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G = C42.298D4order 128 = 27

280th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.298D4, C4.62- 1+4, C42.432C23, C4.252+ 1+4, Q8.Q84C2, D4.Q84C2, C88D430C2, C87D4.7C2, D4.2D44C2, C8.18D47C2, Q8.D44C2, C4⋊C4.189C23, C4⋊C8.316C22, (C2×C4).448C24, (C2×C8).173C23, (C2×D8).29C22, C22.5(C4○D8), C23.407(C2×D4), (C22×C4).174D4, C2.D8.47C22, C4.Q8.94C22, (C2×D4).190C23, (C4×D4).128C22, (C4×Q8).125C22, (C2×Q16).31C22, (C2×Q8).178C23, C4⋊D4.210C22, (C2×C42).905C22, (C22×C8).155C22, (C2×SD16).90C22, C22.708(C22×D4), C22⋊Q8.215C22, D4⋊C4.114C22, C2.71(D8⋊C22), (C22×C4).1581C23, Q8⋊C4.110C22, C4.4D4.165C22, C42.C2.142C22, C23.36C2314C2, C2.67(C22.31C24), (C2×C4⋊C8)⋊32C2, C2.51(C2×C4○D8), (C2×C4).572(C2×D4), SmallGroup(128,1982)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.298D4
C1C2C4C2×C4C42C4×D4C23.36C23 — C42.298D4
C1C2C2×C4 — C42.298D4
C1C22C2×C42 — C42.298D4
C1C2C2C2×C4 — C42.298D4

Generators and relations for C42.298D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, ac=ca, dad=a-1b2, cbc-1=b-1, dbd=a2b-1, dcd=a2c3 >

Subgroups: 340 in 180 conjugacy classes, 86 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×15], D4 [×8], Q8 [×4], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], D8, SD16 [×2], Q16, C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2 [×2], C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C42.C2 [×2], C422C2 [×2], C22×C8 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C2×C4⋊C8, D4.2D4 [×2], Q8.D4 [×2], C88D4 [×2], C87D4, C8.18D4, D4.Q8 [×2], Q8.Q8 [×2], C23.36C23 [×2], C42.298D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C2×C4○D8, D8⋊C22, C42.298D4

Smallest permutation representation of C42.298D4
On 64 points
Generators in S64
(1 61 5 57)(2 62 6 58)(3 63 7 59)(4 64 8 60)(9 34 13 38)(10 35 14 39)(11 36 15 40)(12 37 16 33)(17 28 21 32)(18 29 22 25)(19 30 23 26)(20 31 24 27)(41 52 45 56)(42 53 46 49)(43 54 47 50)(44 55 48 51)
(1 28 55 10)(2 11 56 29)(3 30 49 12)(4 13 50 31)(5 32 51 14)(6 15 52 25)(7 26 53 16)(8 9 54 27)(17 44 39 57)(18 58 40 45)(19 46 33 59)(20 60 34 47)(21 48 35 61)(22 62 36 41)(23 42 37 63)(24 64 38 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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 40)(24 39)(25 26)(27 32)(28 31)(29 30)(41 63)(42 62)(43 61)(44 60)(45 59)(46 58)(47 57)(48 64)(49 52)(50 51)(53 56)(54 55)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I4J4K···4P8A···8H
order122222224···4444···48···8
size111122882···2448···84···4

32 irreducible representations

dim1111111111222444
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D82+ 1+42- 1+4D8⋊C22
kernelC42.298D4C2×C4⋊C8D4.2D4Q8.D4C88D4C87D4C8.18D4D4.Q8Q8.Q8C23.36C23C42C22×C4C22C4C4C2
# reps1122211222228112

Matrix representation of C42.298D4 in GL6(𝔽17)

0160000
100000
004000
000400
000040
000004
,
040000
1300000
0004015
0013020
00016013
001040
,
3140000
330000
00116116
0011111111
00134611
00131366
,
3140000
14140000
00116116
006666
00134611
00441111

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,1,0,0,4,0,16,0,0,0,0,2,0,4,0,0,15,0,13,0],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,11,11,13,13,0,0,6,11,4,13,0,0,11,11,6,6,0,0,6,11,11,6],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,11,6,13,4,0,0,6,6,4,4,0,0,11,6,6,11,0,0,6,6,11,11] >;

C42.298D4 in GAP, Magma, Sage, TeX

C_4^2._{298}D_4
% in TeX

G:=Group("C4^2.298D4");
// GroupNames label

G:=SmallGroup(128,1982);
// by ID

G=gap.SmallGroup(128,1982);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,219,675,304,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,a*c=c*a,d*a*d=a^-1*b^2,c*b*c^-1=b^-1,d*b*d=a^2*b^-1,d*c*d=a^2*c^3>;
// generators/relations

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