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## G = C2×D12⋊C4order 192 = 26·3

### Direct product of C2 and D12⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×D12⋊C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — C2×C4○D12 — C2×D12⋊C4
 Lower central C3 — C6 — C12 — C2×D12⋊C4
 Upper central C1 — C2×C4 — C22×C4 — C2×M4(2)

Generators and relations for C2×D12⋊C4
G = < a,b,c,d | a2=b12=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b7c >

Subgroups: 472 in 170 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C4×Dic3, C4×Dic3, C2×C24, C3×M4(2), C3×M4(2), C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C22×Dic3, C2×C3⋊D4, C22×C12, C2×C4≀C2, D12⋊C4, C2×C4×Dic3, C6×M4(2), C2×C4○D12, C2×D12⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C4≀C2, C2×C22⋊C4, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C2×C4≀C2, D12⋊C4, C2×D6⋊C4, C2×D12⋊C4

Smallest permutation representation of C2×D12⋊C4
On 48 points
Generators in S48
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 44)(14 45)(15 46)(16 47)(17 48)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)
(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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 48)(10 47)(11 46)(12 45)(13 25)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)
(1 7)(2 12)(3 5)(4 10)(6 8)(9 11)(13 18 19 24)(14 23 20 17)(15 16 21 22)(25 31)(26 36)(27 29)(28 34)(30 32)(33 35)(37 38 43 44)(39 48 45 42)(40 41 46 47)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 4P 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 ··· 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 12 12 2 1 1 1 1 2 2 6 ··· 6 12 12 2 2 2 4 4 4 4 4 4 2 2 2 2 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D4 D4 D6 D6 C4×S3 D12 C3⋊D4 D12 C4≀C2 D12⋊C4 kernel C2×D12⋊C4 D12⋊C4 C2×C4×Dic3 C6×M4(2) C2×C4○D12 C2×Dic6 C2×D12 C4○D12 C2×M4(2) C2×C12 C22×C6 M4(2) C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 4 1 1 1 2 2 4 1 3 1 2 1 4 2 4 2 8 4

Matrix representation of C2×D12⋊C4 in GL5(𝔽73)

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

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,72,72,0,0,0,1,0,0,0,0,0,0,27,0,0,0,0,0,46],[72,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,27,0],[27,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,46] >;

C2×D12⋊C4 in GAP, Magma, Sage, TeX

C_2\times D_{12}\rtimes C_4
% in TeX

G:=Group("C2xD12:C4");
// GroupNames label

G:=SmallGroup(192,697);
// by ID

G=gap.SmallGroup(192,697);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,58,136,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^7*c>;
// generators/relations

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