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G = C12.C24order 192 = 26·3

35th non-split extension by C12 of C24 acting via C24/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.35C24, D12.31C23, Dic6.30C23, C4oD4:19D6, (C2xD4):42D6, (C2xQ8):34D6, C3:C8.14C23, D4:D6:13C2, D4:S3:19C22, Q8.13D6:7C2, (C2xC12).218D4, C12.427(C2xD4), (C6xD4):46C22, C4.35(S3xC23), (C6xQ8):38C22, Q8.14D6:13C2, D12:6C22:13C2, C4oD12:21C22, (C2xD12):59C22, C3:5(D8:C22), D4.S3:17C22, C3:Q16:16C22, (C3xD4).23C23, D4.23(C22xS3), (C22xC6).124D4, (C22xC4).298D6, C6.160(C22xD4), (C3xQ8).23C23, Q8.33(C22xS3), Q8.11D6:13C2, (C2xC12).557C23, Q8:2S3:18C22, (C2xDic6):69C22, C23.41(C3:D4), C4.Dic3:37C22, (C22xC12).292C22, (C6xC4oD4):4C2, (C2xC4oD4):8S3, (C2xC3:C8):23C22, (C2xC4oD12):31C2, (C2xC6).591(C2xD4), C4.121(C2xC3:D4), (C3xC4oD4):18C22, (C2xC4.Dic3):31C2, C22.21(C2xC3:D4), C2.33(C22xC3:D4), (C2xC4).203(C3:D4), (C2xC4).246(C22xS3), SmallGroup(192,1381)

Series: Derived Chief Lower central Upper central

C1C12 — C12.C24
C1C3C6C12D12C2xD12C2xC4oD12 — C12.C24
C3C6C12 — C12.C24
C1C4C22xC4C2xC4oD4

Generators and relations for C12.C24
 G = < a,b,c,d,e | a12=b2=c2=e2=1, d2=a6, bab=a-1, ac=ca, ad=da, eae=a7, bc=cb, bd=db, ebe=a9b, cd=dc, ece=a6c, de=ed >

Subgroups: 616 in 262 conjugacy classes, 107 normal (45 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C2xC8, M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C4oD4, C4oD4, C3:C8, Dic6, Dic6, C4xS3, D12, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C3xQ8, C22xS3, C22xC6, C22xC6, C2xM4(2), C4oD8, C8:C22, C8.C22, C2xC4oD4, C2xC4oD4, C2xC3:C8, C4.Dic3, D4:S3, D4.S3, Q8:2S3, C3:Q16, C2xDic6, S3xC2xC4, C2xD12, C4oD12, C4oD12, C2xC3:D4, C22xC12, C22xC12, C6xD4, C6xD4, C6xQ8, C3xC4oD4, C3xC4oD4, D8:C22, C2xC4.Dic3, D12:6C22, Q8.11D6, D4:D6, Q8.13D6, Q8.14D6, C2xC4oD12, C6xC4oD4, C12.C24
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C24, C3:D4, C22xS3, C22xD4, C2xC3:D4, S3xC23, D8:C22, C22xC3:D4, C12.C24

Smallest permutation representation of C12.C24
On 48 points
Generators in S48
(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 12)(3 11)(4 10)(5 9)(6 8)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)(26 36)(27 35)(28 34)(29 33)(30 32)(37 46)(38 45)(39 44)(40 43)(41 42)(47 48)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 31 7 25)(2 32 8 26)(3 33 9 27)(4 34 10 28)(5 35 11 29)(6 36 12 30)(13 43 19 37)(14 44 20 38)(15 45 21 39)(16 46 22 40)(17 47 23 41)(18 48 24 42)
(1 13)(2 20)(3 15)(4 22)(5 17)(6 24)(7 19)(8 14)(9 21)(10 16)(11 23)(12 18)(25 37)(26 44)(27 39)(28 46)(29 41)(30 48)(31 43)(32 38)(33 45)(34 40)(35 47)(36 42)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H 3 4A4B4C4D4E4F4G4H4I6A6B6C6D···6I8A8B8C8D12A12B12C12D12E···12J
order12222222234444444446666···688881212121212···12
size112224412122112224412122224···41212121222224···4

42 irreducible representations

dim11111111122222222244
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D4D4D6D6D6D6C3:D4C3:D4D8:C22C12.C24
kernelC12.C24C2xC4.Dic3D12:6C22Q8.11D6D4:D6Q8.13D6Q8.14D6C2xC4oD12C6xC4oD4C2xC4oD4C2xC12C22xC6C22xC4C2xD4C2xQ8C4oD4C2xC4C23C3C1
# reps11222421113111146224

Matrix representation of C12.C24 in GL4(F73) generated by

14700
66700
005966
00766
,
07200
72000
005966
00714
,
72000
07200
0010
0001
,
46000
04600
00460
00046
,
0010
0001
1000
0100
G:=sub<GL(4,GF(73))| [14,66,0,0,7,7,0,0,0,0,59,7,0,0,66,66],[0,72,0,0,72,0,0,0,0,0,59,7,0,0,66,14],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[46,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;

C12.C24 in GAP, Magma, Sage, TeX

C_{12}.C_2^4
% in TeX

G:=Group("C12.C2^4");
// GroupNames label

G:=SmallGroup(192,1381);
// by ID

G=gap.SmallGroup(192,1381);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,570,1684,235,102,6278]);
// Polycyclic

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

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