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G = C2×C23.4Q8order 128 = 27

Direct product of C2 and C23.4Q8

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

Aliases: C2×C23.4Q8, C24.14Q8, C25.31C22, C23.293C24, C24.236C23, C23.59(C2×Q8), C23.834(C2×D4), (C22×C4).368D4, (C23×C4).66C22, C23.371(C4○D4), C22.58(C22×Q8), C22.46(C41D4), (C22×C4).781C23, C22.176(C22×D4), C22.94(C22⋊Q8), C2.C4264C22, C22.104(C22.D4), C2.5(C2×C41D4), (C22×C4⋊C4)⋊15C2, (C2×C4).294(C2×D4), (C2×C4⋊C4)⋊108C22, C2.11(C2×C22⋊Q8), C22.173(C2×C4○D4), (C2×C2.C42)⋊27C2, (C22×C22⋊C4).21C2, C2.10(C2×C22.D4), (C2×C22⋊C4).487C22, SmallGroup(128,1125)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C2×C23.4Q8
C1C2C22C23C24C25C22×C22⋊C4 — C2×C23.4Q8
C1C23 — C2×C23.4Q8
C1C24 — C2×C23.4Q8
C1C23 — C2×C23.4Q8

Generators and relations for C2×C23.4Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=ce2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >

Subgroups: 868 in 454 conjugacy classes, 164 normal (8 characteristic)
C1, C2, C2 [×14], C2 [×4], C4 [×18], C22, C22 [×34], C22 [×36], C2×C4 [×12], C2×C4 [×66], C23, C23 [×18], C23 [×52], C22⋊C4 [×24], C4⋊C4 [×24], C22×C4 [×24], C22×C4 [×30], C24, C24 [×6], C24 [×12], C2.C42 [×4], C2×C22⋊C4 [×12], C2×C22⋊C4 [×12], C2×C4⋊C4 [×12], C2×C4⋊C4 [×12], C23×C4 [×6], C25, C2×C2.C42, C23.4Q8 [×8], C22×C22⋊C4 [×3], C22×C4⋊C4 [×3], C2×C23.4Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], Q8 [×4], C23 [×15], C2×D4 [×18], C2×Q8 [×6], C4○D4 [×6], C24, C22⋊Q8 [×12], C22.D4 [×12], C41D4 [×4], C22×D4 [×3], C22×Q8, C2×C4○D4 [×3], C23.4Q8 [×8], C2×C22⋊Q8 [×3], C2×C22.D4 [×3], C2×C41D4, C2×C23.4Q8

Smallest permutation representation of C2×C23.4Q8
On 64 points
Generators in S64
(1 47)(2 48)(3 45)(4 46)(5 35)(6 36)(7 33)(8 34)(9 21)(10 22)(11 23)(12 24)(13 49)(14 50)(15 51)(16 52)(17 42)(18 43)(19 44)(20 41)(25 56)(26 53)(27 54)(28 55)(29 59)(30 60)(31 57)(32 58)(37 62)(38 63)(39 64)(40 61)
(2 44)(4 42)(5 54)(6 32)(7 56)(8 30)(10 50)(12 52)(14 22)(16 24)(17 46)(19 48)(25 33)(26 63)(27 35)(28 61)(29 37)(31 39)(34 60)(36 58)(38 53)(40 55)(57 64)(59 62)
(1 11)(2 12)(3 9)(4 10)(5 31)(6 32)(7 29)(8 30)(13 20)(14 17)(15 18)(16 19)(21 45)(22 46)(23 47)(24 48)(25 62)(26 63)(27 64)(28 61)(33 59)(34 60)(35 57)(36 58)(37 56)(38 53)(39 54)(40 55)(41 49)(42 50)(43 51)(44 52)
(1 43)(2 44)(3 41)(4 42)(5 39)(6 40)(7 37)(8 38)(9 49)(10 50)(11 51)(12 52)(13 21)(14 22)(15 23)(16 24)(17 46)(18 47)(19 48)(20 45)(25 59)(26 60)(27 57)(28 58)(29 56)(30 53)(31 54)(32 55)(33 62)(34 63)(35 64)(36 61)
(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 58 9 34)(2 35 10 59)(3 60 11 36)(4 33 12 57)(5 22 29 48)(6 45 30 23)(7 24 31 46)(8 47 32 21)(13 38 18 55)(14 56 19 39)(15 40 20 53)(16 54 17 37)(25 44 64 50)(26 51 61 41)(27 42 62 52)(28 49 63 43)

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

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

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

44 conjugacy classes

class 1 2A···2O2P2Q2R2S4A···4X
order12···222224···4
size11···144444···4

44 irreducible representations

dim11111222
type++++++-
imageC1C2C2C2C2D4Q8C4○D4
kernelC2×C23.4Q8C2×C2.C42C23.4Q8C22×C22⋊C4C22×C4⋊C4C22×C4C24C23
# reps1183312412

Matrix representation of C2×C23.4Q8 in GL8(𝔽5)

10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
04000000
00100000
00040000
00001000
00000100
00000010
00000034
,
10000000
01000000
00100000
00010000
00001000
00000100
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000040
00000004
,
01000000
40000000
00040000
00100000
00002000
00004300
00000033
00000002
,
20000000
03000000
00200000
00030000
00002300
00000300
00000011
00000004

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2],[2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,4] >;

C2×C23.4Q8 in GAP, Magma, Sage, TeX

C_2\times C_2^3._4Q_8
% in TeX

G:=Group("C2xC2^3.4Q8");
// GroupNames label

G:=SmallGroup(128,1125);
// by ID

G=gap.SmallGroup(128,1125);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723]);
// Polycyclic

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

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