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G = C12.76C24  order 192 = 26·3

23rd non-split extension by C12 of C24 acting via C24/C23=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.76C24, C4○D4.61D6, (C6×D4).13C4, C3⋊C8.34C23, (C6×Q8).13C4, C3⋊3(Q8○M4(2)), D4.Dic3⋊9C2, C6.49(C23×C4), C4.75(S3×C23), C4○D4.5Dic3, D4.9(C2×Dic3), C12.98(C22×C4), (C2×D4).10Dic3, Q8.15(C2×Dic3), (C2×Q8).11Dic3, (C22×C4).296D6, (C2×C12).554C23, C4.Dic3⋊35C22, C23.16(C2×Dic3), C2.11(C23×Dic3), C4.20(C22×Dic3), C22.2(C22×Dic3), (C22×C12).289C22, (C2×C3⋊C8)⋊21C22, (C3×C4○D4).4C4, (C2×C4○D4).15S3, (C6×C4○D4).10C2, (C3×D4).26(C2×C4), (C3×Q8).28(C2×C4), (C2×C12).136(C2×C4), (C2×C4.Dic3)⋊29C2, (C2×C6).29(C22×C4), (C22×C6).81(C2×C4), (C2×C4).31(C2×Dic3), (C2×C4).635(C22×S3), (C3×C4○D4).49C22, SmallGroup(192,1378)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C12.76C24
Chief series
C1 — C3 — C6 — C12 — C3⋊C8 — C2×C3⋊C8 — D4.Dic3 — C12.76C24
Lower central
C3 — C6 — C12.76C24

Subgroups: 392 in 258 conjugacy classes, 187 normal (17 characteristic)
C1, C2, C2 [×7], C3, C4 [×2], C4 [×6], C22, C22 [×6], C22 [×3], C6, C6 [×7], C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C12 [×2], C12 [×6], C2×C6, C2×C6 [×6], C2×C6 [×3], C2×C8 [×12], M4(2) [×16], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3⋊C8 [×8], C2×C12, C2×C12 [×15], C3×D4 [×12], C3×Q8 [×4], C22×C6 [×3], C2×M4(2) [×6], C8○D4 [×8], C2×C4○D4, C2×C3⋊C8 [×12], C4.Dic3 [×16], C22×C12 [×3], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], Q8○M4(2), C2×C4.Dic3 [×6], D4.Dic3 [×8], C6×C4○D4, C12.76C24

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], Dic3 [×8], D6 [×7], C22×C4 [×14], C24, C2×Dic3 [×28], C22×S3 [×7], C23×C4, C22×Dic3 [×14], S3×C23, Q8○M4(2), C23×Dic3, C12.76C24

Generators and relations
 G = < a,b,c,d,e | a12=c2=d2=e2=1, b2=a9, bab-1=a5, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=a6b, dcd=a6c, ce=ec, de=ed >

Smallest permutation representation
►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)

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

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

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

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

Matrix representation ►G ⊆ GL4(𝔽73) generated by

24000
02400
0030
0003
,
0010
0001
46000
04600
,
464300
342700
004643
003427
,
72000
31100
00720
00311
,
1000
0100
00720
00072
G:=sub<GL(4,GF(73))| [24,0,0,0,0,24,0,0,0,0,3,0,0,0,0,3],[0,0,46,0,0,0,0,46,1,0,0,0,0,1,0,0],[46,34,0,0,43,27,0,0,0,0,46,34,0,0,43,27],[72,31,0,0,0,1,0,0,0,0,72,31,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72] >;
 

54 conjugacy classes

class 1 2A2B···2H 3 4A4B4C···4I6A6B6C6D···6I8A···8P12A12B12C12D12E···12J
order122···23444···46666···68···81212121212···12
size112···22112···22224···46···622224···4

54 irreducible representations

dim111111122222244
type++++++---+
imageC1C2C2C2C4C4C4S3D6Dic3Dic3Dic3D6Q8○M4(2)C12.76C24
kernelC12.76C24C2×C4.Dic3D4.Dic3C6×C4○D4C6×D4C6×Q8C3×C4○D4C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C4○D4C3C1
# reps168162813314424

In GAP, Magma, Sage, TeX 

C_{12}._{76}C_2^4
 
% in TeX
 
G:=Group("C12.76C2^4");
 
// GroupNames label
 
G:=SmallGroup(192,1378);
 
// by ID
 
G=gap.SmallGroup(192,1378);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,387,1123,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^12=c^2=d^2=e^2=1,b^2=a^9,b*a*b^-1=a^5,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=a^6*b,d*c*d=a^6*c,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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