Copied to
clipboard

G = C2×D4⋊D4order 128 = 27

Direct product of C2 and D4⋊D4

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2×D4⋊D4, C24.101D4, D47(C2×D4), Q87(C2×D4), (C2×D4)⋊48D4, (C2×Q8)⋊35D4, (C22×D8)⋊6C2, C4⋊C4.3C23, C4.37C22≀C2, (C2×D8)⋊36C22, C4.37(C22×D4), C4⋊D448C22, C22⋊C856C22, (C2×C4).219C24, (C2×C8).127C23, (C2×D4).25C23, C23.850(C2×D4), (C22×C4).712D4, D4⋊C470C22, Q8⋊C464C22, (C2×SD16)⋊67C22, (C22×SD16)⋊17C2, C22.86(C4○D8), (C2×Q8).352C23, C22.116C22≀C2, (C22×C8).134C22, (C23×C4).539C22, (C22×C4).957C23, C22.479(C22×D4), C22.111(C8⋊C22), (C22×D4).325C22, (C22×Q8).465C22, C2.7(C2×C4○D8), C2.9(C2×C8⋊C22), (C2×C4⋊D4)⋊44C2, (C2×C22⋊C8)⋊23C2, (C22×C4○D4)⋊7C2, (C2×D4⋊C4)⋊36C2, (C2×Q8⋊C4)⋊22C2, C2.37(C2×C22≀C2), (C2×C4).1093(C2×D4), (C2×C4○D4)⋊63C22, (C2×C4⋊C4).580C22, SmallGroup(128,1732)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×D4⋊D4
C1C2C22C2×C4C22×C4C23×C4C22×C4○D4 — C2×D4⋊D4
C1C2C2×C4 — C2×D4⋊D4
C1C23C23×C4 — C2×D4⋊D4
C1C2C2C2×C4 — C2×D4⋊D4

Generators and relations for C2×D4⋊D4
 G = < a,b,c,d,e | a2=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 844 in 406 conjugacy classes, 116 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×36], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×36], D4 [×4], D4 [×36], Q8 [×4], Q8 [×6], C23, C23 [×2], C23 [×24], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×4], D8 [×8], SD16 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×18], C2×D4 [×8], C2×D4 [×28], C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×32], C24, C24 [×2], C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C2×C22⋊C4, C2×C4⋊C4, C4⋊D4 [×4], C4⋊D4 [×2], C22×C8 [×2], C2×D8 [×4], C2×D8 [×4], C2×SD16 [×4], C2×SD16 [×4], C23×C4, C23×C4, C22×D4 [×2], C22×D4 [×2], C22×Q8, C2×C4○D4 [×4], C2×C4○D4 [×10], C2×C22⋊C8, C2×D4⋊C4, C2×Q8⋊C4, D4⋊D4 [×8], C2×C4⋊D4, C22×D8, C22×SD16, C22×C4○D4, C2×D4⋊D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C4○D8 [×2], C8⋊C22 [×2], C22×D4 [×3], D4⋊D4 [×4], C2×C22≀C2, C2×C4○D8, C2×C8⋊C22, C2×D4⋊D4

Smallest permutation representation of C2×D4⋊D4
On 64 points
Generators in S64
(1 17)(2 18)(3 19)(4 20)(5 56)(6 53)(7 54)(8 55)(9 16)(10 13)(11 14)(12 15)(21 33)(22 34)(23 35)(24 36)(25 32)(26 29)(27 30)(28 31)(37 51)(38 52)(39 49)(40 50)(41 47)(42 48)(43 45)(44 46)(57 63)(58 64)(59 61)(60 62)
(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 51)(2 50)(3 49)(4 52)(5 21)(6 24)(7 23)(8 22)(9 45)(10 48)(11 47)(12 46)(13 42)(14 41)(15 44)(16 43)(17 37)(18 40)(19 39)(20 38)(25 61)(26 64)(27 63)(28 62)(29 58)(30 57)(31 60)(32 59)(33 56)(34 55)(35 54)(36 53)
(1 31 9 35)(2 30 10 34)(3 29 11 33)(4 32 12 36)(5 40 64 42)(6 39 61 41)(7 38 62 44)(8 37 63 43)(13 22 18 27)(14 21 19 26)(15 24 20 25)(16 23 17 28)(45 55 51 57)(46 54 52 60)(47 53 49 59)(48 56 50 58)
(2 4)(5 63)(6 62)(7 61)(8 64)(10 12)(13 15)(18 20)(21 26)(22 25)(23 28)(24 27)(29 33)(30 36)(31 35)(32 34)(37 40)(38 39)(41 44)(42 43)(45 48)(46 47)(49 52)(50 51)(53 60)(54 59)(55 58)(56 57)

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2M2N2O4A···4H4I4J4K4L4M4N8A···8H
order12···22···2224···44444448···8
size11···14···4882···24444884···4

38 irreducible representations

dim111111111222224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4D4C4○D8C8⋊C22
kernelC2×D4⋊D4C2×C22⋊C8C2×D4⋊C4C2×Q8⋊C4D4⋊D4C2×C4⋊D4C22×D8C22×SD16C22×C4○D4C22×C4C2×D4C2×Q8C24C22C22
# reps111181111344182

Matrix representation of C2×D4⋊D4 in GL6(𝔽17)

100000
010000
0016000
0001600
0000160
0000016
,
1600000
0160000
0016000
0001600
000001
0000160
,
010000
100000
0016200
000100
0000314
00001414
,
0160000
100000
001000
0011600
0000013
0000130
,
100000
0160000
001000
0011600
000010
0000016

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,2,1,0,0,0,0,0,0,3,14,0,0,0,0,14,14],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C2×D4⋊D4 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes D_4
% in TeX

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

G:=SmallGroup(128,1732);
// by ID

G=gap.SmallGroup(128,1732);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,2804,1411,172]);
// 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=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽