Copied to
clipboard

G = C42.198D4order 128 = 27

180th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.198D4, C24.55C23, C23.564C24, C22.3382+ 1+4, C22.2532- 1+4, (C2×C42).628C22, (C22×C4).169C23, C22.376(C22×D4), C23.4Q8.17C2, C23.Q8.24C2, C23.11D4.32C2, C23.83C2372C2, C23.81C2373C2, C24.C22.46C2, C23.65C23111C2, C2.C42.278C22, C2.53(C22.26C24), C2.50(C23.38C23), C2.54(C22.33C24), C2.35(C22.31C24), C2.36(C22.35C24), (C4×C4⋊C4)⋊116C2, (C2×C4).408(C2×D4), (C2×C42.C2)⋊19C2, (C2×C4).184(C4○D4), (C2×C4⋊C4).897C22, C22.431(C2×C4○D4), (C2×C422C2).11C2, (C2×C22⋊C4).241C22, SmallGroup(128,1396)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.198D4
C1C2C22C23C22×C4C2×C4⋊C4C23.65C23 — C42.198D4
C1C23 — C42.198D4
C1C23 — C42.198D4
C1C23 — C42.198D4

Generators and relations for C42.198D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 388 in 217 conjugacy classes, 96 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2, C4 [×19], C22 [×3], C22 [×4], C22 [×7], C2×C4 [×10], C2×C4 [×37], C23, C23 [×7], C42 [×4], C42 [×2], C22⋊C4 [×12], C4⋊C4 [×28], C22×C4 [×6], C22×C4 [×8], C24, C2.C42 [×2], C2.C42 [×8], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×3], C2×C22⋊C4 [×4], C2×C4⋊C4 [×5], C2×C4⋊C4 [×10], C42.C2 [×4], C422C2 [×4], C4×C4⋊C4, C24.C22 [×2], C23.65C23 [×2], C23.Q8 [×2], C23.11D4, C23.81C23, C23.81C23 [×2], C23.4Q8, C23.83C23, C2×C42.C2, C2×C422C2, C42.198D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4, 2- 1+4 [×3], C22.26C24, C23.38C23, C22.31C24, C22.33C24 [×2], C22.35C24 [×2], C42.198D4

Smallest permutation representation of C42.198D4
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 53 59 25)(2 54 60 26)(3 55 57 27)(4 56 58 28)(5 62 31 35)(6 63 32 36)(7 64 29 33)(8 61 30 34)(9 22 39 50)(10 23 40 51)(11 24 37 52)(12 21 38 49)(13 20 43 46)(14 17 44 47)(15 18 41 48)(16 19 42 45)
(1 22 61 48)(2 21 62 47)(3 24 63 46)(4 23 64 45)(5 44 26 12)(6 43 27 11)(7 42 28 10)(8 41 25 9)(13 55 37 32)(14 54 38 31)(15 53 39 30)(16 56 40 29)(17 60 49 35)(18 59 50 34)(19 58 51 33)(20 57 52 36)
(1 54 3 56)(2 25 4 27)(5 36 7 34)(6 62 8 64)(9 45 11 47)(10 18 12 20)(13 49 15 51)(14 24 16 22)(17 39 19 37)(21 41 23 43)(26 57 28 59)(29 61 31 63)(30 33 32 35)(38 46 40 48)(42 50 44 52)(53 58 55 60)

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

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

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

32 conjugacy classes

class 1 2A···2G2H4A4B4C4D4E···4P4Q···4W
order12···2244444···44···4
size11···1822224···48···8

32 irreducible representations

dim111111111112244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC42.198D4C4×C4⋊C4C24.C22C23.65C23C23.Q8C23.11D4C23.81C23C23.4Q8C23.83C23C2×C42.C2C2×C422C2C42C2×C4C22C22
# reps112221311114813

Matrix representation of C42.198D4 in GL8(𝔽5)

31000000
22000000
00100000
00010000
00000414
00001020
00000023
00000003
,
20000000
02000000
00100000
00010000
00000400
00004000
00000123
00002143
,
10000000
01000000
00040000
00100000
00002040
00000323
00000030
00000012
,
43000000
01000000
00400000
00010000
00004030
00000414
00001010
00001201

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

C42.198D4 in GAP, Magma, Sage, TeX

C_4^2._{198}D_4
% in TeX

G:=Group("C4^2.198D4");
// GroupNames label

G:=SmallGroup(128,1396);
// by ID

G=gap.SmallGroup(128,1396);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,100,185,136]);
// Polycyclic

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

׿
×
𝔽