 8: $C_8$ x 2, $C_2\times C_4$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 74112: $C_2\times F_{193}$ ($C_1$)

Order 37056: $F_{193}$ ($C_2$) x 2, $C_{386}:C_{96}$ ($C_2$)

Order 24704: $C_{386}:C_{64}$ ($C_3$)

Order 18528: $C_{193}:C_{96}$ ($C_2^2$), $C_{193}:C_{96}$ ($C_4$), $C_{386}:C_{48}$ ($C_4$)

Order 12352: $C_{193}:C_{64}$ ($C_6$) x 2, $C_{386}:C_{32}$ ($C_6$)

Order 9264: $C_{386}:C_{24}$ ($C_8$), $C_{193}:C_{48}$ ($C_2\times C_4$), $C_{193}:C_{48}$ ($C_8$)

Order 6176: $C_{193}:C_{32}$ ($C_2\times C_6$), $C_{193}:C_{32}$ ($C_{12}$), $C_{386}:C_{16}$ ($C_{12}$)

Order 4632: $C_{193}:C_{24}$ ($C_2\times C_8$), $C_{193}:C_{24}$ ($C_{16}$), $C_{386}:C_{12}$ ($C_{16}$)

Order 3088: $C_{193}:C_{16}$ ($C_2\times C_{12}$), $C_{193}:C_{16}$ ($C_{24}$), $C_{386}:C_8$ ($C_{24}$)

Order 2316: $C_{386}:C_6$ ($C_{32}$), $C_{193}:C_{12}$ ($C_2\times C_{16}$), $C_{193}:C_{12}$ ($C_{32}$)

Order 1544: $C_{193}:C_8$ ($C_2\times C_{24}$), $C_{193}:C_8$ ($C_{48}$), $C_{386}:C_4$ ($C_{48}$)

Order 1158: $C_{193}:C_6$ ($C_{64}$), $C_{193}:C_6$ ($C_2\times C_{32}$), $C_{193}:C_6$ ($C_{64}$)

Order 772: $D_{386}$ ($C_{96}$), $C_{193}:C_4$ ($C_2\times C_{48}$), $C_{193}:C_4$ ($C_{96}$)

Order 579: $C_{193}:C_3$ ($C_2\times C_{64}$)

Order 386: $C_{386}$ ($C_{192}$), $D_{193}$ ($C_{192}$), $D_{193}$ ($C_2\times C_{96}$)

Order 193: $C_{193}$ ($C_2\times C_{192}$)

Order 2: $C_2$ ($F_{193}$)

Order 1: $C_1$ ($C_2\times F_{193}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 74112: $C_2\times F_{193}$ ($C_1$)

Order 37056: $C_{386}:C_{96}$ ($C_2$), $F_{193}$ ($C_2$)

Order 24704: $C_{386}:C_{64}$ ($C_3$)

Order 18528: $C_{193}:C_{96}$ ($C_2^2$), $C_{193}:C_{96}$ ($C_4$), $C_{386}:C_{48}$ ($C_4$)

Order 12352: $C_{193}:C_{64}$ ($C_6$), $C_{386}:C_{32}$ ($C_6$)

Order 9264: $C_{386}:C_{24}$ ($C_8$), $C_{193}:C_{48}$ ($C_2\times C_4$), $C_{193}:C_{48}$ ($C_8$)

Order 6176: $C_{193}:C_{32}$ ($C_2\times C_6$), $C_{193}:C_{32}$ ($C_{12}$), $C_{386}:C_{16}$ ($C_{12}$)

Order 4632: $C_{193}:C_{24}$ ($C_2\times C_8$), $C_{193}:C_{24}$ ($C_{16}$), $C_{386}:C_{12}$ ($C_{16}$)

Order 3088: $C_{193}:C_{16}$ ($C_2\times C_{12}$), $C_{193}:C_{16}$ ($C_{24}$), $C_{386}:C_8$ ($C_{24}$)

Order 2316: $C_{386}:C_6$ ($C_{32}$), $C_{193}:C_{12}$ ($C_2\times C_{16}$), $C_{193}:C_{12}$ ($C_{32}$)

Order 1544: $C_{193}:C_8$ ($C_2\times C_{24}$), $C_{193}:C_8$ ($C_{48}$), $C_{386}:C_4$ ($C_{48}$)

Order 1158: $C_{193}:C_6$ ($C_{64}$), $C_{193}:C_6$ ($C_2\times C_{32}$), $C_{193}:C_6$ ($C_{64}$)

Order 772: $D_{386}$ ($C_{96}$), $C_{193}:C_4$ ($C_2\times C_{48}$), $C_{193}:C_4$ ($C_{96}$)

Order 579: $C_{193}:C_3$ ($C_2\times C_{64}$)

Order 386: $C_{386}$ ($C_{192}$), $D_{193}$ ($C_{192}$), $D_{193}$ ($C_2\times C_{96}$)

Order 193: $C_{193}$ ($C_2\times C_{192}$)

Order 2: $C_2$ ($F_{193}$)

Order 1: $C_1$ ($C_2\times F_{193}$)

Series

Derived series

$C_2\times F_{193}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Chief series

$C_2\times F_{193}$

$\rhd$

$C_{386}:C_{96}$

$\rhd$

$C_{386}:C_{48}$

$\rhd$

$C_{386}:C_{24}$

$\rhd$

$C_{386}:C_{12}$

$\rhd$

$C_{386}:C_6$

$\rhd$

$C_{193}:C_6$

$\rhd$

$C_{386}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Lower central series

$C_2\times F_{193}$

$\rhd$

$C_{193}$

Upper central series

$C_1$

$\lhd$

$C_2$

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 $386 \times 386$ 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 $30 \times 30$ rational character table.