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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 [×5], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×5], C6, C6 [×5], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×4], D4 [×6], Q8 [×2], C23, C23 [×2], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×5], C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C3⋊C8 [×4], C2×C12 [×2], C2×C12 [×6], C2×C12 [×4], C3×D4 [×6], C3×Q8 [×2], C22×C6, C22×C6 [×2], C4.D4 [×2], C4.10D4 [×2], C2×M4(2) [×2], C2×C4○D4, C2×C3⋊C8 [×2], C4.Dic3 [×4], C4.Dic3 [×2], C22×C12, C22×C12 [×2], C6×D4, C6×D4 [×2], C6×Q8, C3×C4○D4 [×4], M4(2).8C22, C12.D4 [×2], C12.10D4 [×2], C2×C4.Dic3 [×2], C6×C4○D4, (C6×D4).16C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], M4(2).8C22, C2×C6.D4, (C6×D4).16C4

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