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G = C2×D126C22order 192 = 26·3

Direct product of C2 and D126C22

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D126C22, D127C23, C12.28C24, Dic66C23, C3⋊C84C23, (C2×D4)⋊34D6, C64(C8⋊C22), (C22×D4)⋊7S3, D4⋊S317C22, C12.249(C2×D4), (C2×C12).207D4, (C6×D4)⋊42C22, C4.28(S3×C23), C4○D1219C22, (C2×D12)⋊55C22, D4.S316C22, D4.20(C22×S3), (C3×D4).20C23, (C22×C6).207D4, C6.137(C22×D4), (C22×C4).284D6, (C2×C12).537C23, (C2×Dic6)⋊65C22, C23.99(C3⋊D4), C4.Dic332C22, (C22×C12).270C22, (D4×C2×C6)⋊3C2, C35(C2×C8⋊C22), (C2×D4⋊S3)⋊30C2, (C2×C3⋊C8)⋊20C22, C4.21(C2×C3⋊D4), (C2×C4○D12)⋊28C2, (C2×D4.S3)⋊30C2, (C2×C6).577(C2×D4), (C2×C4).92(C3⋊D4), (C2×C4.Dic3)⋊26C2, C2.10(C22×C3⋊D4), (C2×C4).235(C22×S3), C22.106(C2×C3⋊D4), SmallGroup(192,1352)

Series: Derived Chief Lower central Upper central

C1C12 — C2×D126C22
C1C3C6C12D12C2×D12C2×C4○D12 — C2×D126C22
C3C6C12 — C2×D126C22
C1C22C22×C4C22×D4

Generators and relations for C2×D126C22
 G = < a,b,c,d,e | a2=b12=c2=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b7, dcd=b6c, ece=b3c, de=ed >

Subgroups: 744 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], S3 [×2], C6, C6 [×2], C6 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×4], D4 [×13], Q8 [×3], C23, C23 [×11], Dic3 [×2], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×18], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×5], C2×Q8, C4○D4 [×6], C24, C3⋊C8 [×4], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C3×D4 [×4], C3×D4 [×6], C22×S3, C22×C6, C22×C6 [×10], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C2×C3⋊C8 [×2], C4.Dic3 [×4], D4⋊S3 [×8], D4.S3 [×8], C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C2×C3⋊D4, C22×C12, C6×D4 [×6], C6×D4 [×3], C23×C6, C2×C8⋊C22, C2×C4.Dic3, C2×D4⋊S3 [×2], D126C22 [×8], C2×D4.S3 [×2], C2×C4○D12, D4×C2×C6, C2×D126C22
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C8⋊C22 [×2], C22×D4, C2×C3⋊D4 [×6], S3×C23, C2×C8⋊C22, D126C22 [×2], C22×C3⋊D4, C2×D126C22

Smallest permutation representation of C2×D126C22
On 48 points
Generators in S48
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 13)(12 14)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 48)
(1 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(1 15)(2 22)(3 17)(4 24)(5 19)(6 14)(7 21)(8 16)(9 23)(10 18)(11 13)(12 20)(25 43)(26 38)(27 45)(28 40)(29 47)(30 42)(31 37)(32 44)(33 39)(34 46)(35 41)(36 48)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D4E4F6A···6G6H···6O8A8B8C8D12A12B12C12D
order12222222222234444446···66···6888812121212
size111122444412122222212122···24···4121212124444

42 irreducible representations

dim1111111222222244
type+++++++++++++
imageC1C2C2C2C2C2C2S3D4D4D6D6C3⋊D4C3⋊D4C8⋊C22D126C22
kernelC2×D126C22C2×C4.Dic3C2×D4⋊S3D126C22C2×D4.S3C2×C4○D12D4×C2×C6C22×D4C2×C12C22×C6C22×C4C2×D4C2×C4C23C6C2
# reps1128211131166224

Matrix representation of C2×D126C22 in GL6(𝔽73)

7200000
0720000
0072000
0007200
0000720
0000072
,
800000
29640000
0086000
00666500
0000931
00005664
,
52400000
62210000
0000931
00005664
0086000
00666500
,
100000
010000
001000
000100
0000720
0000072
,
100000
010000
00722900
000100
0000720
0000101

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[8,29,0,0,0,0,0,64,0,0,0,0,0,0,8,66,0,0,0,0,60,65,0,0,0,0,0,0,9,56,0,0,0,0,31,64],[52,62,0,0,0,0,40,21,0,0,0,0,0,0,0,0,8,66,0,0,0,0,60,65,0,0,9,56,0,0,0,0,31,64,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,29,1,0,0,0,0,0,0,72,10,0,0,0,0,0,1] >;

C2×D126C22 in GAP, Magma, Sage, TeX

C_2\times D_{12}\rtimes_6C_2^2
% in TeX

G:=Group("C2xD12:6C2^2");
// GroupNames label

G:=SmallGroup(192,1352);
// by ID

G=gap.SmallGroup(192,1352);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,1684,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^12=c^2=d^2=e^2=1,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,e*b*e=b^7,d*c*d=b^6*c,e*c*e=b^3*c,d*e=e*d>;
// generators/relations

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