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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×D4).302D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C23.33C23 — (C2×D4).302D4
 Lower central C1 — C2 — C2×C4 — (C2×D4).302D4
 Upper central C1 — C22 — C2×C4○D4 — (C2×D4).302D4
 Jennings C1 — C2 — C2 — C2×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 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 8A 8B 8C 8D 8E 8F order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 8 8 8 8 8 8 size 1 1 1 1 2 2 4 4 2 2 2 2 4 ··· 4 8 8 8 8 4 4 4 4 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 D4○SD16 Q8○D8 kernel (C2×D4).302D4 (C22×C8)⋊C2 C2×Q8⋊C4 C23.38D4 C23.25D4 M4(2)⋊C4 C23.47D4 C23.48D4 C23.20D4 C23.33C23 C23.38C23 C2×D4 C2×Q8 C2×C4 C2 C2 # reps 1 1 1 1 1 1 2 2 4 1 1 3 1 8 2 2

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

 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 16 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 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 9 0 0 0 0 2 0 0 0 0 0 0 0 5 12 0 0 0 0 5 5 0 0 0 0 0 0 5 12 0 0 0 0 5 5
,
 0 15 0 0 0 0 8 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

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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