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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, C4○D419D6, (C2×D4)⋊42D6, (C2×Q8)⋊34D6, C3⋊C8.14C23, D4⋊D613C2, D4⋊S319C22, Q8.13D67C2, (C2×C12).218D4, C12.427(C2×D4), (C6×D4)⋊46C22, C4.35(S3×C23), (C6×Q8)⋊38C22, Q8.14D613C2, D126C2213C2, C4○D1221C22, (C2×D12)⋊59C22, C35(D8⋊C22), D4.S317C22, C3⋊Q1616C22, (C3×D4).23C23, D4.23(C22×S3), (C22×C6).124D4, (C22×C4).298D6, C6.160(C22×D4), (C3×Q8).23C23, Q8.33(C22×S3), Q8.11D613C2, (C2×C12).557C23, Q82S318C22, (C2×Dic6)⋊69C22, C23.41(C3⋊D4), C4.Dic337C22, (C22×C12).292C22, (C6×C4○D4)⋊4C2, (C2×C4○D4)⋊8S3, (C2×C3⋊C8)⋊23C22, (C2×C4○D12)⋊31C2, (C2×C6).591(C2×D4), C4.121(C2×C3⋊D4), (C3×C4○D4)⋊18C22, (C2×C4.Dic3)⋊31C2, C22.21(C2×C3⋊D4), C2.33(C22×C3⋊D4), (C2×C4).203(C3⋊D4), (C2×C4).246(C22×S3), SmallGroup(192,1381)

Series: Derived Chief Lower central Upper central

C1C12 — C12.C24
C1C3C6C12D12C2×D12C2×C4○D12 — C12.C24
C3C6C12 — C12.C24
C1C4C22×C4C2×C4○D4

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 [×7], C3, C4 [×4], C4 [×4], C22 [×3], C22 [×9], S3 [×2], C6, C6 [×5], C8 [×4], C2×C4 [×6], C2×C4 [×10], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23, C23 [×2], Dic3 [×2], C12 [×4], C12 [×2], D6 [×4], C2×C6 [×3], C2×C6 [×5], C2×C8 [×2], M4(2) [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×4], C4○D4 [×8], C3⋊C8 [×4], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×6], C2×C12 [×5], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8, C22×S3, C22×C6, C22×C6, C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4, C2×C4○D4, C2×C3⋊C8 [×2], C4.Dic3 [×4], D4⋊S3 [×4], D4.S3 [×4], Q82S3 [×4], C3⋊Q16 [×4], C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4 [×4], C3×C4○D4 [×2], D8⋊C22, C2×C4.Dic3, D126C22 [×2], Q8.11D6 [×2], D4⋊D6 [×2], Q8.13D6 [×4], Q8.14D6 [×2], C2×C4○D12, C6×C4○D4, C12.C24
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C3⋊D4 [×6], S3×C23, D8⋊C22, C22×C3⋊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 24)(14 23)(15 22)(16 21)(17 20)(18 19)(26 36)(27 35)(28 34)(29 33)(30 32)(37 42)(38 41)(39 40)(43 48)(44 47)(45 46)
(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 40 19 46)(14 41 20 47)(15 42 21 48)(16 43 22 37)(17 44 23 38)(18 45 24 39)
(1 20)(2 15)(3 22)(4 17)(5 24)(6 19)(7 14)(8 21)(9 16)(10 23)(11 18)(12 13)(25 41)(26 48)(27 43)(28 38)(29 45)(30 40)(31 47)(32 42)(33 37)(34 44)(35 39)(36 46)

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

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

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

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.C24C2×C4.Dic3D126C22Q8.11D6D4⋊D6Q8.13D6Q8.14D6C2×C4○D12C6×C4○D4C2×C4○D4C2×C12C22×C6C22×C4C2×D4C2×Q8C4○D4C2×C4C23C3C1
# 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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