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G = C2×C232D4order 128 = 27

Direct product of C2 and C232D4

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

Aliases: C2×C232D4, C246D4, C25.26C22, C24.231C23, C23.284C24, C234(C2×D4), (D4×C23)⋊1C2, (C22×C4)⋊15D4, (C22×D4)⋊53C22, C22.108C22≀C2, C23.364(C4○D4), C22.45(C41D4), (C22×C4).775C23, (C23×C4).316C22, C22.167(C22×D4), C22.160(C4⋊D4), C2.C4261C22, (C2×C4)⋊6(C2×D4), C2.4(C2×C41D4), C2.7(C2×C4⋊D4), C2.5(C2×C22≀C2), (C2×C22⋊C4)⋊79C22, (C22×C22⋊C4)⋊15C2, C22.164(C2×C4○D4), (C2×C2.C42)⋊23C2, SmallGroup(128,1116)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C2×C232D4
C1C2C22C23C24C25D4×C23 — C2×C232D4
C1C23 — C2×C232D4
C1C24 — C2×C232D4
C1C23 — C2×C232D4

Generators and relations for C2×C232D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, bd=db, ebe-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1908 in 906 conjugacy classes, 180 normal (8 characteristic)
C1, C2, C2 [×14], C2 [×12], C4 [×14], C22, C22 [×34], C22 [×108], C2×C4 [×12], C2×C4 [×46], D4 [×96], C23, C23 [×26], C23 [×156], C22⋊C4 [×24], C22×C4 [×20], C22×C4 [×18], C2×D4 [×168], C24, C24 [×18], C24 [×36], C2.C42 [×4], C2×C22⋊C4 [×12], C2×C22⋊C4 [×12], C23×C4, C23×C4 [×3], C22×D4 [×12], C22×D4 [×36], C25 [×3], C2×C2.C42, C232D4 [×8], C22×C22⋊C4 [×3], D4×C23 [×3], C2×C232D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×24], C23 [×15], C2×D4 [×36], C4○D4 [×2], C24, C22≀C2 [×12], C4⋊D4 [×12], C41D4 [×4], C22×D4 [×6], C2×C4○D4, C232D4 [×8], C2×C22≀C2 [×3], C2×C4⋊D4 [×3], C2×C41D4, C2×C232D4

Smallest permutation representation of C2×C232D4
On 64 points
Generators in S64
(1 32)(2 29)(3 30)(4 31)(5 60)(6 57)(7 58)(8 59)(9 51)(10 52)(11 49)(12 50)(13 18)(14 19)(15 20)(16 17)(21 39)(22 40)(23 37)(24 38)(25 61)(26 62)(27 63)(28 64)(33 56)(34 53)(35 54)(36 55)(41 47)(42 48)(43 45)(44 46)
(1 57)(2 28)(3 59)(4 26)(5 16)(6 32)(7 14)(8 30)(9 22)(10 47)(11 24)(12 45)(13 63)(15 61)(17 60)(18 27)(19 58)(20 25)(21 56)(23 54)(29 64)(31 62)(33 39)(34 44)(35 37)(36 42)(38 49)(40 51)(41 52)(43 50)(46 53)(48 55)
(1 11)(2 12)(3 9)(4 10)(5 37)(6 38)(7 39)(8 40)(13 36)(14 33)(15 34)(16 35)(17 54)(18 55)(19 56)(20 53)(21 58)(22 59)(23 60)(24 57)(25 46)(26 47)(27 48)(28 45)(29 50)(30 51)(31 52)(32 49)(41 62)(42 63)(43 64)(44 61)
(1 55)(2 56)(3 53)(4 54)(5 41)(6 42)(7 43)(8 44)(9 20)(10 17)(11 18)(12 19)(13 49)(14 50)(15 51)(16 52)(21 28)(22 25)(23 26)(24 27)(29 33)(30 34)(31 35)(32 36)(37 62)(38 63)(39 64)(40 61)(45 58)(46 59)(47 60)(48 57)
(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 4)(2 3)(5 42)(6 41)(7 44)(8 43)(9 12)(10 11)(13 16)(14 15)(17 18)(19 20)(21 25)(22 28)(23 27)(24 26)(29 30)(31 32)(33 34)(35 36)(37 63)(38 62)(39 61)(40 64)(45 59)(46 58)(47 57)(48 60)(49 52)(50 51)(53 56)(54 55)

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

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

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

44 conjugacy classes

class 1 2A···2O2P···2AA4A···4P
order12···22···24···4
size11···14···44···4

44 irreducible representations

dim11111222
type+++++++
imageC1C2C2C2C2D4D4C4○D4
kernelC2×C232D4C2×C2.C42C232D4C22×C22⋊C4D4×C23C22×C4C24C23
# reps1183312124

Matrix representation of C2×C232D4 in GL8(ℤ)

10000000
01000000
00100000
00010000
0000-1000
00000-100
00000010
00000001
,
12000000
0-1000000
00-1-20000
00010000
00000-100
0000-1000
00000010
0000000-1
,
10000000
01000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
00000010
00000001
,
-10000000
11000000
00100000
00010000
00001000
00000100
00000001
000000-10
,
-10000000
0-1000000
00100000
00-1-10000
00001000
00000-100
00000001
00000010

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C2×C232D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes_2D_4
% in TeX

G:=Group("C2xC2^3:2D4");
// GroupNames label

G:=SmallGroup(128,1116);
// by ID

G=gap.SmallGroup(128,1116);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723]);
// 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,f*b*f=b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,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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