Abstract group 912.174: $C_{12}\times D_{38}$

• Label: 912.174
• Order: \( 2^{4} \cdot 3 \cdot 19 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 19 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \cdot 19 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{2} \)
• Perm deg.: $28$
• Trans deg.: $456$
• Rank: $3$

Group information

Description:	$C_{12}\times D_{38}$
Order:	\(912\)\(\medspace = 2^{4} \cdot 3 \cdot 19 \)
Exponent:	\(228\)\(\medspace = 2^{2} \cdot 3 \cdot 19 \)
Automorphism group:	$C_{38}.C_{18}.C_2^5$, of order \(21888\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 19 \)
Composition factors:	$C_2$ x 4, $C_3$, $C_{19}$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.

Group statistics

Order	1	2	3	4	6	12	19	38	57	76	114	228
Elements	1	79	2	80	158	160	18	54	36	72	108	144	912
Conjugacy classes	1	7	2	8	14	16	9	27	18	36	54	72	264
Divisions	1	7	1	4	7	4	1	3	1	2	3	2	36
Autjugacy classes	1	3	1	2	3	2	1	2	1	1	2	1	20

Dimension	1	2	4	18	36	72
Irr. complex chars.	48	216	0	0	0	0	264
Irr. rational chars.	8	12	4	4	6	2	36

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$456$
Rank:	$3$
Inequivalent generating triples:	$10920$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	4	22

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{4}=c^{114}=[a,b]=[b,c]=1, c^{a}=c^{37} \rangle$
Permutation group:	Degree $28$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(23,24)(25,26)(27,28) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 20 & 19 \\ 19 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right), \left(\begin{array}{rr} 20 & 0 \\ 0 & 20 \end{array}\right), \left(\begin{array}{rr} 1 & 42 \\ 0 & 37 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/57\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \times\, $ $C_3$ $\, \times\, $ $D_{19}$
Semidirect product:	$C_{228}$ $\,\rtimes\,$ $C_2^2$ (2)	$(C_2\times C_{76})$ $\,\rtimes\,$ $C_6$	$(C_{38}:C_4)$ $\,\rtimes\,$ $C_6$	$C_{76}$ $\,\rtimes\,$ $(C_2\times C_6)$ (2)	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{114}$ . $C_2^3$	$(C_6\times D_{38})$ . $C_2$	$(C_2\times D_{38})$ . $C_6$	$D_{38}$ . $(C_2\times C_6)$ (2)	all 12

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

Homology

Abelianization:	$C_{2}^{2} \times C_{12} \simeq C_{2}^{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 792 subgroups in 108 conjugacy classes, 70 normal (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{12}$	$G/Z \simeq$ $D_{19}$
Commutator:	$G' \simeq$ $C_{19}$	$G/G' \simeq$ $C_2^2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_6\times D_{38}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{228}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{12}\times D_{38}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{114}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $C_{19}$

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 912: $C_{12}\times D_{38}$

Order 456: $C_{12}\times D_{19}$ x 4, $C_6\times D_{38}$, $C_2\times C_{228}$, $C_{114}:C_4$

Order 304: $C_4\times D_{38}$

Order 228: $C_3\times D_{38}$ x 6, $C_{228}$ x 2, $C_{19}:C_{12}$ x 2, $C_2\times C_{114}$

Order 152: $C_4\times D_{19}$ x 4, $C_2\times C_{76}$, $C_{38}:C_4$, $C_2\times D_{38}$

Order 114: $C_3\times D_{19}$ x 4, $C_{114}$ x 3

Order 76: $D_{38}$ x 6, $C_{76}$ x 2, $C_{19}:C_4$ x 2, $C_2\times C_{38}$

Order 57: $C_{57}$

Order 48: $C_2^2\times C_{12}$

Order 38: $D_{19}$ x 4, $C_{38}$ x 3

Order 24: $C_2\times C_{12}$ x 6, $C_2^2\times C_6$

Order 19: $C_{19}$

Order 16: $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 7, $C_{12}$ x 4

Order 8: $C_2\times C_4$ x 6, $C_2^3$

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 7, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 912: $C_{12}\times D_{38}$

Order 456: $C_6\times D_{38}$, $C_2\times C_{228}$, $C_{114}:C_4$, $C_{12}\times D_{19}$

Order 304: $C_4\times D_{38}$

Order 228: $C_3\times D_{38}$ x 2, $C_{228}$, $C_{19}:C_{12}$, $C_2\times C_{114}$

Order 152: $C_2\times C_{76}$, $C_{38}:C_4$, $C_4\times D_{19}$, $C_2\times D_{38}$

Order 114: $C_{114}$ x 2, $C_3\times D_{19}$

Order 76: $D_{38}$ x 2, $C_2\times C_{38}$, $C_{76}$, $C_{19}:C_4$

Order 57: $C_{57}$

Order 48: $C_2^2\times C_{12}$

Order 38: $C_{38}$ x 2, $D_{19}$

Order 24: $C_2\times C_{12}$ x 3, $C_2^2\times C_6$

Order 19: $C_{19}$

Order 16: $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 3, $C_{12}$ x 2

Order 8: $C_2\times C_4$ x 3, $C_2^3$

Order 6: $C_6$ x 3

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 912: $C_{12}\times D_{38}$ ($C_1$)

Order 456: $C_{12}\times D_{19}$ ($C_2$) x 4, $C_6\times D_{38}$ ($C_2$), $C_2\times C_{228}$ ($C_2$), $C_{114}:C_4$ ($C_2$)

Order 304: $C_4\times D_{38}$ ($C_3$)

Order 228: $C_3\times D_{38}$ ($C_4$) x 4, $C_{228}$ ($C_2^2$) x 2, $C_{19}:C_{12}$ ($C_2^2$) x 2, $C_3\times D_{38}$ ($C_2^2$) x 2, $C_2\times C_{114}$ ($C_2^2$)

Order 152: $C_4\times D_{19}$ ($C_6$) x 4, $C_2\times C_{76}$ ($C_6$), $C_{38}:C_4$ ($C_6$), $C_2\times D_{38}$ ($C_6$)

Order 114: $C_3\times D_{19}$ ($C_2\times C_4$) x 4, $C_{114}$ ($C_2\times C_4$) x 2, $C_{114}$ ($C_2^3$)

Order 76: $D_{38}$ ($C_{12}$) x 4, $D_{38}$ ($C_2\times C_6$) x 2, $C_{76}$ ($C_2\times C_6$) x 2, $C_{19}:C_4$ ($C_2\times C_6$) x 2, $C_2\times C_{38}$ ($C_2\times C_6$)

Order 57: $C_{57}$ ($C_2^2\times C_4$)

Order 38: $D_{19}$ ($C_2\times C_{12}$) x 4, $C_{38}$ ($C_2\times C_{12}$) x 2, $C_{38}$ ($C_2^2\times C_6$)

Order 24: $C_2\times C_{12}$ ($D_{19}$)

Order 19: $C_{19}$ ($C_2^2\times C_{12}$)

Order 12: $C_{12}$ ($D_{38}$) x 2, $C_2\times C_6$ ($D_{38}$)

Order 8: $C_2\times C_4$ ($C_3\times D_{19}$)

Order 6: $C_6$ ($C_4\times D_{19}$) x 2, $C_6$ ($C_2\times D_{38}$)

Order 4: $C_4$ ($C_3\times D_{38}$) x 2, $C_2^2$ ($C_3\times D_{38}$)

Order 3: $C_3$ ($C_4\times D_{38}$)

Order 2: $C_2$ ($C_{12}\times D_{19}$) x 2, $C_2$ ($C_6\times D_{38}$)

Order 1: $C_1$ ($C_{12}\times D_{38}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 912: $C_{12}\times D_{38}$ ($C_1$)

Order 456: $C_6\times D_{38}$ ($C_2$), $C_2\times C_{228}$ ($C_2$), $C_{114}:C_4$ ($C_2$), $C_{12}\times D_{19}$ ($C_2$)

Order 304: $C_4\times D_{38}$ ($C_3$)

Order 228: $C_{228}$ ($C_2^2$), $C_{19}:C_{12}$ ($C_2^2$), $C_2\times C_{114}$ ($C_2^2$), $C_3\times D_{38}$ ($C_2^2$), $C_3\times D_{38}$ ($C_4$)

Order 152: $C_2\times C_{76}$ ($C_6$), $C_{38}:C_4$ ($C_6$), $C_4\times D_{19}$ ($C_6$), $C_2\times D_{38}$ ($C_6$)

Order 114: $C_{114}$ ($C_2^3$), $C_{114}$ ($C_2\times C_4$), $C_3\times D_{19}$ ($C_2\times C_4$)

Order 76: $C_2\times C_{38}$ ($C_2\times C_6$), $D_{38}$ ($C_2\times C_6$), $D_{38}$ ($C_{12}$), $C_{76}$ ($C_2\times C_6$), $C_{19}:C_4$ ($C_2\times C_6$)

Order 57: $C_{57}$ ($C_2^2\times C_4$)

Order 38: $C_{38}$ ($C_2\times C_{12}$), $C_{38}$ ($C_2^2\times C_6$), $D_{19}$ ($C_2\times C_{12}$)

Order 24: $C_2\times C_{12}$ ($D_{19}$)

Order 19: $C_{19}$ ($C_2^2\times C_{12}$)

Order 12: $C_2\times C_6$ ($D_{38}$), $C_{12}$ ($D_{38}$)

Order 8: $C_2\times C_4$ ($C_3\times D_{19}$)

Order 6: $C_6$ ($C_4\times D_{19}$), $C_6$ ($C_2\times D_{38}$)

Order 4: $C_2^2$ ($C_3\times D_{38}$), $C_4$ ($C_3\times D_{38}$)

Order 3: $C_3$ ($C_4\times D_{38}$)

Order 2: $C_2$ ($C_6\times D_{38}$), $C_2$ ($C_{12}\times D_{19}$)

Order 1: $C_1$ ($C_{12}\times D_{38}$)

Series

Derived series

$C_{12}\times D_{38}$

$\rhd$

$C_{19}$

$\rhd$

$C_1$

Chief series

$C_{12}\times D_{38}$

$\rhd$

$C_{12}\times D_{19}$

$\rhd$

$C_3\times D_{38}$

$\rhd$

$C_{114}$

$\rhd$

$C_{57}$

$\rhd$

$C_{19}$

$\rhd$

$C_1$

Lower central series

$C_{12}\times D_{38}$

$\rhd$

$C_{19}$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{12}$

Supergroups

This group is a maximal subgroup of 57 larger groups in the database.

This group is a maximal quotient of 21 larger groups in the database.

Character theory

Complex character table

See the $264 \times 264$ 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 $36 \times 36$ rational character table.