C3xC2^3.37D4 - GroupNames

G = C₃×C₂³.₃₇D₄ order 192 = 2⁶·3

Direct product of C₃ and C₂³.₃₇D₄

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₃×C₂³.₃₇D₄, (C₆×D₄)⋊₂₀C₄, (C₂×D₄)⋊₈C₁₂, C₄.₅₅(C₆×D₄), D₄.₆(C₂×C₁₂), D₄⋊C₄⋊₁₅C₆, (C₂×C₂₄)⋊₃₈C₂², C₄²⋊C₂⋊₃C₆, (C₂×C₁₂).₃₁₅D₄, C₁₂.₄₆₂(C₂×D₄), C₄.₅(C₂²×C₁₂), (C₂²×D₄).₈C₆, C₂³.₄₂(C₃×D₄), C₂².₄₅(C₆×D₄), (C₆×M₄(2))⋊₂₉C₂, (C₂×M₄(2))⋊₁₁C₆, (C₂²×C₆).₁₅₉D₄, C₆.₁₂₇(C₈⋊C₂²), C₁₂.₈₂(C₂²⋊C₄), C₁₂.₁₅₀(C₂²×C₄), (C₂×C₁₂).₈₉₄C₂³, (C₆×D₄).₂₈₉C₂², (C₂²×C₁₂).₄₁₁C₂², C₄⋊C₄⋊₈(C₂×C₆), (C₂×C₈)⋊₈(C₂×C₆), (D₄×C₂×C₆).₁₉C₂, (C₂×C₄).₂₃(C₃×D₄), C₂.₂(C₃×C₈⋊C₂²), (C₃×C₄⋊C₄)⋊₆₄C₂², (C₂×C₄).₂₁(C₂×C₁₂), (C₂×D₄).₄₇(C₂×C₆), (C₃×D₄).₂₈(C₂×C₄), (C₂×C₆).₆₂₁(C₂×D₄), C₄.₁₄(C₃×C₂²⋊C₄), C₂.₂₁(C₆×C₂²⋊C₄), (C₃×D₄⋊C₄)⋊₃₈C₂, (C₂×C₁₂).₁₉₄(C₂×C₄), C₆.₁₀₉(C₂×C₂²⋊C₄), (C₂²×C₄).₃₅(C₂×C₆), (C₂×C₄).₆₉(C₂²×C₆), (C₃×C₄²⋊C₂)⋊₂₄C₂, (C₂×C₆).₈₁(C₂²⋊C₄), C₂².₂₀(C₃×C₂²⋊C₄), SmallGroup(192,851)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄ — C₃×C₂³.₃₇D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂×C₁₂ — C₃×C₄⋊C₄ — C₃×D₄⋊C₄ — C₃×C₂³.₃₇D₄

Lower central

C₁ — C₂ — C₄ — C₃×C₂³.₃₇D₄

Upper central

C₁ — C₂×C₆ — C₂²×C₁₂ — C₃×C₂³.₃₇D₄

Generators and relations for C₃×C₂³.₃₇D₄
G = < a,b,c,d,e,f | a³=b²=c²=d²=1, e⁴=d, f²=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe^-1=fbf^-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=cde³ >

Subgroups: 386 in 190 conjugacy classes, 82 normal (26 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², C₂², C₆, C₆, C₆, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, C₂³, C₂³, C₁₂, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂₄, C₂×C₁₂, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₂²×C₆, C₂²×C₆, D₄⋊C₄, C₄²⋊C₂, C₂×M₄(2), C₂²×D₄, C₄×C₁₂, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₂×C₂₄, C₃×M₄(2), C₂²×C₁₂, C₆×D₄, C₆×D₄, C₂³×C₆, C₂³.₃₇D₄, C₃×D₄⋊C₄, C₃×C₄²⋊C₂, C₆×M₄(2), D₄×C₂×C₆, C₃×C₂³.₃₇D₄
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, D₄, C₂³, C₁₂, C₂×C₆, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₂×C₂²⋊C₄, C₈⋊C₂², C₃×C₂²⋊C₄, C₂²×C₁₂, C₆×D₄, C₂³.₃₇D₄, C₆×C₂²⋊C₄, C₃×C₈⋊C₂², C₃×C₂³.₃₇D₄

Smallest permutation representation of C₃×C₂³.₃₇D₄
►On 48 points

Generators in S₄₈

(1 34 15)(2 35 16)(3 36 9)(4 37 10)(5 38 11)(6 39 12)(7 40 13)(8 33 14)(17 26 44)(18 27 45)(19 28 46)(20 29 47)(21 30 48)(22 31 41)(23 32 42)(24 25 43)

(1 24)(2 21)(3 18)(4 23)(5 20)(6 17)(7 22)(8 19)(9 45)(10 42)(11 47)(12 44)(13 41)(14 46)(15 43)(16 48)(25 34)(26 39)(27 36)(28 33)(29 38)(30 35)(31 40)(32 37)

(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)

(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)

(1 8 20 19)(2 18 21 7)(3 6 22 17)(4 24 23 5)(9 12 41 44)(10 43 42 11)(13 16 45 48)(14 47 46 15)(25 32 38 37)(26 36 39 31)(27 30 40 35)(28 34 33 29)

G:=sub<Sym(48)| (1,34,15)(2,35,16)(3,36,9)(4,37,10)(5,38,11)(6,39,12)(7,40,13)(8,33,14)(17,26,44)(18,27,45)(19,28,46)(20,29,47)(21,30,48)(22,31,41)(23,32,42)(24,25,43), (1,24)(2,21)(3,18)(4,23)(5,20)(6,17)(7,22)(8,19)(9,45)(10,42)(11,47)(12,44)(13,41)(14,46)(15,43)(16,48)(25,34)(26,39)(27,36)(28,33)(29,38)(30,35)(31,40)(32,37), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,8,20,19)(2,18,21,7)(3,6,22,17)(4,24,23,5)(9,12,41,44)(10,43,42,11)(13,16,45,48)(14,47,46,15)(25,32,38,37)(26,36,39,31)(27,30,40,35)(28,34,33,29)>;

G:=Group( (1,34,15)(2,35,16)(3,36,9)(4,37,10)(5,38,11)(6,39,12)(7,40,13)(8,33,14)(17,26,44)(18,27,45)(19,28,46)(20,29,47)(21,30,48)(22,31,41)(23,32,42)(24,25,43), (1,24)(2,21)(3,18)(4,23)(5,20)(6,17)(7,22)(8,19)(9,45)(10,42)(11,47)(12,44)(13,41)(14,46)(15,43)(16,48)(25,34)(26,39)(27,36)(28,33)(29,38)(30,35)(31,40)(32,37), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,8,20,19)(2,18,21,7)(3,6,22,17)(4,24,23,5)(9,12,41,44)(10,43,42,11)(13,16,45,48)(14,47,46,15)(25,32,38,37)(26,36,39,31)(27,30,40,35)(28,34,33,29) );

G=PermutationGroup([[(1,34,15),(2,35,16),(3,36,9),(4,37,10),(5,38,11),(6,39,12),(7,40,13),(8,33,14),(17,26,44),(18,27,45),(19,28,46),(20,29,47),(21,30,48),(22,31,41),(23,32,42),(24,25,43)], [(1,24),(2,21),(3,18),(4,23),(5,20),(6,17),(7,22),(8,19),(9,45),(10,42),(11,47),(12,44),(13,41),(14,46),(15,43),(16,48),(25,34),(26,39),(27,36),(28,33),(29,38),(30,35),(31,40),(32,37)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,17),(7,18),(8,19),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(25,38),(26,39),(27,40),(28,33),(29,34),(30,35),(31,36),(32,37)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,8,20,19),(2,18,21,7),(3,6,22,17),(4,24,23,5),(9,12,41,44),(10,43,42,11),(13,16,45,48),(14,47,46,15),(25,32,38,37),(26,36,39,31),(27,30,40,35),(28,34,33,29)]])

66 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	3A	3B	4A	4B	4C	4D	4E	4F	4G	4H	6A	···	6F	6G	6H	6I	6J	6K	···	6R	8A	8B	8C	8D	12A	···	12H	12I	···	12P	24A	···	24H
order	1	2	2	2	2	2	2	2	2	2	3	3	4	4	4	4	4	4	4	4	6	···	6	6	6	6	6	6	···	6	8	8	8	8	12	···	12	12	···	12	24	···	24
size	1	1	1	1	2	2	4	4	4	4	1	1	2	2	2	2	4	4	4	4	1	···	1	2	2	2	2	4	···	4	4	4	4	4	2	···	2	4	···	4	4	···	4

66 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+	+								+	+			+
image	C₁	C₂	C₂	C₂	C₂	C₃	C₄	C₆	C₆	C₆	C₆	C₁₂	D₄	D₄	C₃×D₄	C₃×D₄	C₈⋊C₂²	C₃×C₈⋊C₂²
kernel	C₃×C₂³.₃₇D₄	C₃×D₄⋊C₄	C₃×C₄²⋊C₂	C₆×M₄(2)	D₄×C₂×C₆	C₂³.₃₇D₄	C₆×D₄	D₄⋊C₄	C₄²⋊C₂	C₂×M₄(2)	C₂²×D₄	C₂×D₄	C₂×C₁₂	C₂²×C₆	C₂×C₄	C₂³	C₆	C₂
# reps	1	4	1	1	1	2	8	8	2	2	2	16	3	1	6	2	2	4

Matrix representation of C₃×C₂³.₃₇D₄ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	8	0	0	0
0	0	0	8	0	0
0	0	0	0	8	0
0	0	0	0	0	8

1	0	0	0	0	0
0	1	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	1	0
0	0	0	0	0	1

72	0	0	0	0	0
0	72	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	72	0
0	0	0	0	0	72

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	72
0	0	0	72	0	0
0	0	72	0	0	0

0	1	0	0	0	0
72	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,72,0,0],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₃×C₂³.₃₇D₄ in GAP, Magma, Sage, TeX

C_3\times C_2^3._{37}D_4

% in TeX

G:=Group("C3xC2^3.37D4");

// GroupNames label

G:=SmallGroup(192,851);

// by ID

G=gap.SmallGroup(192,851);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,1059,520,4204,2111,172]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^3>;

// generators/relations

G = C3×C23.37D4 order 192 = 26·3

Direct product of C3 and C23.37D4

G = C₃×C₂³.₃₇D₄ order 192 = 2⁶·3

Direct product of C₃ and C₂³.₃₇D₄