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G = C2×D4⋊3Q8  order 128 = 27

Direct product of C2 and D4⋊3Q8

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

Aliases: C2×D4⋊3Q8, C22.61C25, C23.126C24, C24.618C23, C42.559C23, C22.1132+ (1+4), D4⋊6(C2×Q8), (C2×D4)⋊24Q8, C4⋊Q8⋊84C22, (C2×C4).60C24, (C4×Q8)⋊95C22, C4.49(C22×Q8), C2.11(Q8×C23), C4⋊C4.471C23, C22⋊Q8⋊89C22, C23.123(C2×Q8), (C2×D4).502C23, (C4×D4).352C22, C22.9(C22×Q8), C22⋊C4.85C23, (C2×Q8).279C23, C42.C2⋊49C22, (C2×C42).930C22, (C23×C4).599C22, C2.20(C2×2+ (1+4)), (C22×C4).1197C23, (C22×D4).616C22, (C22×Q8).354C22, D4○3(C2×C4⋊C4), C4⋊C4○4(C2×D4), (C2×C4×Q8)⋊53C2, (C2×C4⋊Q8)⋊52C2, (C2×C4×D4).88C2, (C22×C4⋊C4)⋊46C2, C4.134(C2×C4○D4), (C2×C22⋊Q8)⋊73C2, (C2×C4).323(C2×Q8), (C2×C4⋊C4)⋊138C22, (C2×C42.C2)⋊44C2, C2.33(C22×C4○D4), (C2×C4).850(C4○D4), C22.159(C2×C4○D4), (C2×C22⋊C4).541C22, SmallGroup(128,2204)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C2×D4⋊3Q8
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C4⋊C4 — C2×D4⋊3Q8
Lower central
C1 — C22 — C2×D4⋊3Q8
Upper central
C1 — C23 — C2×D4⋊3Q8
Jennings
C1 — C22 — C2×D4⋊3Q8

Subgroups: 828 in 600 conjugacy classes, 444 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×8], C4 [×22], C22, C22 [×14], C22 [×24], C2×C4 [×34], C2×C4 [×54], D4 [×16], Q8 [×16], C23, C23 [×12], C23 [×8], C42 [×12], C22⋊C4 [×24], C4⋊C4 [×64], C22×C4 [×3], C22×C4 [×34], C22×C4 [×16], C2×D4 [×12], C2×Q8 [×12], C2×Q8 [×8], C24 [×2], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×2], C2×C4⋊C4 [×30], C4×D4 [×24], C4×Q8 [×8], C22⋊Q8 [×48], C42.C2 [×16], C4⋊Q8 [×8], C23×C4 [×6], C22×D4, C22×Q8, C22×Q8 [×2], C22×C4⋊C4 [×2], C2×C4×D4, C2×C4×D4 [×2], C2×C4×Q8, C2×C22⋊Q8 [×6], C2×C42.C2 [×2], C2×C4⋊Q8, D4⋊3Q8 [×16], C2×D4⋊3Q8

Quotients:
C1, C2 [×31], C22 [×155], Q8 [×8], C23 [×155], C2×Q8 [×28], C4○D4 [×4], C24 [×31], C22×Q8 [×14], C2×C4○D4 [×6], 2+ (1+4) [×2], C25, D4⋊3Q8 [×4], Q8×C23, C22×C4○D4, C2×2+ (1+4), C2×D4⋊3Q8

Generators and relations
 G = < a,b,c,d,e | a2=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b2c, ede-1=d-1 >

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

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

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

Matrix representation ►G ⊆ GL5(𝔽5)

40000
01000
00100
00010
00001
,
10000
04000
00400
00012
00044
,
10000
01000
00100
00040
00011
,
40000
00400
01000
00040
00004
,
40000
02000
00300
00024
00033

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

50 conjugacy classes

class 1 2A···2G2H···2O4A···4P4Q···4AH
order12···22···24···44···4
size11···12···22···24···4

50 irreducible representations

dim11111111224
type++++++++-+
imageC1C2C2C2C2C2C2C2Q8C4○D42+ (1+4)
kernelC2×D4⋊3Q8C22×C4⋊C4C2×C4×D4C2×C4×Q8C2×C22⋊Q8C2×C42.C2C2×C4⋊Q8D4⋊3Q8C2×D4C2×C4C22
# reps123162116882

In GAP, Magma, Sage, TeX 

C_2\times D_4\rtimes_3Q_8
 
% in TeX
 
G:=Group("C2xD4:3Q8");
 
// GroupNames label
 
G:=SmallGroup(128,2204);
 
// by ID
 
G=gap.SmallGroup(128,2204);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,680,1430,570,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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