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## G = C22×C3⋊D4order 96 = 25·3

### Direct product of C22 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C22×C3⋊D4
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — C22×C3⋊D4
 Lower central C3 — C6 — C22×C3⋊D4
 Upper central C1 — C23 — C24

Generators and relations for C22×C3⋊D4
G = < a,b,c,d,e | a2=b2=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 498 in 236 conjugacy classes, 105 normal (11 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×11], C22 [×28], S3 [×4], C6, C6 [×6], C6 [×4], C2×C4 [×6], D4 [×16], C23, C23 [×6], C23 [×14], Dic3 [×4], D6 [×4], D6 [×12], C2×C6 [×11], C2×C6 [×12], C22×C4, C2×D4 [×12], C24, C24, C2×Dic3 [×6], C3⋊D4 [×16], C22×S3 [×6], C22×S3 [×4], C22×C6, C22×C6 [×6], C22×C6 [×4], C22×D4, C22×Dic3, C2×C3⋊D4 [×12], S3×C23, C23×C6, C22×C3⋊D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C3⋊D4 [×6], S3×C23, C22×C3⋊D4

Smallest permutation representation of C22×C3⋊D4
On 48 points
Generators in S48
(1 44)(2 41)(3 42)(4 43)(5 46)(6 47)(7 48)(8 45)(9 18)(10 19)(11 20)(12 17)(13 25)(14 26)(15 27)(16 28)(21 35)(22 36)(23 33)(24 34)(29 40)(30 37)(31 38)(32 39)
(1 5)(2 6)(3 7)(4 8)(9 23)(10 24)(11 21)(12 22)(13 32)(14 29)(15 30)(16 31)(17 36)(18 33)(19 34)(20 35)(25 39)(26 40)(27 37)(28 38)(41 47)(42 48)(43 45)(44 46)
(1 15 20)(2 17 16)(3 13 18)(4 19 14)(5 30 35)(6 36 31)(7 32 33)(8 34 29)(9 42 25)(10 26 43)(11 44 27)(12 28 41)(21 46 37)(22 38 47)(23 48 39)(24 40 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)
(1 46)(2 45)(3 48)(4 47)(5 44)(6 43)(7 42)(8 41)(9 32)(10 31)(11 30)(12 29)(13 23)(14 22)(15 21)(16 24)(17 40)(18 39)(19 38)(20 37)(25 33)(26 36)(27 35)(28 34)

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

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

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

36 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 6A ··· 6O order 1 2 ··· 2 2 2 2 2 2 2 2 2 3 4 4 4 4 6 ··· 6 size 1 1 ··· 1 2 2 2 2 6 6 6 6 2 6 6 6 6 2 ··· 2

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 C3⋊D4 kernel C22×C3⋊D4 C22×Dic3 C2×C3⋊D4 S3×C23 C23×C6 C24 C2×C6 C23 C22 # reps 1 1 12 1 1 1 4 7 8

Matrix representation of C22×C3⋊D4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 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 1
,
 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 0 12 0 0 0 0 1 12
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 0 0 4 11 0 0 0 0 2 9
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,4,2,0,0,0,0,11,9],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C22×C3⋊D4 in GAP, Magma, Sage, TeX

C_2^2\times C_3\rtimes D_4
% in TeX

G:=Group("C2^2xC3:D4");
// GroupNames label

G:=SmallGroup(96,219);
// by ID

G=gap.SmallGroup(96,219);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,579,2309]);
// Polycyclic

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

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