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G = C2×D46D4order 128 = 27

Direct product of C2 and D46D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×D46D4, C22.53C25, C42.552C23, C24.491C23, C23.124C24, C22.792- 1+4, (C2×D4)⋊56D4, D411(C2×D4), C4⋊Q881C22, (C2×C4).55C24, C2.20(D4×C23), (C4×D4)⋊102C22, C4⋊C4.465C23, C4⋊D470C22, C23.484(C2×D4), C4.109(C22×D4), C22⋊Q883C22, C22.7(C22×D4), (C2×D4).450C23, C22⋊C4.12C23, (C2×Q8).429C23, (C2×C42).924C22, (C23×C4).593C22, C2.11(C2×2- 1+4), (C22×C4).1190C23, (C22×D4).614C22, C22.D441C22, (C22×Q8).491C22, C4⋊C43(C2×D4), D42(C2×C4⋊C4), (C2×C4×D4)⋊80C2, C42(C2×C4○D4), (C2×C4⋊Q8)⋊50C2, C4⋊C42(C22×D4), (C2×C4)⋊17(C4○D4), (C2×C4⋊D4)⋊60C2, (C22×C4⋊C4)⋊43C2, (C2×C22⋊Q8)⋊68C2, (C2×C4⋊C4)⋊133C22, (C2×C4).1109(C2×D4), (C22×C4○D4)⋊17C2, (C2×C4○D4)⋊71C22, C2.26(C22×C4○D4), C22.157(C2×C4○D4), (C2×C22.D4)⋊55C2, (C2×C22⋊C4).376C22, SmallGroup(128,2196)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×D46D4
C1C2C22C23C24C23×C4C22×C4○D4 — C2×D46D4
C1C22 — C2×D46D4
C1C23 — C2×D46D4
C1C22 — C2×D46D4

Generators and relations for C2×D46D4
 G = < a,b,c,d,e | a2=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 1212 in 816 conjugacy classes, 444 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×12], C4 [×8], C4 [×18], C22, C22 [×14], C22 [×44], C2×C4 [×30], C2×C4 [×82], D4 [×16], D4 [×40], Q8 [×16], C23, C23 [×16], C23 [×20], C42 [×4], C22⋊C4 [×32], C4⋊C4 [×40], C22×C4 [×3], C22×C4 [×40], C22×C4 [×32], C2×D4 [×32], C2×D4 [×20], C2×Q8 [×8], C2×Q8 [×8], C4○D4 [×64], C24 [×4], C2×C42, C2×C22⋊C4 [×8], C2×C4⋊C4 [×2], C2×C4⋊C4 [×24], C4×D4 [×16], C4⋊D4 [×16], C22⋊Q8 [×16], C22.D4 [×32], C4⋊Q8 [×8], C23×C4 [×8], C22×D4 [×2], C22×D4 [×4], C22×Q8 [×2], C2×C4○D4 [×16], C2×C4○D4 [×16], C22×C4⋊C4 [×2], C2×C4×D4 [×2], C2×C4⋊D4 [×2], C2×C22⋊Q8 [×2], C2×C22.D4 [×4], C2×C4⋊Q8, D46D4 [×16], C22×C4○D4 [×2], C2×D46D4
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C4○D4 [×4], C24 [×31], C22×D4 [×14], C2×C4○D4 [×6], 2- 1+4 [×2], C25, D46D4 [×4], D4×C23, C22×C4○D4, C2×2- 1+4, C2×D46D4

Smallest permutation representation of C2×D46D4
On 64 points
Generators in S64
(1 15)(2 16)(3 13)(4 14)(5 32)(6 29)(7 30)(8 31)(9 35)(10 36)(11 33)(12 34)(17 39)(18 40)(19 37)(20 38)(21 27)(22 28)(23 25)(24 26)(41 59)(42 60)(43 57)(44 58)(45 55)(46 56)(47 53)(48 54)(49 62)(50 63)(51 64)(52 61)
(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)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 34)(2 33)(3 36)(4 35)(5 25)(6 28)(7 27)(8 26)(9 14)(10 13)(11 16)(12 15)(17 53)(18 56)(19 55)(20 54)(21 30)(22 29)(23 32)(24 31)(37 45)(38 48)(39 47)(40 46)(41 62)(42 61)(43 64)(44 63)(49 59)(50 58)(51 57)(52 60)
(1 51 25 19)(2 52 26 20)(3 49 27 17)(4 50 28 18)(5 55 34 57)(6 56 35 58)(7 53 36 59)(8 54 33 60)(9 44 29 46)(10 41 30 47)(11 42 31 48)(12 43 32 45)(13 62 21 39)(14 63 22 40)(15 64 23 37)(16 61 24 38)
(1 62)(2 63)(3 64)(4 61)(5 45)(6 46)(7 47)(8 48)(9 58)(10 59)(11 60)(12 57)(13 51)(14 52)(15 49)(16 50)(17 23)(18 24)(19 21)(20 22)(25 39)(26 40)(27 37)(28 38)(29 56)(30 53)(31 54)(32 55)(33 42)(34 43)(35 44)(36 41)

G:=sub<Sym(64)| (1,15)(2,16)(3,13)(4,14)(5,32)(6,29)(7,30)(8,31)(9,35)(10,36)(11,33)(12,34)(17,39)(18,40)(19,37)(20,38)(21,27)(22,28)(23,25)(24,26)(41,59)(42,60)(43,57)(44,58)(45,55)(46,56)(47,53)(48,54)(49,62)(50,63)(51,64)(52,61), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,34)(2,33)(3,36)(4,35)(5,25)(6,28)(7,27)(8,26)(9,14)(10,13)(11,16)(12,15)(17,53)(18,56)(19,55)(20,54)(21,30)(22,29)(23,32)(24,31)(37,45)(38,48)(39,47)(40,46)(41,62)(42,61)(43,64)(44,63)(49,59)(50,58)(51,57)(52,60), (1,51,25,19)(2,52,26,20)(3,49,27,17)(4,50,28,18)(5,55,34,57)(6,56,35,58)(7,53,36,59)(8,54,33,60)(9,44,29,46)(10,41,30,47)(11,42,31,48)(12,43,32,45)(13,62,21,39)(14,63,22,40)(15,64,23,37)(16,61,24,38), (1,62)(2,63)(3,64)(4,61)(5,45)(6,46)(7,47)(8,48)(9,58)(10,59)(11,60)(12,57)(13,51)(14,52)(15,49)(16,50)(17,23)(18,24)(19,21)(20,22)(25,39)(26,40)(27,37)(28,38)(29,56)(30,53)(31,54)(32,55)(33,42)(34,43)(35,44)(36,41)>;

G:=Group( (1,15)(2,16)(3,13)(4,14)(5,32)(6,29)(7,30)(8,31)(9,35)(10,36)(11,33)(12,34)(17,39)(18,40)(19,37)(20,38)(21,27)(22,28)(23,25)(24,26)(41,59)(42,60)(43,57)(44,58)(45,55)(46,56)(47,53)(48,54)(49,62)(50,63)(51,64)(52,61), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,34)(2,33)(3,36)(4,35)(5,25)(6,28)(7,27)(8,26)(9,14)(10,13)(11,16)(12,15)(17,53)(18,56)(19,55)(20,54)(21,30)(22,29)(23,32)(24,31)(37,45)(38,48)(39,47)(40,46)(41,62)(42,61)(43,64)(44,63)(49,59)(50,58)(51,57)(52,60), (1,51,25,19)(2,52,26,20)(3,49,27,17)(4,50,28,18)(5,55,34,57)(6,56,35,58)(7,53,36,59)(8,54,33,60)(9,44,29,46)(10,41,30,47)(11,42,31,48)(12,43,32,45)(13,62,21,39)(14,63,22,40)(15,64,23,37)(16,61,24,38), (1,62)(2,63)(3,64)(4,61)(5,45)(6,46)(7,47)(8,48)(9,58)(10,59)(11,60)(12,57)(13,51)(14,52)(15,49)(16,50)(17,23)(18,24)(19,21)(20,22)(25,39)(26,40)(27,37)(28,38)(29,56)(30,53)(31,54)(32,55)(33,42)(34,43)(35,44)(36,41) );

G=PermutationGroup([(1,15),(2,16),(3,13),(4,14),(5,32),(6,29),(7,30),(8,31),(9,35),(10,36),(11,33),(12,34),(17,39),(18,40),(19,37),(20,38),(21,27),(22,28),(23,25),(24,26),(41,59),(42,60),(43,57),(44,58),(45,55),(46,56),(47,53),(48,54),(49,62),(50,63),(51,64),(52,61)], [(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),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,34),(2,33),(3,36),(4,35),(5,25),(6,28),(7,27),(8,26),(9,14),(10,13),(11,16),(12,15),(17,53),(18,56),(19,55),(20,54),(21,30),(22,29),(23,32),(24,31),(37,45),(38,48),(39,47),(40,46),(41,62),(42,61),(43,64),(44,63),(49,59),(50,58),(51,57),(52,60)], [(1,51,25,19),(2,52,26,20),(3,49,27,17),(4,50,28,18),(5,55,34,57),(6,56,35,58),(7,53,36,59),(8,54,33,60),(9,44,29,46),(10,41,30,47),(11,42,31,48),(12,43,32,45),(13,62,21,39),(14,63,22,40),(15,64,23,37),(16,61,24,38)], [(1,62),(2,63),(3,64),(4,61),(5,45),(6,46),(7,47),(8,48),(9,58),(10,59),(11,60),(12,57),(13,51),(14,52),(15,49),(16,50),(17,23),(18,24),(19,21),(20,22),(25,39),(26,40),(27,37),(28,38),(29,56),(30,53),(31,54),(32,55),(33,42),(34,43),(35,44),(36,41)])

50 conjugacy classes

class 1 2A···2G2H···2O2P2Q2R2S4A···4P4Q···4AD
order12···22···222224···44···4
size11···12···244442···24···4

50 irreducible representations

dim111111111224
type++++++++++-
imageC1C2C2C2C2C2C2C2C2D4C4○D42- 1+4
kernelC2×D46D4C22×C4⋊C4C2×C4×D4C2×C4⋊D4C2×C22⋊Q8C2×C22.D4C2×C4⋊Q8D46D4C22×C4○D4C2×D4C2×C4C22
# reps1222241162882

Matrix representation of C2×D46D4 in GL5(𝔽5)

40000
01000
00100
00010
00001
,
10000
04000
00400
00022
00003
,
40000
01000
00100
00022
00013
,
40000
02200
00300
00010
00001
,
40000
03300
04200
00044
00001

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,2,3],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,1,0,0,0,2,3],[4,0,0,0,0,0,2,0,0,0,0,2,3,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,3,4,0,0,0,3,2,0,0,0,0,0,4,0,0,0,0,4,1] >;

C2×D46D4 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_6D_4
% in TeX

G:=Group("C2xD4:6D4");
// GroupNames label

G:=SmallGroup(128,2196);
// by ID

G=gap.SmallGroup(128,2196);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,184,570]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^4=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,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

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