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G = C3×C22.D4  order 96 = 25·3

Direct product of C3 and C22.D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C3×C22.D4, C4⋊C4⋊4C6, C2.7(C6×D4), C22⋊C4⋊4C6, (C22×C4)⋊5C6, (C2×D4).4C6, (C2×C6).23D4, C6.70(C2×D4), (C22×C12)⋊5C2, (C6×D4).11C2, C22.4(C3×D4), C6.43(C4○D4), C23.13(C2×C6), (C2×C6).78C23, (C2×C12).65C22, (C22×C6).29C22, C22.13(C22×C6), (C3×C4⋊C4)⋊13C2, (C2×C4).5(C2×C6), C2.6(C3×C4○D4), (C3×C22⋊C4)⋊12C2, SmallGroup(96,170)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3×C22.D4
Chief series
C1 — C2 — C22 — C2×C6 — C22×C6 — C6×D4 — C3×C22.D4
Lower central
C1 — C22 — C3×C22.D4
Upper central
C1 — C2×C6 — C3×C22.D4

Generators and relations for C3×C22.D4
 G = < a,b,c,d,e | a3=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >

Subgroups: 116 in 78 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C22.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C3×C22.D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C22.D4, C6×D4, C3×C4○D4, C3×C22.D4

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

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

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

(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)

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

C3×C22.D4 is a maximal subgroup of
 (C22×C12)⋊C4  C22⋊C4⋊D6  C6.792- 1+4  C4⋊C4.197D6  C6.802- 1+4  C6.812- 1+4  C6.1202+ 1+4  C6.1212+ 1+4  C6.822- 1+4  C4⋊C4⋊28D6  C6.612+ 1+4  C6.1222+ 1+4  C6.622+ 1+4  C6.632+ 1+4  C6.642+ 1+4  C6.652+ 1+4  C6.662+ 1+4  C6.672+ 1+4  C6.852- 1+4  C6.682+ 1+4  C6.692+ 1+4

42 conjugacy classes

class 1 2A2B2C2D2E2F3A3B4A4B4C4D4E4F4G6A···6F6G6H6I6J6K6L12A···12H12I···12N
order12222223344444446···666666612···1212···12
size11112241122224441···12222442···24···4

42 irreducible representations

dim11111111112222
type++++++
imageC1C2C2C2C2C3C6C6C6C6D4C4○D4C3×D4C3×C4○D4
kernelC3×C22.D4C3×C22⋊C4C3×C4⋊C4C22×C12C6×D4C22.D4C22⋊C4C4⋊C4C22×C4C2×D4C2×C6C6C22C2
# reps13211264222448

Matrix representation of C3×C22.D4 ►in GL4(𝔽13) generated by

3000
0300
0010
0001
,
71100
11600
0010
0001
,
12000
01200
0010
0001
,
5000
9800
0012
001212
,
1000
71200
0010
001212
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[7,11,0,0,11,6,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[5,9,0,0,0,8,0,0,0,0,1,12,0,0,2,12],[1,7,0,0,0,12,0,0,0,0,1,12,0,0,0,12] >;
 

C3×C22.D4 in GAP, Magma, Sage, TeX 

C_3\times C_2^2.D_4
 
% in TeX
 
G:=Group("C3xC2^2.D4");
 
// GroupNames label
 
G:=SmallGroup(96,170);
 
// by ID
 
G=gap.SmallGroup(96,170);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,938,122]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;
 
// generators/relations
 

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