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G = C2×C4.Dic3order 96 = 25·3

Direct product of C2 and C4.Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C4.Dic3
 Chief series C1 — C3 — C6 — C12 — C3⋊C8 — C2×C3⋊C8 — C2×C4.Dic3
 Lower central C3 — C6 — C2×C4.Dic3
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×C4.Dic3
G = < a,b,c,d | a2=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 98 in 68 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C3⋊C8 [×4], C2×C12 [×2], C2×C12 [×4], C22×C6, C2×M4(2), C2×C3⋊C8 [×2], C4.Dic3 [×4], C22×C12, C2×C4.Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], M4(2) [×2], C22×C4, C2×Dic3 [×6], C22×S3, C2×M4(2), C4.Dic3 [×2], C22×Dic3, C2×C4.Dic3

Smallest permutation representation of C2×C4.Dic3
On 48 points
Generators in S48
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 37)(33 38)(34 39)(35 40)(36 41)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 40 43 46)(38 41 44 47)(39 42 45 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)
(1 25 10 34 7 31 4 28)(2 30 11 27 8 36 5 33)(3 35 12 32 9 29 6 26)(13 45 22 42 19 39 16 48)(14 38 23 47 20 44 17 41)(15 43 24 40 21 37 18 46)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C2 C4 C4 S3 Dic3 D6 Dic3 M4(2) C4.Dic3 kernel C2×C4.Dic3 C2×C3⋊C8 C4.Dic3 C22×C12 C2×C12 C22×C6 C22×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 2 4 1 6 2 1 3 3 1 4 8

Matrix representation of C2×C4.Dic3 in GL3(𝔽73) generated by

 72 0 0 0 72 0 0 0 72
,
 1 0 0 0 46 0 0 0 27
,
 72 0 0 0 3 0 0 0 24
,
 46 0 0 0 0 1 0 46 0
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[1,0,0,0,46,0,0,0,27],[72,0,0,0,3,0,0,0,24],[46,0,0,0,0,46,0,1,0] >;

C2×C4.Dic3 in GAP, Magma, Sage, TeX

C_2\times C_4.{\rm Dic}_3
% in TeX

G:=Group("C2xC4.Dic3");
// GroupNames label

G:=SmallGroup(96,128);
// by ID

G=gap.SmallGroup(96,128);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,362,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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