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G = (C2xC12).Q8order 192 = 26·3

8th non-split extension by C2xC12 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2xC12).8Q8, C4.Dic3:4C4, C12.29(C4:C4), (C2xC6).2C42, (C2xC4).3Dic6, C4.47(D6:C4), (C2xC12).105D4, (C2xC4).127D12, (C22xC4).74D6, C4.9(Dic3:C4), C42:C2.3S3, C22:C4.1Dic3, C22.2(C4xDic3), C12.10(C22:C4), C3:1(M4(2):4C4), C22.2(C4:Dic3), C23.11(C2xDic3), C6.9(C2.C42), C2.10(C6.C42), (C22xC12).122C22, C22.11(C6.D4), (C2xC3:C8):2C4, (C2xC6).5(C4:C4), (C2xC4).19(C4xS3), (C2xC12).58(C2xC4), (C3xC22:C4).1C4, (C22xC6).30(C2xC4), (C2xC4.Dic3).9C2, (C2xC4).232(C3:D4), (C2xC6).91(C22:C4), (C3xC42:C2).3C2, SmallGroup(192,92)

Series: Derived Chief Lower central Upper central

C1C2xC6 — (C2xC12).Q8
C1C3C6C12C2xC12C22xC12C2xC4.Dic3 — (C2xC12).Q8
C3C6C2xC6 — (C2xC12).Q8
C1C4C22xC4C42:C2

Generators and relations for (C2xC12).Q8
 G = < a,b,c,d | a2=b12=c4=1, d2=ab9c2, ab=ba, cac-1=dad-1=ab6, cbc-1=b7, dbd-1=b5, dcd-1=ab6c-1 >

Subgroups: 184 in 90 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, C23, C12, C12, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, M4(2), C22xC4, C3:C8, C2xC12, C2xC12, C22xC6, C42:C2, C2xM4(2), C2xC3:C8, C2xC3:C8, C4.Dic3, C4.Dic3, C4xC12, C3xC22:C4, C3xC4:C4, C22xC12, M4(2):4C4, C2xC4.Dic3, C3xC42:C2, (C2xC12).Q8
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Q8, Dic3, D6, C42, C22:C4, C4:C4, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2.C42, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, M4(2):4C4, C6.C42, (C2xC12).Q8

Smallest permutation representation of (C2xC12).Q8
On 48 points
Generators in S48
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 37)(36 38)
(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 8)(4 10)(6 12)(14 20)(16 22)(18 24)(25 45 31 39)(26 40 32 46)(27 47 33 41)(28 42 34 48)(29 37 35 43)(30 44 36 38)
(1 46 23 35 7 40 17 29)(2 39 24 28 8 45 18 34)(3 44 13 33 9 38 19 27)(4 37 14 26 10 43 20 32)(5 42 15 31 11 48 21 25)(6 47 16 36 12 41 22 30)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E8A···8H12A12B12C12D12E···12N
order122223444444444666668···81212121212···12
size1122221122244442224412···1222224···4

42 irreducible representations

dim11111122222222244
type+++++--+-+
imageC1C2C2C4C4C4S3D4Q8Dic3D6Dic6C4xS3D12C3:D4M4(2):4C4(C2xC12).Q8
kernel(C2xC12).Q8C2xC4.Dic3C3xC42:C2C2xC3:C8C4.Dic3C3xC22:C4C42:C2C2xC12C2xC12C22:C4C22xC4C2xC4C2xC4C2xC4C2xC4C3C1
# reps12144413121242424

Matrix representation of (C2xC12).Q8 in GL4(F73) generated by

1000
77200
00720
00661
,
49000
512400
00700
00523
,
13100
07200
00131
00772
,
0010
0001
27000
02700
G:=sub<GL(4,GF(73))| [1,7,0,0,0,72,0,0,0,0,72,66,0,0,0,1],[49,51,0,0,0,24,0,0,0,0,70,52,0,0,0,3],[1,0,0,0,31,72,0,0,0,0,1,7,0,0,31,72],[0,0,27,0,0,0,0,27,1,0,0,0,0,1,0,0] >;

(C2xC12).Q8 in GAP, Magma, Sage, TeX

(C_2\times C_{12}).Q_8
% in TeX

G:=Group("(C2xC12).Q8");
// GroupNames label

G:=SmallGroup(192,92);
// by ID

G=gap.SmallGroup(192,92);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,184,1123,851,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=a*b^9*c^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^6,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=a*b^6*c^-1>;
// generators/relations

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