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## G = D4×C24order 128 = 27

### Direct product of C24 and D4

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: D4×C24, C4⋊C25, C263C2, C22⋊C25, C2.1C26, C233C24, C2412C23, C2511C22, (C2×C4)⋊4C24, (C24×C4)⋊10C2, (C22×C4)⋊25C23, (C23×C4)⋊60C22, SmallGroup(128,2320)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — D4×C24
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C26 — D4×C24
 Lower central C1 — C2 — D4×C24
 Upper central C1 — C25 — D4×C24
 Jennings C1 — C2 — D4×C24

Generators and relations for D4×C24
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 7420 in 5276 conjugacy classes, 3132 normal (5 characteristic)
C1, C2, C2 [×30], C2 [×32], C4 [×16], C22 [×187], C22 [×480], C2×C4 [×120], D4 [×256], C23 [×395], C23 [×1120], C22×C4 [×140], C2×D4 [×960], C24 [×311], C24 [×480], C23×C4 [×30], C22×D4 [×560], C25, C25 [×60], C25 [×32], C24×C4, D4×C23 [×60], C26 [×2], D4×C24
Quotients: C1, C2 [×63], C22 [×651], D4 [×16], C23 [×1395], C2×D4 [×120], C24 [×651], C22×D4 [×140], C25 [×63], D4×C23 [×30], C26, D4×C24

Smallest permutation representation of D4×C24
On 64 points
Generators in S64
(1 17)(2 18)(3 19)(4 20)(5 24)(6 21)(7 22)(8 23)(9 28)(10 25)(11 26)(12 27)(13 32)(14 29)(15 30)(16 31)(33 50)(34 51)(35 52)(36 49)(37 54)(38 55)(39 56)(40 53)(41 58)(42 59)(43 60)(44 57)(45 62)(46 63)(47 64)(48 61)
(1 34)(2 35)(3 36)(4 33)(5 63)(6 64)(7 61)(8 62)(9 41)(10 42)(11 43)(12 44)(13 37)(14 38)(15 39)(16 40)(17 51)(18 52)(19 49)(20 50)(21 47)(22 48)(23 45)(24 46)(25 59)(26 60)(27 57)(28 58)(29 55)(30 56)(31 53)(32 54)
(1 31)(2 32)(3 29)(4 30)(5 26)(6 27)(7 28)(8 25)(9 22)(10 23)(11 24)(12 21)(13 18)(14 19)(15 20)(16 17)(33 56)(34 53)(35 54)(36 55)(37 52)(38 49)(39 50)(40 51)(41 48)(42 45)(43 46)(44 47)(57 64)(58 61)(59 62)(60 63)
(1 64)(2 61)(3 62)(4 63)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 47)(18 48)(19 45)(20 46)(21 51)(22 52)(23 49)(24 50)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)
(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 27)(2 26)(3 25)(4 28)(5 32)(6 31)(7 30)(8 29)(9 20)(10 19)(11 18)(12 17)(13 24)(14 23)(15 22)(16 21)(33 58)(34 57)(35 60)(36 59)(37 46)(38 45)(39 48)(40 47)(41 50)(42 49)(43 52)(44 51)(53 64)(54 63)(55 62)(56 61)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2AE 2AF ··· 2BK 4A ··· 4P order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 2 type + + + + + image C1 C2 C2 C2 D4 kernel D4×C24 C24×C4 D4×C23 C26 C24 # reps 1 1 60 2 16

Matrix representation of D4×C24 in GL6(ℤ)

 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 0
,
 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1] >;

D4×C24 in GAP, Magma, Sage, TeX

D_4\times C_2^4
% in TeX

G:=Group("D4xC2^4");
// GroupNames label

G:=SmallGroup(128,2320);
// by ID

G=gap.SmallGroup(128,2320);
# by ID

G:=PCGroup([7,-2,2,2,2,2,2,-2,925]);
// Polycyclic

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

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