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## G = (C22×C12)⋊C4order 192 = 26·3

### 2nd semidirect product of C22×C12 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C22×C12)⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C6×D4 — C23.7D6 — (C22×C12)⋊C4
 Lower central C3 — C6 — C2×C6 — C22×C6 — (C22×C12)⋊C4
 Upper central C1 — C2 — C22 — C2×D4 — C22.D4

Generators and relations for (C22×C12)⋊C4
G = < a,b,c,d | a2=b2=c12=d4=1, ab=ba, ac=ca, dad-1=abc6, bc=cb, dbd-1=bc6, dcd-1=abc-1 >

Subgroups: 208 in 68 conjugacy classes, 23 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×4], C22, C22 [×4], C6, C6 [×3], C8, C2×C4, C2×C4 [×4], D4, C23 [×2], Dic3, C12 [×3], C2×C6, C2×C6 [×4], C22⋊C4, C22⋊C4 [×2], C4⋊C4, M4(2), C22×C4, C2×D4, C3⋊C8, C2×Dic3, C2×C12, C2×C12 [×3], C3×D4, C22×C6 [×2], C23⋊C4, C4.D4, C22.D4, C4.Dic3, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C23.D4, C12.D4, C23.7D6, C3×C22.D4, (C22×C12)⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, C2×Dic3, C3⋊D4 [×2], C23⋊C4, C6.D4, C23.D4, C23.7D6, (C22×C12)⋊C4

Character table of (C22×C12)⋊C4

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 8A 8B 12A 12B 12C 12D 12E 12F 12G size 1 1 2 4 4 2 4 4 4 8 24 24 2 2 2 4 4 8 24 24 4 4 4 4 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 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 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 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 1 -1 -1 linear of order 2 ρ5 1 1 1 -1 1 1 -1 1 1 -1 -i i 1 1 1 1 1 -1 -i i 1 1 1 1 -1 -1 -1 linear of order 4 ρ6 1 1 1 -1 1 1 -1 -1 -1 1 -i i 1 1 1 1 1 -1 i -i -1 -1 -1 -1 -1 1 1 linear of order 4 ρ7 1 1 1 -1 1 1 -1 1 1 -1 i -i 1 1 1 1 1 -1 i -i 1 1 1 1 -1 -1 -1 linear of order 4 ρ8 1 1 1 -1 1 1 -1 -1 -1 1 i -i 1 1 1 1 1 -1 -i i -1 -1 -1 -1 -1 1 1 linear of order 4 ρ9 2 2 2 2 2 -1 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 -2 -2 2 2 0 0 0 0 0 2 2 2 -2 -2 -2 0 0 0 0 0 0 2 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 2 -2 0 0 0 0 0 2 2 2 -2 -2 2 0 0 0 0 0 0 -2 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 2 -1 2 -2 -2 -2 0 0 -1 -1 -1 -1 -1 -1 0 0 1 1 1 1 -1 1 1 orthogonal lifted from D6 ρ13 2 2 2 -2 2 -1 -2 -2 -2 2 0 0 -1 -1 -1 -1 -1 1 0 0 1 1 1 1 1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ14 2 2 2 -2 2 -1 -2 2 2 -2 0 0 -1 -1 -1 -1 -1 1 0 0 -1 -1 -1 -1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ15 2 2 2 -2 -2 -1 2 0 0 0 0 0 -1 -1 -1 1 1 1 0 0 √-3 -√-3 √-3 -√-3 -1 √-3 -√-3 complex lifted from C3⋊D4 ρ16 2 2 2 2 -2 -1 -2 0 0 0 0 0 -1 -1 -1 1 1 -1 0 0 √-3 -√-3 √-3 -√-3 1 -√-3 √-3 complex lifted from C3⋊D4 ρ17 2 2 2 2 -2 -1 -2 0 0 0 0 0 -1 -1 -1 1 1 -1 0 0 -√-3 √-3 -√-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 1 0 0 -√-3 √-3 -√-3 √-3 -1 -√-3 √-3 complex lifted from C3⋊D4 ρ19 4 4 -4 0 0 4 0 0 0 0 0 0 -4 4 -4 0 0 0 0 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 2 2√-3 -2√-3 0 0 0 0 0 0 0 0 0 0 complex lifted from C23.7D6 ρ21 4 4 -4 0 0 -2 0 0 0 0 0 0 2 -2 2 -2√-3 2√-3 0 0 0 0 0 0 0 0 0 0 complex lifted from C23.7D6 ρ22 4 -4 0 0 0 4 0 2i -2i 0 0 0 0 -4 0 0 0 0 0 0 2i 2i -2i -2i 0 0 0 complex lifted from C23.D4 ρ23 4 -4 0 0 0 4 0 -2i 2i 0 0 0 0 -4 0 0 0 0 0 0 -2i -2i 2i 2i 0 0 0 complex lifted from C23.D4 ρ24 4 -4 0 0 0 -2 0 2i -2i 0 0 0 2√-3 2 -2√-3 0 0 0 0 0 2ζ4ζ3 2ζ4ζ32 2ζ43ζ3 2ζ43ζ32 0 0 0 complex faithful ρ25 4 -4 0 0 0 -2 0 2i -2i 0 0 0 -2√-3 2 2√-3 0 0 0 0 0 2ζ4ζ32 2ζ4ζ3 2ζ43ζ32 2ζ43ζ3 0 0 0 complex faithful ρ26 4 -4 0 0 0 -2 0 -2i 2i 0 0 0 -2√-3 2 2√-3 0 0 0 0 0 2ζ43ζ32 2ζ43ζ3 2ζ4ζ32 2ζ4ζ3 0 0 0 complex faithful ρ27 4 -4 0 0 0 -2 0 -2i 2i 0 0 0 2√-3 2 -2√-3 0 0 0 0 0 2ζ43ζ3 2ζ43ζ32 2ζ4ζ3 2ζ4ζ32 0 0 0 complex faithful

Smallest permutation representation of (C22×C12)⋊C4
On 48 points
Generators in S48
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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)
(2 36 20 42)(3 11)(4 34 22 40)(5 9)(6 32 24 38)(8 30 14 48)(10 28 16 46)(12 26 18 44)(13 19)(15 17)(21 23)(25 37 31 43)(27 47 33 41)(29 45 35 39)

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

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

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

Matrix representation of (C22×C12)⋊C4 in GL4(𝔽73) generated by

 72 0 71 0 0 72 0 71 0 0 1 0 0 0 0 1
,
 30 60 0 0 13 43 0 0 0 0 30 60 0 0 13 43
,
 59 66 59 39 7 66 34 20 0 0 0 27 0 0 46 46
,
 1 0 0 0 72 72 0 0 21 43 43 13 22 52 43 30
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,71,0,1,0,0,71,0,1],[30,13,0,0,60,43,0,0,0,0,30,13,0,0,60,43],[59,7,0,0,66,66,0,0,59,34,0,46,39,20,27,46],[1,72,21,22,0,72,43,52,0,0,43,43,0,0,13,30] >;

(C22×C12)⋊C4 in GAP, Magma, Sage, TeX

(C_2^2\times C_{12})\rtimes C_4
% in TeX

G:=Group("(C2^2xC12):C4");
// GroupNames label

G:=SmallGroup(192,98);
// by ID

G=gap.SmallGroup(192,98);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,219,675,297,1684,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^12=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c^6,b*c=c*b,d*b*d^-1=b*c^6,d*c*d^-1=a*b*c^-1>;
// generators/relations

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