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G = C42.233D4order 128 = 27

215th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.233D4, C42.349C23, D44(C4○D4), D8⋊C48C2, C4⋊D822C2, Q84(C4○D4), C4⋊SD166C2, Q8⋊Q86C2, D4⋊D419C2, D4⋊Q823C2, C4⋊C8.51C22, C4⋊C4.68C23, (C2×C8).42C23, SD16⋊C49C2, C8⋊C4.8C22, (C2×C4).313C24, C42.6C46C2, (C2×D8).60C22, (C4×D4).78C22, (C2×D4).92C23, (C22×C4).453D4, C23.253(C2×D4), C4⋊Q8.269C22, C4.125(C8⋊C22), C4.Q8.17C22, C2.D8.88C22, C22⋊C8.26C22, (C4×Q8).303C22, (C2×Q8).378C23, D4⋊C4.32C22, C41D4.142C22, C4⋊D4.168C22, C23.19D418C2, C22.26C247C2, (C2×C42).840C22, Q8⋊C4.33C22, (C2×SD16).14C22, C22.573(C22×D4), C2.32(D8⋊C22), (C22×C4).1029C23, C42⋊C2.323C22, C2.114(C22.19C24), (C4×C4○D4)⋊12C2, C4.198(C2×C4○D4), C2.36(C2×C8⋊C22), (C2×C4).1221(C2×D4), (C2×C4○D4).314C22, SmallGroup(128,1847)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.233D4
C1C2C4C2×C4C42C4×D4C4×C4○D4 — C42.233D4
C1C2C2×C4 — C42.233D4
C1C22C2×C42 — C42.233D4
C1C2C2C2×C4 — C42.233D4

Generators and relations for C42.233D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, bd=db, dcd=a2c3 >

Subgroups: 428 in 215 conjugacy classes, 90 normal (44 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×13], C8 [×4], C2×C4 [×6], C2×C4 [×22], D4 [×2], D4 [×17], Q8 [×2], Q8 [×3], C23, C23 [×3], C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×4], D8 [×4], SD16 [×4], C22×C4 [×3], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C2×Q8, C4○D4 [×8], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×D4 [×3], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×D8 [×2], C2×SD16 [×2], C2×C4○D4, C2×C4○D4, C42.6C4, SD16⋊C4 [×2], D8⋊C4 [×2], D4⋊D4 [×2], C4⋊D8, C4⋊SD16, D4⋊Q8, Q8⋊Q8, C23.19D4 [×2], C4×C4○D4, C22.26C24, C42.233D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C8⋊C22, D8⋊C22, C42.233D4

Smallest permutation representation of C42.233D4
On 64 points
Generators in S64
(1 20 5 24)(2 47 6 43)(3 22 7 18)(4 41 8 45)(9 28 13 32)(10 56 14 52)(11 30 15 26)(12 50 16 54)(17 63 21 59)(19 57 23 61)(25 34 29 38)(27 36 31 40)(33 55 37 51)(35 49 39 53)(42 62 46 58)(44 64 48 60)
(1 26 58 49)(2 31 59 54)(3 28 60 51)(4 25 61 56)(5 30 62 53)(6 27 63 50)(7 32 64 55)(8 29 57 52)(9 48 37 18)(10 45 38 23)(11 42 39 20)(12 47 40 17)(13 44 33 22)(14 41 34 19)(15 46 35 24)(16 43 36 21)
(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 35)(2 34)(3 33)(4 40)(5 39)(6 38)(7 37)(8 36)(9 64)(10 63)(11 62)(12 61)(13 60)(14 59)(15 58)(16 57)(17 25)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 56)(48 55)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4J4K···4Q4R4S8A8B8C8D
order1222222224···44···4448888
size1111444882···24···4888888

32 irreducible representations

dim111111111111222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D4C8⋊C22D8⋊C22
kernelC42.233D4C42.6C4SD16⋊C4D8⋊C4D4⋊D4C4⋊D8C4⋊SD16D4⋊Q8Q8⋊Q8C23.19D4C4×C4○D4C22.26C24C42C22×C4D4Q8C4C2
# reps112221111211224422

Matrix representation of C42.233D4 in GL6(𝔽17)

0160000
1600000
004000
000400
000040
000004
,
1300000
0130000
0001150
0010015
0010016
0001160
,
400000
0130000
00101088
0016198
0001167
0010017
,
0130000
400000
000490
0013008
000004
0000130

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,1,0,0,0,1,0,0,1,0,0,15,0,0,16,0,0,0,15,16,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,10,16,0,10,0,0,10,1,1,0,0,0,8,9,16,1,0,0,8,8,7,7],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,9,0,0,13,0,0,0,8,4,0] >;

C42.233D4 in GAP, Magma, Sage, TeX

C_4^2._{233}D_4
% in TeX

G:=Group("C4^2.233D4");
// GroupNames label

G:=SmallGroup(128,1847);
// by ID

G=gap.SmallGroup(128,1847);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,521,304,4037,1027,124]);
// Polycyclic

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

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