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G = C4214D4order 128 = 27

8th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C4214D4, C24.200C23, C23.210C24, C22.482+ 1+4, C4.39(C4×D4), C41D419C4, C4219(C2×C4), C428C413C2, C424C412C2, C23.23D48C2, C23.13(C22×C4), (C23×C4).48C22, C22.98(C22×D4), (C22×C4).475C23, (C2×C42).417C22, C22.101(C23×C4), C24.3C2215C2, C2.7(C22.29C24), (C22×D4).106C22, C2.16(C22.11C24), C2.C42.46C22, C2.4(C22.34C24), (C2×C4×D4)⋊8C2, C2.27(C2×C4×D4), (C2×D4)⋊16(C2×C4), (C2×C4).1190(C2×D4), (C2×C41D4).12C2, C22.95(C2×C4○D4), (C2×C4).650(C4○D4), (C2×C4⋊C4).809C22, (C2×C4).226(C22×C4), (C2×C22⋊C4).29C22, SmallGroup(128,1060)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4214D4
C1C2C22C23C22×C4C2×C42C424C4 — C4214D4
C1C22 — C4214D4
C1C23 — C4214D4
C1C23 — C4214D4

Generators and relations for C4214D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=ab2, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 764 in 370 conjugacy classes, 148 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×4], C4 [×14], C22 [×3], C22 [×4], C22 [×40], C2×C4 [×14], C2×C4 [×42], D4 [×32], C23, C23 [×8], C23 [×24], C42 [×8], C42 [×2], C22⋊C4 [×16], C4⋊C4 [×4], C22×C4, C22×C4 [×10], C22×C4 [×16], C2×D4 [×24], C2×D4 [×16], C24 [×4], C2.C42 [×8], C2×C42 [×3], C2×C22⋊C4 [×12], C2×C4⋊C4 [×2], C4×D4 [×8], C41D4 [×8], C23×C4 [×4], C22×D4 [×6], C424C4, C428C4, C23.23D4 [×8], C24.3C22 [×2], C2×C4×D4 [×2], C2×C41D4, C4214D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, 2+ 1+4 [×4], C2×C4×D4, C22.11C24 [×2], C22.29C24 [×2], C22.34C24 [×2], C4214D4

Smallest permutation representation of C4214D4
On 64 points
Generators in S64
(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 15 10 50)(2 16 11 51)(3 13 12 52)(4 14 9 49)(5 63 37 36)(6 64 38 33)(7 61 39 34)(8 62 40 35)(17 44 46 22)(18 41 47 23)(19 42 48 24)(20 43 45 21)(25 57 54 30)(26 58 55 31)(27 59 56 32)(28 60 53 29)
(1 63 55 47)(2 33 56 19)(3 61 53 45)(4 35 54 17)(5 58 41 50)(6 32 42 16)(7 60 43 52)(8 30 44 14)(9 62 25 46)(10 36 26 18)(11 64 27 48)(12 34 28 20)(13 39 29 21)(15 37 31 23)(22 49 40 57)(24 51 38 59)
(1 58)(2 59)(3 60)(4 57)(5 63)(6 64)(7 61)(8 62)(9 30)(10 31)(11 32)(12 29)(13 28)(14 25)(15 26)(16 27)(17 22)(18 23)(19 24)(20 21)(33 38)(34 39)(35 40)(36 37)(41 47)(42 48)(43 45)(44 46)(49 54)(50 55)(51 56)(52 53)

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2O4A···4L4M···4AB
order12···22···24···44···4
size11···14···42···24···4

44 irreducible representations

dim11111111224
type+++++++++
imageC1C2C2C2C2C2C2C4D4C4○D42+ 1+4
kernelC4214D4C424C4C428C4C23.23D4C24.3C22C2×C4×D4C2×C41D4C41D4C42C2×C4C22
# reps111822116444

Matrix representation of C4214D4 in GL8(𝔽5)

20000000
02000000
00100000
00010000
00000030
00000003
00002000
00000200
,
10000000
01000000
00400000
00040000
00004300
00001100
00000043
00000011
,
43000000
11000000
00010000
00400000
00000010
00000001
00001000
00000100
,
40000000
11000000
00100000
00040000
00004300
00000100
00000043
00000001

G:=sub<GL(8,GF(5))| [2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,3,1],[4,1,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1] >;

C4214D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{14}D_4
% in TeX

G:=Group("C4^2:14D4");
// GroupNames label

G:=SmallGroup(128,1060);
// by ID

G=gap.SmallGroup(128,1060);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,219,268,675,80]);
// Polycyclic

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

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