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## G = (C6×D4).16C4order 192 = 26·3

### 10th non-split extension by C6×D4 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C6×D4).16C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4.Dic3 — C2×C4.Dic3 — (C6×D4).16C4
 Lower central C3 — C6 — C2×C6 — (C6×D4).16C4
 Upper central C1 — C4 — C22×C4 — C2×C4○D4

Generators and relations for (C6×D4).16C4
G = < a,b,c,d | a6=b4=c2=1, d4=b2, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, dbd-1=a3b, dcd-1=b2c >

Subgroups: 296 in 150 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C4.D4, C4.10D4, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3, C4.Dic3, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, M4(2).8C22, C12.D4, C12.10D4, C2×C4.Dic3, C6×C4○D4, (C6×D4).16C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, M4(2).8C22, C2×C6.D4, (C6×D4).16C4

Smallest permutation representation of (C6×D4).16C4
On 48 points
Generators in S48
(1 22 28 5 18 32)(2 29 19)(3 24 30 7 20 26)(4 31 21)(6 25 23)(8 27 17)(9 39 43 13 35 47)(10 44 36)(11 33 45 15 37 41)(12 46 38)(14 48 40)(16 42 34)
(1 13 5 9)(2 10 6 14)(3 11 7 15)(4 16 8 12)(17 38 21 34)(18 39 22 35)(19 36 23 40)(20 37 24 33)(25 48 29 44)(26 41 30 45)(27 46 31 42)(28 47 32 43)
(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)(17 36)(18 33)(19 38)(20 35)(21 40)(22 37)(23 34)(24 39)(25 42)(26 47)(27 44)(28 41)(29 46)(30 43)(31 48)(32 45)
(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)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - + - + + image C1 C2 C2 C2 C2 C4 C4 S3 D4 Dic3 D6 Dic3 D6 D6 C3⋊D4 M4(2).8C22 (C6×D4).16C4 kernel (C6×D4).16C4 C12.D4 C12.10D4 C2×C4.Dic3 C6×C4○D4 C22×C12 C6×D4 C2×C4○D4 C2×C12 C22×C4 C22×C4 C2×D4 C2×D4 C2×Q8 C2×C4 C3 C1 # reps 1 2 2 2 1 4 4 1 4 2 1 2 1 1 8 2 4

Matrix representation of (C6×D4).16C4 in GL4(𝔽73) generated by

 9 0 0 8 0 9 0 65 0 0 8 0 0 0 0 8
,
 46 0 0 6 0 27 0 0 0 0 46 46 0 0 0 27
,
 0 46 6 0 27 0 67 67 0 0 46 46 0 0 54 27
,
 57 0 1 55 16 0 0 18 72 1 0 57 2 0 0 16
G:=sub<GL(4,GF(73))| [9,0,0,0,0,9,0,0,0,0,8,0,8,65,0,8],[46,0,0,0,0,27,0,0,0,0,46,0,6,0,46,27],[0,27,0,0,46,0,0,0,6,67,46,54,0,67,46,27],[57,16,72,2,0,0,1,0,1,0,0,0,55,18,57,16] >;

(C6×D4).16C4 in GAP, Magma, Sage, TeX

(C_6\times D_4)._{16}C_4
% in TeX

G:=Group("(C6xD4).16C4");
// GroupNames label

G:=SmallGroup(192,796);
// by ID

G=gap.SmallGroup(192,796);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,387,297,136,1684,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^4=c^2=1,d^4=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c=b^-1,d*b*d^-1=a^3*b,d*c*d^-1=b^2*c>;
// generators/relations

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