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## G = C3×C22⋊Q8order 96 = 25·3

### Direct product of C3 and C22⋊Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C22⋊Q8
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C6×Q8 — C3×C22⋊Q8
 Lower central C1 — C22 — C3×C22⋊Q8
 Upper central C1 — C2×C6 — C3×C22⋊Q8

Generators and relations for C3×C22⋊Q8
G = < a,b,c,d,e | a3=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 100 in 74 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×C12, C2×C12, C2×C12, C3×Q8, C22×C6, C22⋊Q8, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C6×Q8, C3×C22⋊Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, C2×D4, C2×Q8, C4○D4, C3×D4, C3×Q8, C22×C6, C22⋊Q8, C6×D4, C6×Q8, C3×C4○D4, C3×C22⋊Q8

Smallest permutation representation of C3×C22⋊Q8
On 48 points
Generators in S48
(1 5 19)(2 6 20)(3 7 17)(4 8 18)(9 27 21)(10 28 22)(11 25 23)(12 26 24)(13 46 36)(14 47 33)(15 48 34)(16 45 35)(29 40 43)(30 37 44)(31 38 41)(32 39 42)
(13 37)(14 38)(15 39)(16 40)(29 35)(30 36)(31 33)(32 34)(41 47)(42 48)(43 45)(44 46)
(1 11)(2 12)(3 9)(4 10)(5 25)(6 26)(7 27)(8 28)(13 37)(14 38)(15 39)(16 40)(17 21)(18 22)(19 23)(20 24)(29 35)(30 36)(31 33)(32 34)(41 47)(42 48)(43 45)(44 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)
(1 31 3 29)(2 30 4 32)(5 38 7 40)(6 37 8 39)(9 35 11 33)(10 34 12 36)(13 28 15 26)(14 27 16 25)(17 43 19 41)(18 42 20 44)(21 45 23 47)(22 48 24 46)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 Q8 C4○D4 C3×D4 C3×Q8 C3×C4○D4 kernel C3×C22⋊Q8 C3×C22⋊C4 C3×C4⋊C4 C22×C12 C6×Q8 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C12 C2×C6 C6 C4 C22 C2 # reps 1 2 3 1 1 2 4 6 2 2 2 2 2 4 4 4

Matrix representation of C3×C22⋊Q8 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 0 1 0 0 12 0 0 0 0 0 12 0 0 0 0 12
,
 9 10 0 0 10 4 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[0,12,0,0,1,0,0,0,0,0,12,0,0,0,0,12],[9,10,0,0,10,4,0,0,0,0,0,1,0,0,1,0] >;

C3×C22⋊Q8 in GAP, Magma, Sage, TeX

C_3\times C_2^2\rtimes Q_8
% in TeX

G:=Group("C3xC2^2:Q8");
// GroupNames label

G:=SmallGroup(96,169);
// by ID

G=gap.SmallGroup(96,169);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,144,313,151,938]);
// Polycyclic

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

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