Copied to
clipboard

## G = (C2×D4).D6order 192 = 26·3

### 2nd non-split extension by C2×D4 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C2×D4).D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C6×D4 — C23.23D6 — (C2×D4).D6
 Lower central C3 — C6 — C2×C6 — C22×C6 — (C2×D4).D6
 Upper central C1 — C2 — C22 — C2×D4 — C23⋊C4

Generators and relations for (C2×D4).D6
G = < a,b,c,d,e | a2=b4=c2=1, d6=ab2c, e2=ba=dbd-1=ab, ece-1=ac=ca, dad-1=eae-1=ab2, cbc=ebe-1=b-1, dcd-1=b2c, ede-1=bcd5 >

Subgroups: 224 in 68 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, M4(2), C22×C4, C2×D4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C23⋊C4, C4.D4, C22.D4, C4.Dic3, Dic3⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C22×Dic3, C6×D4, C23.D4, C12.D4, C3×C23⋊C4, C23.23D6, (C2×D4).D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C23⋊C4, D6⋊C4, C23.D4, C23.6D6, (C2×D4).D6

Character table of (C2×D4).D6

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E size 1 1 2 4 4 2 4 8 8 12 12 24 2 4 4 4 8 24 24 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 linear of order 2 ρ5 1 1 1 -1 1 1 -1 -i i 1 1 -1 1 -1 -1 1 1 -i i i i -1 -i -i linear of order 4 ρ6 1 1 1 -1 1 1 -1 -i i -1 -1 1 1 -1 -1 1 1 i -i i i -1 -i -i linear of order 4 ρ7 1 1 1 -1 1 1 -1 i -i -1 -1 1 1 -1 -1 1 1 -i i -i -i -1 i i linear of order 4 ρ8 1 1 1 -1 1 1 -1 i -i 1 1 -1 1 -1 -1 1 1 i -i -i -i -1 i i linear of order 4 ρ9 2 2 2 -2 -2 2 2 0 0 0 0 0 2 -2 -2 2 -2 0 0 0 0 2 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 -1 2 2 2 0 0 0 -1 -1 -1 -1 -1 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 2 -2 2 -2 0 0 0 0 0 2 2 2 2 -2 0 0 0 0 -2 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 2 -1 2 -2 -2 0 0 0 -1 -1 -1 -1 -1 0 0 1 1 -1 1 1 orthogonal lifted from D6 ρ13 2 2 2 -2 -2 -1 2 0 0 0 0 0 -1 1 1 -1 1 0 0 -√3 √3 -1 √3 -√3 orthogonal lifted from D12 ρ14 2 2 2 -2 -2 -1 2 0 0 0 0 0 -1 1 1 -1 1 0 0 √3 -√3 -1 -√3 √3 orthogonal lifted from D12 ρ15 2 2 2 -2 2 -1 -2 2i -2i 0 0 0 -1 1 1 -1 -1 0 0 i i 1 -i -i complex lifted from C4×S3 ρ16 2 2 2 -2 2 -1 -2 -2i 2i 0 0 0 -1 1 1 -1 -1 0 0 -i -i 1 i i complex lifted from C4×S3 ρ17 2 2 2 2 -2 -1 -2 0 0 0 0 0 -1 -1 -1 -1 1 0 0 √-3 -√-3 1 √-3 -√-3 complex lifted from C3⋊D4 ρ18 2 2 2 2 -2 -1 -2 0 0 0 0 0 -1 -1 -1 -1 1 0 0 -√-3 √-3 1 -√-3 √-3 complex lifted from C3⋊D4 ρ19 4 4 -4 0 0 4 0 0 0 0 0 0 4 0 0 -4 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ20 4 4 -4 0 0 -2 0 0 0 0 0 0 -2 -2√-3 2√-3 2 0 0 0 0 0 0 0 0 complex lifted from C23.6D6 ρ21 4 4 -4 0 0 -2 0 0 0 0 0 0 -2 2√-3 -2√-3 2 0 0 0 0 0 0 0 0 complex lifted from C23.6D6 ρ22 4 -4 0 0 0 4 0 0 0 -2i 2i 0 -4 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C23.D4 ρ23 4 -4 0 0 0 4 0 0 0 2i -2i 0 -4 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C23.D4 ρ24 8 -8 0 0 0 -4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of (C2×D4).D6
On 48 points
Generators in S48
(2 29)(4 31)(6 33)(8 35)(10 25)(12 27)(14 38)(16 40)(18 42)(20 44)(22 46)(24 48)
(1 47 28 23)(2 48 29 24)(3 13 30 37)(4 14 31 38)(5 39 32 15)(6 40 33 16)(7 17 34 41)(8 18 35 42)(9 43 36 19)(10 44 25 20)(11 21 26 45)(12 22 27 46)
(1 34)(2 8)(3 36)(4 10)(5 26)(6 12)(7 28)(9 30)(11 32)(13 43)(14 20)(15 45)(16 22)(17 47)(18 24)(19 37)(21 39)(23 41)(25 31)(27 33)(29 35)(38 44)(40 46)(42 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 12 47 46 28 27 23 22)(2 45 24 11 29 21 48 26)(3 25 13 44 30 10 37 20)(4 43 38 36 31 19 14 9)(5 8 39 42 32 35 15 18)(6 41 16 7 33 17 40 34)

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

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

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

Matrix representation of (C2×D4).D6 in GL6(𝔽73)

 1 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 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 46 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 46
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 46 0 0 0 0 27 0 0 0 0 0 0 0 0 27 0 0 0 0 46 0
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 27 0 0 0 0 46 0 0 0
,
 1 72 0 0 0 0 0 72 0 0 0 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 1 0 0 0 0 0 0 72 0 0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,46],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,46,0,0,0,0,0,0,46,0,0] >;

(C2×D4).D6 in GAP, Magma, Sage, TeX

(C_2\times D_4).D_6
% in TeX

G:=Group("(C2xD4).D6");
// GroupNames label

G:=SmallGroup(192,31);
// by ID

G=gap.SmallGroup(192,31);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,422,346,297,851,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^6=a*b^2*c,e^2=b*a=d*b*d^-1=a*b,e*c*e^-1=a*c=c*a,d*a*d^-1=e*a*e^-1=a*b^2,c*b*c=e*b*e^-1=b^-1,d*c*d^-1=b^2*c,e*d*e^-1=b*c*d^5>;
// generators/relations

Export

׿
×
𝔽