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## G = C3×2- 1+4order 96 = 25·3

### Direct product of C3 and 2- 1+4

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

Aliases: C3×2- 1+4, C6.20C24, C12.52C23, Q8(C3×D4), D4(C3×Q8), C4○D46C6, (C2×Q8)⋊7C6, (C6×Q8)⋊12C2, D4.4(C2×C6), Q8.7(C2×C6), (C2×C6).8C23, C2.5(C23×C6), C4.10(C22×C6), (C2×C12).71C22, (C3×D4).14C22, C22.3(C22×C6), (C3×Q8).15C22, (C3×D4)(C3×Q8), (C3×C4○D4)⋊9C2, (C2×C4).12(C2×C6), SmallGroup(96,225)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×2- 1+4
 Chief series C1 — C2 — C6 — C2×C6 — C3×D4 — C3×C4○D4 — C3×2- 1+4
 Lower central C1 — C2 — C3×2- 1+4
 Upper central C1 — C6 — C3×2- 1+4

Generators and relations for C3×2- 1+4
G = < a,b,c,d,e | a3=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 156 in 146 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2 [×5], C3, C4 [×10], C22 [×5], C6, C6 [×5], C2×C4 [×15], D4 [×10], Q8 [×10], C12 [×10], C2×C6 [×5], C2×Q8 [×5], C4○D4 [×10], C2×C12 [×15], C3×D4 [×10], C3×Q8 [×10], 2- 1+4, C6×Q8 [×5], C3×C4○D4 [×10], C3×2- 1+4
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], C2×C6 [×35], C24, C22×C6 [×15], 2- 1+4, C23×C6, C3×2- 1+4

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

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

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

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

C3×2- 1+4 is a maximal subgroup of   2- 1+44S3  2- 1+4.2S3  D12.34C23  D12.35C23  D12.39C23  2- 1+4⋊C9
C3×2- 1+4 is a maximal quotient of   C3×D4×Q8

51 conjugacy classes

 class 1 2A 2B ··· 2F 3A 3B 4A ··· 4J 6A 6B 6C ··· 6L 12A ··· 12T order 1 2 2 ··· 2 3 3 4 ··· 4 6 6 6 ··· 6 12 ··· 12 size 1 1 2 ··· 2 1 1 2 ··· 2 1 1 2 ··· 2 2 ··· 2

51 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + - image C1 C2 C2 C3 C6 C6 2- 1+4 C3×2- 1+4 kernel C3×2- 1+4 C6×Q8 C3×C4○D4 2- 1+4 C2×Q8 C4○D4 C3 C1 # reps 1 5 10 2 10 20 1 2

Matrix representation of C3×2- 1+4 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 5 6 2 6 5 2 1 4 0 0 4 6 0 0 3 3
,
 5 0 6 0 1 5 4 1 3 0 2 0 6 4 1 2
,
 5 6 5 0 5 2 5 5 0 0 3 1 0 0 4 4
,
 1 6 1 4 1 3 2 6 2 5 4 6 1 5 1 6
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[5,5,0,0,6,2,0,0,2,1,4,3,6,4,6,3],[5,1,3,6,0,5,0,4,6,4,2,1,0,1,0,2],[5,5,0,0,6,2,0,0,5,5,3,4,0,5,1,4],[1,1,2,1,6,3,5,5,1,2,4,1,4,6,6,6] >;

C3×2- 1+4 in GAP, Magma, Sage, TeX

C_3\times 2_-^{1+4}
% in TeX

G:=Group("C3xES-(2,2)");
// GroupNames label

G:=SmallGroup(96,225);
// by ID

G=gap.SmallGroup(96,225);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-2,601,295,476,230,1347]);
// Polycyclic

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

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