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G = C2×D12⋊C4  order 192 = 26·3

Direct product of C2 and D12⋊C4

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

Aliases: C2×D12⋊C4, M4(2)⋊23D6, C23.55D12, C6⋊2C4≀C2, C4○D12⋊7C4, (C2×D12)⋊15C4, D12⋊21(C2×C4), C4.14(D6⋊C4), Dic6⋊20(C2×C4), (C2×Dic6)⋊15C4, (C2×C12).177D4, C12.445(C2×D4), (C2×C4).155D12, (C2×M4(2))⋊14S3, (C6×M4(2))⋊22C2, C12.69(C22×C4), (C22×C6).106D4, (C22×C4).367D6, C22.16(C2×D12), C12.68(C22⋊C4), (C2×C12).419C23, C22.52(D6⋊C4), (C4×Dic3)⋊62C22, C4○D12.42C22, (C3×M4(2))⋊35C22, (C22×C12).192C22, C3⋊3(C2×C4≀C2), C4.54(S3×C2×C4), (C2×C4×Dic3)⋊2C2, (C2×C4).85(C4×S3), C2.34(C2×D6⋊C4), (C2×C6).32(C2×D4), C4.136(C2×C3⋊D4), C6.62(C2×C22⋊C4), (C2×C12).112(C2×C4), (C2×C4○D12).14C2, (C2×C4).278(C3⋊D4), (C2×C6).68(C22⋊C4), (C2×C4).512(C22×S3), SmallGroup(192,697)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C2×D12⋊C4
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — C2×C4○D12 — C2×D12⋊C4
Lower central
C3 — C6 — C12 — C2×D12⋊C4
Upper central
C1 — C2×C4 — C22×C4 — C2×M4(2)

Generators and relations for C2×D12⋊C4
 G = < a,b,c,d | a2=b12=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b7c >

Subgroups: 472 in 170 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C4×Dic3, C4×Dic3, C2×C24, C3×M4(2), C3×M4(2), C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C22×Dic3, C2×C3⋊D4, C22×C12, C2×C4≀C2, D12⋊C4, C2×C4×Dic3, C6×M4(2), C2×C4○D12, C2×D12⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C4≀C2, C2×C22⋊C4, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C2×C4≀C2, D12⋊C4, C2×D6⋊C4, C2×D12⋊C4

Smallest permutation representation of C2×D12⋊C4
►On 48 points
Generators in S48
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 44)(14 45)(15 46)(16 47)(17 48)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G···4N4O4P6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F24A···24H
order1222222234444444···44466666888812121212121224···24
size111122121221111226···612122224444442222444···4

48 irreducible representations

dim1111111122222222224
type++++++++++++
imageC1C2C2C2C2C4C4C4S3D4D4D6D6C4×S3D12C3⋊D4D12C4≀C2D12⋊C4
kernelC2×D12⋊C4D12⋊C4C2×C4×Dic3C6×M4(2)C2×C4○D12C2×Dic6C2×D12C4○D12C2×M4(2)C2×C12C22×C6M4(2)C22×C4C2×C4C2×C4C2×C4C23C6C2
# reps1411122413121424284

Matrix representation of C2×D12⋊C4 ►in GL5(𝔽73)

720000
01000
00100
00010
00001
,
10000
072100
072000
000270
000046
,
720000
072000
072100
000027
000460
,
270000
00100
01000
000720
000046

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,72,72,0,0,0,1,0,0,0,0,0,0,27,0,0,0,0,0,46],[72,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,27,0],[27,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,46] >;
 

C2×D12⋊C4 in GAP, Magma, Sage, TeX 

C_2\times D_{12}\rtimes C_4
 
% in TeX
 
G:=Group("C2xD12:C4");
 
// GroupNames label
 
G:=SmallGroup(192,697);
 
// by ID
 
G=gap.SmallGroup(192,697);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,58,136,1684,438,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^12=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^7*c>;
 
// generators/relations
 

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