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G = C4215D4order 128 = 27

9th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C4215D4, C24.235C23, C23.292C24, (C22×C4)⋊18D4, C42(C232D4), C23.145(C2×D4), C4(C23.Q8), C4.209(C4⋊D4), C4(C23.11D4), C23.15(C4○D4), C232D4.39C2, C42(C23.10D4), (C23×C4).323C22, (C2×C42).455C22, (C22×C4).496C23, C23.Q8102C2, C22.175(C22×D4), C23.10D4117C2, C23.11D4139C2, C4(C23.81C23), (C22×D4).494C22, C2.10(C22.19C24), C2.9(C22.26C24), C23.81C23145C2, C2.C42.532C22, C2.12(C23.36C23), (C2×C4×D4)⋊17C2, (C4×C4⋊C4)⋊51C2, (C4×C22⋊C4)⋊47C2, (C2×C4).293(C2×D4), C2.12(C2×C4⋊D4), (C2×C4).87(C4○D4), (C2×C42⋊C2)⋊16C2, (C2×C4)2(C232D4), (C2×C4⋊C4).836C22, C22.172(C2×C4○D4), (C2×C4)(C23.Q8), (C2×C4)(C23.10D4), (C2×C22⋊C4).486C22, (C22×C4)(C23.Q8), SmallGroup(128,1124)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4215D4
C1C2C22C23C22×C4C23×C4C2×C4×D4 — C4215D4
C1C23 — C4215D4
C1C22×C4 — C4215D4
C1C23 — C4215D4

Generators and relations for C4215D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, bc=cb, bd=db, dcd=c-1 >

Subgroups: 644 in 348 conjugacy classes, 116 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×30], C2×C4 [×16], C2×C4 [×44], D4 [×24], C23, C23 [×6], C23 [×18], C42 [×4], C42 [×6], C22⋊C4 [×24], C4⋊C4 [×14], C22×C4 [×6], C22×C4 [×10], C22×C4 [×16], C2×D4 [×18], C24, C24 [×2], C2.C42 [×2], C2.C42 [×4], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×3], C2×C4⋊C4 [×4], C42⋊C2 [×4], C4×D4 [×12], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C4×C22⋊C4 [×2], C4×C4⋊C4, C232D4, C23.10D4, C23.10D4 [×2], C23.Q8 [×2], C23.11D4, C23.81C23, C2×C42⋊C2, C2×C4×D4, C2×C4×D4 [×2], C4215D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×10], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4 [×5], C2×C4⋊D4, C22.19C24 [×3], C23.36C23 [×2], C22.26C24, C4215D4

Smallest permutation representation of C4215D4
On 64 points
Generators in S64
(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 7 23 9)(2 8 24 10)(3 5 21 11)(4 6 22 12)(13 31 27 37)(14 32 28 38)(15 29 25 39)(16 30 26 40)(17 42 64 60)(18 43 61 57)(19 44 62 58)(20 41 63 59)(33 49 48 54)(34 50 45 55)(35 51 46 56)(36 52 47 53)
(1 58 25 35)(2 43 26 45)(3 60 27 33)(4 41 28 47)(5 17 37 49)(6 63 38 53)(7 19 39 51)(8 61 40 55)(9 62 29 56)(10 18 30 50)(11 64 31 54)(12 20 32 52)(13 48 21 42)(14 36 22 59)(15 46 23 44)(16 34 24 57)
(1 45)(2 35)(3 47)(4 33)(5 53)(6 49)(7 55)(8 51)(9 50)(10 56)(11 52)(12 54)(13 59)(14 42)(15 57)(16 44)(17 38)(18 29)(19 40)(20 31)(21 36)(22 48)(23 34)(24 46)(25 43)(26 58)(27 41)(28 60)(30 62)(32 64)(37 63)(39 61)

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2M4A···4H4I···4AD
order12···22···24···44···4
size11···14···41···14···4

44 irreducible representations

dim11111111112222
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D4
kernelC4215D4C4×C22⋊C4C4×C4⋊C4C232D4C23.10D4C23.Q8C23.11D4C23.81C23C2×C42⋊C2C2×C4×D4C42C22×C4C2×C4C23
# reps121132111344128

Matrix representation of C4215D4 in GL6(𝔽5)

040000
100000
004000
004100
000040
000004
,
300000
030000
002000
000200
000010
000001
,
040000
100000
001300
001400
000012
000044
,
100000
040000
004200
000100
000043
000001

G:=sub<GL(6,GF(5))| [0,1,0,0,0,0,4,0,0,0,0,0,0,0,4,4,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,1,1,0,0,0,0,3,4,0,0,0,0,0,0,1,4,0,0,0,0,2,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,2,1,0,0,0,0,0,0,4,0,0,0,0,0,3,1] >;

C4215D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{15}D_4
% in TeX

G:=Group("C4^2:15D4");
// GroupNames label

G:=SmallGroup(128,1124);
// by ID

G=gap.SmallGroup(128,1124);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,80]);
// Polycyclic

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

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