Copied to
clipboard

G = C2×C19⋊C6order 228 = 22·3·19

Direct product of C2 and C19⋊C6

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C2×C19⋊C6, C38⋊C6, D38⋊C3, D19⋊C6, C19⋊(C2×C6), C19⋊C3⋊C22, (C2×C19⋊C3)⋊C2, SmallGroup(228,7)

Series: Derived Chief Lower central Upper central

C1C19 — C2×C19⋊C6
C1C19C19⋊C3C19⋊C6 — C2×C19⋊C6
C19 — C2×C19⋊C6
C1C2

Generators and relations for C2×C19⋊C6
 G = < a,b,c | a2=b19=c6=1, ab=ba, ac=ca, cbc-1=b12 >

19C2
19C2
19C3
19C22
19C6
19C6
19C6
19C2×C6

Character table of C2×C19⋊C6

 class 12A2B2C3A3B6A6B6C6D6E6F19A19B19C38A38B38C
 size 1119191919191919191919666666
ρ1111111111111111111    trivial
ρ211-1-111-1-1-11-11111111    linear of order 2
ρ31-11-111-111-1-1-1111-1-1-1    linear of order 2
ρ41-1-11111-1-1-11-1111-1-1-1    linear of order 2
ρ51-11-1ζ3ζ32ζ65ζ32ζ3ζ65ζ6ζ6111-1-1-1    linear of order 6
ρ61-1-11ζ3ζ32ζ3ζ6ζ65ζ65ζ32ζ6111-1-1-1    linear of order 6
ρ71111ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ3111111    linear of order 3
ρ81-1-11ζ32ζ3ζ32ζ65ζ6ζ6ζ3ζ65111-1-1-1    linear of order 6
ρ911-1-1ζ3ζ32ζ65ζ6ζ65ζ3ζ6ζ32111111    linear of order 6
ρ101111ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ32111111    linear of order 3
ρ111-11-1ζ32ζ3ζ6ζ3ζ32ζ6ζ65ζ65111-1-1-1    linear of order 6
ρ1211-1-1ζ32ζ3ζ6ζ65ζ6ζ32ζ65ζ3111111    linear of order 6
ρ136-60000000000ζ191519131910199196194ζ19181912191119819719ζ19171916191419519319219151913191019919619419181912191119819719191719161914195193192    orthogonal faithful
ρ14660000000000ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192    orthogonal lifted from C19⋊C6
ρ15660000000000ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719    orthogonal lifted from C19⋊C6
ρ166-60000000000ζ191719161914195193192ζ191519131910199196194ζ1918191219111981971919171916191419519319219151913191019919619419181912191119819719    orthogonal faithful
ρ17660000000000ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194    orthogonal lifted from C19⋊C6
ρ186-60000000000ζ19181912191119819719ζ191719161914195193192ζ19151913191019919619419181912191119819719191719161914195193192191519131910199196194    orthogonal faithful

Smallest permutation representation of C2×C19⋊C6
On 38 points
Generators in S38
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)
(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)
(2 9 8 19 12 13)(3 17 15 18 4 6)(5 14 10 16 7 11)(21 28 27 38 31 32)(22 36 34 37 23 25)(24 33 29 35 26 30)

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

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

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

C2×C19⋊C6 is a maximal subgroup of   D76⋊C3  D38⋊C6
C2×C19⋊C6 is a maximal quotient of   Dic38⋊C3  D76⋊C3  D38⋊C6

Matrix representation of C2×C19⋊C6 in GL6(𝔽229)

22800000
02280000
00228000
00022800
00002280
00000228
,
10110683106101228
100000
010000
001000
000100
000010
,
100000
2102099999209210
000001
2261644124128
1024772204107
000100

G:=sub<GL(6,GF(229))| [228,0,0,0,0,0,0,228,0,0,0,0,0,0,228,0,0,0,0,0,0,228,0,0,0,0,0,0,228,0,0,0,0,0,0,228],[101,1,0,0,0,0,106,0,1,0,0,0,83,0,0,1,0,0,106,0,0,0,1,0,101,0,0,0,0,1,228,0,0,0,0,0],[1,210,0,2,102,0,0,209,0,26,4,0,0,99,0,16,77,0,0,99,0,44,2,1,0,209,0,124,204,0,0,210,1,128,107,0] >;

C2×C19⋊C6 in GAP, Magma, Sage, TeX

C_2\times C_{19}\rtimes C_6
% in TeX

G:=Group("C2xC19:C6");
// GroupNames label

G:=SmallGroup(228,7);
// by ID

G=gap.SmallGroup(228,7);
# by ID

G:=PCGroup([4,-2,-2,-3,-19,3459,347]);
// Polycyclic

G:=Group<a,b,c|a^2=b^19=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^12>;
// generators/relations

Export

Subgroup lattice of C2×C19⋊C6 in TeX
Character table of C2×C19⋊C6 in TeX

׿
×
𝔽