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G = (C2×D4).302D4order 128 = 27

55th non-split extension by C2×D4 of D4 acting via D4/C22=C2

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

Aliases: (C2×D4).302D4, (C2×Q8).237D4, C2.15(Q8○D8), C4⋊C4.395C23, (C2×C8).312C23, (C2×C4).295C24, C23.246(C2×D4), (C2×Q8).70C23, C4.Q8.12C22, C2.D8.83C22, C2.24(D4○SD16), C23.47D43C2, C23.25D48C2, C22⋊C8.17C22, M4(2)⋊C427C2, C22⋊Q8.23C22, C23.38D411C2, C23.20D414C2, C23.48D413C2, (C22×C8).185C22, C22.555(C22×D4), (C22×C4).1011C23, Q8⋊C4.150C22, C4.83(C22.D4), (C2×M4(2)).77C22, (C22×Q8).293C22, C42⋊C2.124C22, C23.33C23.9C2, C23.38C23.12C2, C22.19(C22.D4), C4.105(C2×C4○D4), (C2×C4).490(C2×D4), (C2×Q8⋊C4)⋊30C2, (C2×C4).297(C4○D4), (C2×C4⋊C4).611C22, (C22×C8)⋊C2.3C2, (C2×C4○D4).140C22, C2.60(C2×C22.D4), SmallGroup(128,1829)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — (C2×D4).302D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C23.33C23 — (C2×D4).302D4
C1C2C2×C4 — (C2×D4).302D4
C1C22C2×C4○D4 — (C2×D4).302D4
C1C2C2C2×C4 — (C2×D4).302D4

Generators and relations for (C2×D4).302D4
 G = < a,b,c,d,e | a2=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=ebe-1=ab-1, dcd-1=ece-1=ab2c, ede-1=ab2d3 >

Subgroups: 340 in 193 conjugacy classes, 92 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×17], D4 [×6], Q8 [×8], C23, C23 [×2], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C22⋊C8 [×4], Q8⋊C4 [×8], C4.Q8 [×2], C4.Q8 [×2], C2.D8 [×2], C2.D8 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×2], C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C22×Q8, C2×C4○D4, (C22×C8)⋊C2, C2×Q8⋊C4, C23.38D4, C23.25D4, M4(2)⋊C4, C23.47D4 [×2], C23.48D4 [×2], C23.20D4 [×4], C23.33C23, C23.38C23, (C2×D4).302D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C22.D4, D4○SD16, Q8○D8, (C2×D4).302D4

Smallest permutation representation of (C2×D4).302D4
On 64 points
Generators in S64
(1 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 63 5 59)(2 8 6 4)(3 57 7 61)(9 26 13 30)(10 16 14 12)(11 28 15 32)(17 49 21 53)(18 20 22 24)(19 51 23 55)(25 31 29 27)(33 35 37 39)(34 45 38 41)(36 47 40 43)(42 44 46 48)(50 52 54 56)(58 64 62 60)
(1 53)(2 24)(3 55)(4 18)(5 49)(6 20)(7 51)(8 22)(9 47)(10 39)(11 41)(12 33)(13 43)(14 35)(15 45)(16 37)(17 59)(19 61)(21 63)(23 57)(25 44)(26 36)(27 46)(28 38)(29 48)(30 40)(31 42)(32 34)(50 62)(52 64)(54 58)(56 60)
(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 46 5 42)(2 40 6 36)(3 44 7 48)(4 38 8 34)(9 22 13 18)(10 55 14 51)(11 20 15 24)(12 53 16 49)(17 25 21 29)(19 31 23 27)(26 54 30 50)(28 52 32 56)(33 57 37 61)(35 63 39 59)(41 62 45 58)(43 60 47 64)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O4P4Q4R8A8B8C8D8E8F
order1222222244444···44444888888
size1111224422224···48888444488

32 irreducible representations

dim1111111111122244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○SD16Q8○D8
kernel(C2×D4).302D4(C22×C8)⋊C2C2×Q8⋊C4C23.38D4C23.25D4M4(2)⋊C4C23.47D4C23.48D4C23.20D4C23.33C23C23.38C23C2×D4C2×Q8C2×C4C2C2
# reps1111112241131822

Matrix representation of (C2×D4).302D4 in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
010000
0001600
001000
000001
0000160
,
1600000
010000
000001
0000160
0001600
001000
,
090000
200000
0051200
005500
0000512
000055
,
0150000
800000
0000125
000055
0012500
005500

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[0,2,0,0,0,0,9,0,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,12,5],[0,8,0,0,0,0,15,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,12,5,0,0,0,0,5,5,0,0] >;

(C2×D4).302D4 in GAP, Magma, Sage, TeX

(C_2\times D_4)._{302}D_4
% in TeX

G:=Group("(C2xD4).302D4");
// GroupNames label

G:=SmallGroup(128,1829);
// by ID

G=gap.SmallGroup(128,1829);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,100,1018,4037,1027,124]);
// Polycyclic

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

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